Algebra

Jacobians of CM curves are abelian varieties with a particularly large endomorphism algebra, which provides them with a rich arithmetic structure. The motivating question for the results in this talk is whether we can find hyperelliptic and non-hyperelliptic curves with maximal CM by a given order whose Jacobians are isogenous.
Joint work with Sorina Ionica, and Jeroen Sijsling considers this question in genus 3 by using the catalogue of CM fields in the LMFDB, and found a (small) list of such isogenous Jacobians. This talk describes the main constructions, some results, and Shimura class groups.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Let A be a finite-dimensional algebra over a number field and let Lambda be an order in A. Under certain hypothesis on A, we give an efficient algorithm that given two Lambda-lattices X and Y, determines whether X and Y are isomorphic, and if so, computes an explicit isomorphism X -> Y. As an application, we give an algorithm for the following long-standing problem: given a positive integer n and two n x n integral matrices A and B, determine whether A and B are similar over Z, and if so, return a matrix C in GL_n(Z) such that B= CAC^-1. This is joint work with Werner Bley and Tommy Hofmann. The preprint is available here: https://arxiv.org/abs/2202.03526

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Given a smooth, geometrically connected and projective curve C defined over a finite field k, let K=k(C) be the function field of rational functions on C. The Tamagawa number \tau(G) of a semisimple K-group G is defined as the covolume of the discrete group G(K) (embedded diagonally) in the adelic group G(A) with respect to the Tamagawa measure. The Weil conjecture, recently proved by Gaitsgory and Lurie, states that if G is simply-connected then \tau(G)=1. 
 

Our aim is to prove, without relying on the Weil conjecture, the following fact: 
Let G be a quasi-split inner form of a split semisimple and simply-connected K-group G_1. Then \tau(G)=\tau(G_1). 
This Theorem can serve as an alternative proof to the Weil conjecture. 

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

The talk is based on joint works with Mikhail Budrevich and Constantine Taranin.

Two important functions in matrix theory, the determinant and the permanent, have similar definitions.  However, while the determinant may be computed in polynomial time, it is an open question whether fast algorithms computing the permanent exist.  Therefore, any bounds on the permanent are of interest.

The class of matrices with entries 1 and -1 is very important in algebra, combinatorics, and their various applications.  In 1974, Wang posed the problem of finding an upper bound for the permanent of a (1, -1)-matrix.  This problem has appeared in several monographs and survey papers.  Kräuter later made a conjecture about the form of this bound.

In this talk we present a complete solution of Wang’s problem, obtained by proving Kräuter’s conjecture.  In particular, we characterize the matrices with the maximal possible permanent among the matrices of a given rank.

Also, we will discuss the permanents of (0,1)-matrices.  In 1965, Brualdi and Newman showed that every integer in the interval [0,2^{n-1}] is the permanent of some n x n (0,1)-matrix.  We improve their bound by showing that this is true for every integer in an interval somewhat larger than [0,2^n].

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

We introduce and study multivariate zeta functions enumerating subrepresentations of integral quiver representations and discuss their connections to various enumeration problems in algebra. This is joint work with Christopher Voll.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

The dimension of the space of multilinear products of higher commutators is equal to the number of derangements, $[e^{-1}n!]$. 
Our search for a combinatorial explanation for this fact led us to study representations of left regular bands, whose resolution is obtained through analysis of cubical partial representations. There are applications in combinatorics, probability, and nonassociative algebra.

This is joint work with Guy Blachar and Louis Rowen.
The lecture is dedicated to Stuart Margolis on the occasion of his retirement, for repeatedly planting semigroup seeds in our minds.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Joint work with Yoav Segev.
``Fusion rules'' are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov, defined as nonassociative algebras generated by semisimple idempotents of degree 3, satisfying fusion rules based on a natural 2-grading . Axial algebras, in turn, are closely related to 3-transposition groups and vertex operator algebras.

We introduce a noncommutative generalization of primitive axial algebras (PAJ for short), and show that they all have Jordan type. Extending the known theory, we bring in the fundamental notion of Miyamoto involutions, and the ensuing topology on the set
of primitive axes.

Accompanying this is the ``axial graph'' on a generating set of axes X, where two axes
are neighbors if and only if their product is nonzero. The axial graph aids us in
decomposing a PAJ into connected components. The PAJ's which are not commutative are easily described, implying that all PAJ's are flexible, and any PAJ is a direct product of noncommutative PAJ 's and a commutative PAJ .

We obtain a Frobenius form for any PAJ which is not quite unique, and prove some properties which previously had been axioms. We give a complete description of all axes of 2-generated PAJ's, thereby enabling a solution of the question of whether primitive axes are conjugate.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Ramanujan in 1916 proved the following notable congruence

$$\tau(n)\equiv \sigma_{11}(n) \pmod{691}, \forall~ n\ge 1$$ 

between the two important arithmetic functions $\tau(n)$ and $\sigma_{11}(n)$. In other words, this says that there is a congruence between the cuspidal Hekce eigenform $\Delta(z)$ and the non-cuspidal eigenform $E_{12}(z)$ modulo the prime $691$. Existence of such congruences opened the door for many modern developments in the theory of modular forms.

There are several well-known ways to prove, interpret, and generalize Ramanujan's congruence. For newforms of prime level, some partial results about the existence of such congruences are known. Recently, using the theory of period polynomials, Gaba-Popa (under some technical assumptions) extended these results by determining also the Atkin-Lehner eigenvalue of the newform involved. In this talk, we refine the result of Gaba-Popa under a mild assumption by using completely different ideas. More precisely, we establish congruences modulo certain primes between a cuspidal newform and an Eisenstein series of weight k and prime level. The main ingredients to establish our result are some classical theorems from the theory of Galois representations attached to newforms. As an application, we derive a lower bound for the largest degree of the coefficients field among Hecke eigenforms. This is joint work with A. Kumar, P. Moree and S. K. Singh.

It was shown by M. Bhargava and P. Harron that for n=3,4,5, the shapes of rings of integers of S_n-number fields of degree n become equidistributed in a certain homogeneous space when the fields are ordered by absolute discriminant.  We present a family of analogous distribution questions in some family of torus bundles over the aforementioned homogeneous space and discuss their answers. Our main tool is a new high dimensional equidistribution result in the flavor of Weyl's equidistribution theorem and the work of Bhargava-Harron.

The details of this work appear in the ArXiv preprint 
https://arxiv.org/abs/2201.10942

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group SUT(n, F) is minimal for every local field F of characteristic distinct from 2. This result is new even for the field R of reals and it leads to some important consequences. We prove criteria for the

minimality and total minimality of the special linear group SL(n, F), where F is a subfield of a local field. One of our main applications is a characterization of

Fermat primes, which asserts that for an odd prime p the following conditions are equivalent:

(1) p is a Fermat prime;

(2) SL(p − 1,Q) is minimal, where Q is the field of rationals equipped with the

p-adic topology;

(3) SL(p − 1,Q(i)) is minimal, where Q(i) ⊂ C is the Gaussian rational field.

https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Let $G$ be a group. Let $g(k,n)$ be the maximum number of length-n words over an arbitrary k-letter subset within $G$. How does $g(k,n)$ behave? Obviously $g(k,n)$ is at most $k^n$, and Semple-Shalev proved that if $G$ is finitely generated and residually finite then $g(k,n)<k^n$ (for some, and hence for all sufficiently large k,n) if and only if G is virtually nilpotent. In this case, it is natural to ask how far can g(k,n) be from $k^n$. For k fixed and n ending to infinity, $g(k,n)$ grows polynomially.

Our group-theoretic result is a quantification of the Semple-Shalev Theorem at the other extreme, where $k=\Theta(n)$. Specifically, for a finitely generated residually finite group G, $lim_{n\rightarrow \infty} g(n,n)/n^n$ is either zero, if and only if G is virtually abelian, or at least 4/5, which is sharp for G being (any) Heisenberg group; for higher free nilpotent groups, this limit is 1.

Along the way, we encounter the following combinatorial problem. The 'pair histogram' of a function $f\colon [n]\rightarrow [n]$ is the data consisting of the quantities $#\{(i,j)|i\leq j, f(i)=a, f(j)=b\}$ for each $a,b\in [n]$. What is the probability that a uniformly random function is uniquely determined by its pair histogram? The answer converges to 2/3, and we moreover calculate the limit distribution of the number of sources of pair histograms.

We also interpret our results by means of random lattice paths in $\mathbb{Z}^n$ and their projected polygons, and provide a model-theoretic characterization (by means of free submodels and polynomial identities) of having suboptimal $g(k,n)$, which is valid in various classes of algebraic structures.

This is a joint work with Hagai Lavner.

https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

See abstract in the attached pdf file.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Let $F$ be a field with absolute Galois group $G_F$, $p$ be a prime, and $\mu_{p^e}$ be the $G_F$-module of roots of unity of order dividing $p^e$ in a fixed algebraic closure of $F$.
Let $\alpha \in H^n(F,\mu_{p^e}^{\otimes n})$ be a symbol (i.e $\alpha=a_1\cup \dots \cup a_n$ where $a_i\in H^1(F, \mu_{p^e})$) with effective exponent $p^{e-1}$ (that is $p^{e-1}\alpha=0 \in H^n(G_F,\mu_p^{\otimes n})$. In this work we show how to write $\alpha$ as a sum of symbols from $H^n(F,\mu_{p^{e-1}}^{\otimes n})$. If $n>3$ and $p\neq 2$ we assume $F$ is prime to $p$ closed.

I will describe my work (some joint with I. Dan-Cohen) to extend the computational boundary of Kim's non-abelian Chabauty's method beyond the highly-studied Quadratic Chabauty. Faltings' Theorem says that the number of rational points on curves of higher genus is finite, and non-abelian Chabauty provides a blueprint both for proving this finiteness and for computing the sets of rational points. We first review classical Chabauty-Coleman, which does the same but works only for certain curves. Then we describe Kim's non-abelian generalization, which replaces abelian varieties in Chabauty-Coleman by Selmer groups (a kind of Galois cohomology) and eventually "non-abelian" Selmer varieties. Finally, we describe recent work in attempting to compute these sets using the theory of Tannakian categories.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

The concept of Hausdorff dimension was defined in the 1930s and
was originally applied to fractals and shapes in nature. However, from the
work of Abercrombie, Barnea and Shalev in the 1990s, the computation of
the Hausdorff dimensions in profinite groups has been made possible.
Starting with Abert and Virag's well-known result that there are groups
acting on a rooted tree with all possible Hausdorff dimensions,
mathematicians have been interested in computing the Hausdorff dimensions
of explicit families of groups acting on rooted trees, and in particular,
of the so-called branch groups. Branch groups first appeared in the
context of the Burnside problem, where they delivered the first explicit
examples of finitely generated infinite torsion groups. Since then, branch
groups have gone on to play a key role in group theory and beyond. In this
talk, we will survey known results concerning the Hausdorff dimensions of
branch groups, in particular mentioning some recent joint work Gustavo
Fernandez-Alcober and Sukran Gul.
 

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

In this talk I will describe some new computations of the second bounded cohomology (with trivial real coefficients) of a large class of groups of homeomorphisms of 1-manifolds. I will also discuss applications of this to some problems concerning the spectrum of stable commutator length of finitely presented groups and simplicial volume of manifolds. This is joint work with Francesco Fournier-Facio.

(Graded) Submodule zeta functions are complex functions that are associated to an algebra of endomorphisms of a module. Pattern algebras are examples of such algebras of endomorphisms. In this talk, I will introduce these terms and discuss some properties of the (graded) submodule zeta functions associated to pattern algebras. I will also explain how the results for pattern algebras relate to general conjectures about submodule zeta functions.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Recall that every division algebra over a number field is cyclic. In this talk, we will show a higher dimensional analogue of this classical fact. More precisely, let $F$ be the function field of a curve over a non-archimedean local field. Let $m \geq 2$ be an integer coprime to the characteristic of the residue field. We will show that every element in $H^{3}(F, \mu_{m}^{\otimes 2})$ is a symbol. This extends a result of Parimala and Suresh where they show this when $m$ is prime and under the assumption that $F$ contains a primitive $m^{th}$ root of unity.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

The strong form of Serre's conjecture states that every two-dimensional continuous, odd, irreducible mod p representation of the absolute Galois group of Q arises from a modular form of a specific minimal weight, level and character. In this talk we show the minimal weight is equal to a notion of minimal weight inspired by work of Buzzard, Diamond and Jarvis. Moreover, using the Breuil-Mézard conjecture we give a third interpretation of this minimal weight as the smallest k>1 such that the representation has a crystalline lift of Hodge-Tate type (0, k-1). After discussing the interplay between these three characterisations of minimal weight in the more general setting of Galois representations over totally real fields, we investigate its consequences for generalised Serre conjectures.

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https://us02web.zoom.us/j/89571187014

We define a class of multivariate rational functions associated with 

hyperplane arrangements called flag Hilbert-Poincaré series, and we show that

these rational functions are closely related to enumeration problems 

from algebra. We report on a general self-reciprocity result and explore 

other connections within algebraic combinatorics via Hilbert series of 

Stanley-Reisner rings. This is joint work with Christopher Voll.

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Topic: BIU Algebra Seminar -- Maglione

Time: Jun 2, 2021 10:30 AM Jerusalem

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https://us02web.zoom.us/j/81198490507

Meeting ID: 811 9849 0507

Let R be a noetherian (commutative) ring of Krull dimension d. A classical theorem of Forster states that a rank-n locally free R-module can be generated by n+d elements. Swan and Chase observed that this upper bound cannot be improved in general. I will discuss a joint work with Zinovy Reichstein and Ben Williams where similar upper and lower bounds are obtained for R-algebras, provided that R is of finite type over an infinite field k. For example, every Azumaya R-algebra of degree n (i.e. an n-by-n matrix algebra bundle over Spec R) can be generated by floor(d/(n-1))+2 elements, and there exist degree-n Azumaya algebras over d-dimensional rings which cannot be generated by fewer than floor(d/(2n-2))+2 elements. The proof reinterprets the problem as a question on "how well" certain algebraic spaces approximate the classifying stack of the automorphism scheme of the algebra in question.

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Topic: BIU Algebra Seminar -- First
Time: Mar 10, 2021 10:30 AM Jerusalem

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Meeting ID: 879 6471 5372

The talk will be split into two parts. The first will be about the notion of Hasse-Schmidt derivation on a classical exterior algebra, which I introduced years ago to deal with Schubert calculus for complex Grassmannians. In this first part, I will focus on the purely combinatorial features of the construction suited to be transferred in the second part of the talk, concerned with some joint work in progress with Louis Rowen and Adam Chapman. The new framework will be the more general one of Rowen's monoidal triples. We will analyze a few weaker  notions of exterior semi-algebra and how much of the theory discussed in the first part can be extended to this more demanding situation. The kind of results proposed suggests the possibility of extending a classical part of representation theory coming from the theory of infinite-dimensional integrable systems, which will be briefly discussed while highlighting its promising potential.

A key goal of the Langlands program is to attach Galois representations to automorphic representations. In general, there are two methods to construct these representations. The first, and the most effective, is to extract the Galois representation from the étale cohomology of a suitable Shimura variety. However, most Galois representations cannot be constructed in this way. The second, more general, method is to construct the Galois representation, via its corresponding pseudocharacter, as a p-adic limit of Galois representations constructed using the first method.

In this talk, I will give an expository overview of the second method. I will then demonstrate how this  construction can be refined by using V. Lafforgue’s G-pseudocharacters in place of classical pseudocharacters. As an application, I will prove that the Galois representations attached to certain irregular automorphic representations of U(a,b) are odd, generalising a result of Bellaïche-Chenevier in the regular case. This work is joint with Tobias Berger.

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Topic: BIU Algebra Seminar -- Weiss

Time: Jan 20, 2021 10:30 AM Jerusalem

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A conic is the Severi-Brauer variety of a quaternion algebra Q, and the question of which anisotropic quadratic forms become isotropic over the function field F_Q of a conic has puzzled algebraists for the last three decades. An anisotropic quadratic form is F_Q-minimal if it becomes isotropic over F_Q but any proper subform remains anisotropic. Minimal forms are known to have odd dimension, and examples of minimal forms of any odd dimension were constructed by Hoffmann and Van Geel in characteristic not 2. In this talk, we shall discuss the new analogous examples in characteristic 2 and dimensions 5 and 7. The 7-dimensional example also gives rise to a degree 8 algebra with involution that has $Q$ as a factor as a central simple algebra but not as an algebra with involution. The talk is based on recent joint work with Anne Quéguiner-Mathieu.

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Topic: BIU Algebra Seminar -- Chapman

Time: Dec 30, 2020 10:30 AM Jerusalem

Meeting ID: 841 3979 5965

Passcode: 795644

A representation zeta function of a group G is a (meromorphic continuation of) a Dirichlet series in a complex variable s whose n-th coefficient is the number of irreducible representations of dimension n of G (supposing that these numbers are finite). In 2006 Jaikin-Zapirain proved one of the most fundamental results in the area, namely that if G is a FAb compact p-adic analytic group (e.g., SL_n(Z_p)) and p > 2, then the representation zeta function of G is "virtually rational" in p^{-s}. Two reasons why such a result is interesting is that it immediately implies meromorphic continuation of the zeta function and that its abscissa of convergence is a rational number.

In the talk, I will explain what FAb and "virtually rational" mean here and outline recent joint work with M. Zordan on a new proof of Jaikin-Zapirain's theorem, valid for all primes p. In particular, this also settles a conjecture of Jaikin-Zapirain that the result holds for p = 2. The proof involves projective representations of finite groups as well as a rationality result from the model theory of the p-adic numbers. Such model theoretic rationality results have been proved and used by Hrushovski, Martin, Rideau and Cluckers to establish, among other things, rationality of twist representation zeta functions of nilpotent groups (counting representations up to one-dimensional twists). Our techniques extend to also prove virtual rationality of twist representation zeta functions of groups such as GL_n(Z_p).

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Topic: BIU Algebra Seminar -- Stasinski
Time: Dec 16, 2020 11:00 AM Jerusalem

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Meeting ID: 822 2968 4281
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 Let A->B be a local homomorphism of (commutative) Noetherian local rings. Bart de Smit conjectured in the late 1990's that if A and B have the same embedding dimension, then any finitely generated B-module that is flat over A, is flat over B. This conjecture was proved in 2017 and allows one in some situations to dispense with patching in the techniques à la Wiles to prove modularity lifting theorems (e.g. in the proof of FLT). We generalize this result as follows: if M is a finitely generated B-module whose flat dimension over A satisfies flat dim_A(M) \leq edim(A)-edim(B), then M is free as a B-module. If moreover M is nonzero this forces the morphism A->B to be a special type of complete intersection. This provides by the way a new and simpler proof of de Smit's conjecture. This is joint work with Srikanth Iyengar and Chandrashekhar Khare.

We present the congruence kernel problem for endomorphism rings of finitely generated projective modules, and give it a positive answer in the case of faithful modules over commutative rings.

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Topic: Bar-Ilan Algebra Seminar -- Bar-On
Time: Nov 18, 2020 10:30 AM Jerusalem

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Meeting ID: 883 5594 7298
Passcode: order of A_9
 

In 1960, Graham Higman formulated his famous PORC conjecture in relation to the function f(p,n) counting the isomorphism classes of p-groups of order p^n . By means of explicit formulas, the PORC conjecture has been verified for n < 8. Despite that, it is still open and has in recent years been questioned. I will discuss (generalizations of) an example of du Sautoy and Vaughan-Lee (2012), together with a conceptualization of the phenomena they observe. Hidden heroes of this story turn out to be Hessian matrices and torsion points of elliptic curves. This is joint work with Christopher Voll.

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Topic: BIU Algebra Seminar -- Stanojkovski
Time: Nov 4, 2020 10:30 AM Jerusalem

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Meeting ID: 871 3155 0161
Password: Order of the group A_10

Given a finite set of Brauer classes B of a fixed period \ell, we define eind(B) to be the minimum of degrees of field extensions L/F such that b \otimes_F L = 0 for every b in B. We provide upper bounds for eind(B) which depend on invariants of fields of lower arithmetic complexity, for B in the Brauer group of a semi-global field. As a simple application of our result, we obtain an upper bound for the splitting index of quadratic forms and finiteness of symbol length for function fields of curves over higher-local fields.

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The Serre weight conjectures are conjectures that, roughly 
speaking, predict the weight of a modular form from which a given mod p 
Galois representation arises. Starting from Serre's original conjecture 
for classical modular forms, I will give a motivated approach towards 
correct generalisations of the Serre weight conjectures to Hilbert 
modular forms. Then I'll talk about new results giving a more explicit 
version of the weight conjectures for Hilbert modular forms.

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Michael Schein is inviting you to a scheduled Zoom meeting.

Topic: BIU Algebra Seminar -- Steinmetz

Time: Aug 26, 2020 10:20 AM Jerusalem

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The notion of admissibility of representations of p-adic groups

   goes back to Harish-Chandra. Jacquet and Vigneras have shown that

   smooth irreducible representations of connected reductive p-adic

   groups over algebraically closed fields of characteristic different

   from p are admissible.

   The smooth irreducible representations of $\mathrm{GL}_2({\mathbb Q}_p)$

   over $\bar{\mathbb F}_p$ are also known to be admissible, by the

   work of Barthel-Livne, Breuil and Berger.  However, recently Daniel Le

   constructed non-admissible smooth irreducible representations of

   $\mathrm{GL}_2({\mathbb Q}_{p^f})$ over $\bar{\mathbb F}_p$

   for f > 2, where ${\mathbb Q}_{p^f}$ is the unramified extension

   of ${\mathbb Q}_p$ of  degree f. His construction uses a

   diagram (in the sense of Breuil and Paskunas) attached to

   an irreducible mod p representation of the Galois group of

   ${\mathbb Q}_{p^f}$.

   We shall speak about a variant of Le's construction in the case f = 2

   which uses instead a diagram attached to a reducible split representation

   of the Galois group of ${\mathbb Q}_{p^2}$. This is joint work

   with Mihir Sheth.

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Topic: BIU Algebra Seminar -- Ghate

Time: Aug 5, 2020 10:00 AM Jerusalem

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Our base field is an uncountable algebraically closed field (of any characteristic). We show that an affine space is determined by the abstract group structure of its regular automorphism group in the category of connected affine varieties. To prove this we use tools from birational geometry and study commutative subgroups of the group of automorphisms of affine varieties.

(This is joint work with Serge Cantat and Junyi Xie).

Gromov proved in 1981 that finitely generated groups of polynomial growth are virtually nilpotent. The Grigorchuk group and far reaching generalizations constructed upon it provide examples of groups with intermediate (namely, super-polynomial but subexponential) growth.

For finitely generated associative algebras, a much wider class of growth functions is possible. In particular, many functions, including intermediate growth functions and oscillating functions, are realizable as growth functions of algebras (Belov, Bartholdi-Smoktunowicz, others); nil algebras (which are analogs of Burnside groups) of polynomial growth exist (Lenagan-Smoktunowicz); using matrix wreath products, many intermediate growth functions can be realized in important classes of algebras, including nil rings (Zelmanov); any `sufficiently regular' growth function which is more rapid than $n^{\log n}$ is the growth function of a simple algebra (simple groups with intermediate growth were only recently constructed), and many intermediate growth function more rapid than $\exp(\sqrt(n))$ are the growth functions of finitely generated domains, yielding quotient division rings with interesting properties (in particular, amenable but of infinite GK-transcendence degree).

We present new results on possible (and impossible) growth functions of important classes of algebras, answer several open questions posed by experts in the area and survey the main open problems in the field.

Let C be an algebraic curve of genus g.  An effective procedure to reduce any non-special divisor on C to an equivalent divisor composed of g points is suggested.  The hyperelliptic case is considered as the simplest model.  The advantage of the proposed procedure is its explicitness: all steps are realized through arithmetic operations on polynomials.  The resulting reduced divisor is obtained in the form of the Jacobi inversion problem, which unambiguously defines the divisor.  At the same time, values of abelian functions on the divisor are obtained.

We classify the types of root systems R in the rings of integers O_K of number fields K such that the Weyl
group W(R) lies in the group generated by field automorphisms and multiplications by nonzero elements of K.
We also classify number fields K such that the lattice O_K provided with a trace-like form is isometric or similar to a root lattice. 

This is a report on a joint work with Vladimir Popov (Steklov Institute, Moscow).

We investigate profinite completions of non-strongly complete profinite groups. We present and study the tower of profinite completions.  Furthermore, we study how far a non-strongly complete profinite group can be from being strongly complete, i.e. how many non-open finite-index subgroups it admits. 

It is a classical fact that free (discrete) groups can be embedded in GL_2(Z).  In 1987, Zubkov showed that for a non-abelian free pro-p group F^(p), the situation changes, and for every p > 2, groups of the form GL_2(R) satisfy a "pro-p identity."  More formally, for every p > 2 there exists a nontrivial element g of F^(p) that vanishes under every (continuous) homomorphism F^(p) --> GL_2(R), when R is a pro-finite commutative ring.  In particular, when p > 2, F^(p) cannot be embedded in GL_2(R).  

In 2005, inspired by the solution of the Specht problem, Zelmanov sketched a proof for the following generalization: if n is a natural number, then for every p >> n, GL_n(R) satisfies a "pro-p identity."

In the talk I will discuss Zelmanov's approach, its connection to the Specht problem, and its implications to the area of polynomial identities of Lie algebras.  In addition, I will discuss a recent result regarding the case p = n = 2, saying that GL_2(R) satisfies a pro-2 identity provided that char(R) = 2 (joint with E. Zelmanov).

See attached file.

A conjugacy limit group is the limit of a sequence of conjugates of the positive diagonal Cartan

subgroup, C \leq SL(n) in the Chabauty topology.   Over R, the group C is naturally associated to a projective n-1 simplex.  We can compute the conjugacy limits of C by collapsing the n-1 simplex in different ways.  In low dimensions, we enumerate all possible ways of doing this.  In higher dimensions we show there are infinitely many non-conjugate limits of C. 

In the Q_p case, SL(n,Q_p) has an associated p+1 regular affine building.  (We'll give a gentle introduction to buildings in the talk).  The group C stabilizes and apartment in this building, and limits are contained in the parabolic subgroups stabilizing the facets in the spherical building at infinity. There is a strong interplay between the conjugacy limit groups and the geometry of the building, which we exploit to extend some of the results above.  The Q_p part is joint work with Corina Ciobotaru and Alain Valette. 

Let S be an integral domain with field of fractions F, and let A be an F-algebra.  An S-subalgebra R of A is called S-nice if R lies over S and the localization of R with respect to S\{0} is A.  Let X be the set of all S-nice subalgebras of A.  We define a notion of open sets on X which makes this set a T_0-Alexandroff space.  This enables us to study the algebraic structure of X from a topological point of view.  We prove that an irreducible subset of X has a supremum with respect to the specialization order.  We present equivalent conditions for an open set of X to be irreducible and characterize the irreducible components of X.  We also characterize quasi-compactness of subsets of a T_0-Alexandroff topological space.

In 2014, Moshe Newman asked the following question:  If two subgroups of a finite solvable group are isomorphic and one is a maximal proper subgroup of G, must the other also be a maximal proper subgroup of G?  This question is still open.  I plan to discuss recent results with Geoffrey Robinson that give some sufficient conditions for an affirmative answer.

For an algebra A over a unitary commutative ring k, we have the Hochschild homology HH_*(A).  One use of it was a generalization of the modules of differential forms to non-commutative algebras.  This gave us HDR_*(A), the non-commutative de Rham homology, developed by Alain Connes in his paper “Non-commutative differential geometry.”  In that paper he also produced cyclic homology, HC_*(A), which is connected to both Hochschild and de Rham homology.  The nicest connection between them is when k contains Q.  Then we get the Karoubi exact sequence

0 —> HDR_n(A) —> HC_n(A) —> HH_{n+1}(A).

In the study of quantum groups, cyclic homology is generalized to twisted cyclic homology for a pair of an algebra together with a given k-algebra automorphism.  I was able to extend Karoubi’s theorem to twisted cyclic homology and also to twisted cyclic homology for crossed product algebras (an algebra together with a group of k-algebra automorphisms).  Another extension of cyclic homology is to coalgebras, producing a cyclic cohomology.  One example would be the coalgebra of a Frobenius algebra.

Let p be a multilinear polynomial in several non-commuting variables, with coefficients in an arbitrary field K.  Kaplansky conjectured that for any n, the image of p evaluated on the set M_n(K) of nxn matrices is a vector space.  We settle the analogous conjecture for a quaternion algebra.

See attached file.

Over the decades that U(ZG), the unit group of the integral group ring of a finite group G, has been studied, many conjectures have been raised on how the structure of G influences the structure of subgroups of U(ZG).  Though it often took considerable time, counterexamples for the strongest of these conjectures were found in the class of solvable groups.  Contrary to this, the arithmetic properties of finite subgroups of U(ZG) are very restricted for solvable G.  For instance, the orders of group elements and orders of torsion units u in U(ZG) coincide, under the natural assumption that u has augmentation 1.

A problem on these arithmetic properties, the Prime Graph Question for integral group rings, asks whether it is true that whenever U(ZG) contains an element of augmentation 1 and order pq, where p and q are distinct primes, that G must also contain an element of order pq.  In contrast to other problems in the area, this question is known to have a reduction to almost simple groups.

Employing the combinatorics of Young tableaux and Brauer’s theory of blocks of defect 1 we show that when the Sylow p-subgroup of G has order p, then U(ZG) contains an element of augmentation 1 and order pq, for any prime q, if and only if G contains an element of order pq.  This directly answers the Prime Graph Question for 22 sporadic simple groups and also for infinite series of almost simple groups of Lie type.

This is joint work with M. Caicedo.

One of the most successful approaches to the inverse Galois problem over Q is via specialization of polynomials f(t,X) in two (or more) variables, or equivalently, specialization of Galois covers of the projective line. The fundamental underlying result, Hilbert's irreducibility theorem, ensures that this specialization process preserves the Galois group "most of the time".

In this talk, I will review a series of recent results under the following general question: Which kinds of "strong" versions of the inverse Galois problem can, or cannot be expected to be solved via a specialization approach as above?
Such strong versions include famous problems such as "Grunwald problems", "Malle's conjecture" on the distribution of Galois groups, and the "Q-admissibility conjecture" about G-crossed product division algebras. (This being a survey talk, I will try to make it accessible for a broad audience.) 

Let A be an algebra over a field k, and let G be a finite group.  We say A is G-graded if there are k-subspaces A_g for all g in G such that A is the direct sum of the subspaces A_g, and A_g A_h is contained in A_gh for all elements g,h of G.  Finite group gradings play an important role in the study of finite-dimensional division algebras and, more generally, in the study of finite-dimensional central simple algebras.  For example, crossed product algebras, which provide the bridge between Brauer groups and Galois cohomology, and symbol algebras, which provide the bridge between Brauer groups and K-theory, are both naturally graded algebras.

We consider the following question: what are all possible (finite) group gradings on finite-dimensional k-central division algebras?

In this talk we give, by means of generic constructions, a complete answer in the case where the center k contains an algebraically closed field of characteristic zero.

This work is joint with Eli Aljadeff and Yakov Karasik.

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. Using intersection theory, we give a formula for the degrees of the algebraic boundaries of Voronoi cells of curves and surfaces. We discuss an application to low-rank matrix approximation. This is joint work with Diego Cifuentes, Kristian Ranestad, and Bernd Sturmfels.

In 1981, Sansuc obtained a formula for Tamagawa numbers of reductive groups over number fields, modulo some then unknown results on the arithmetic of simply connected groups which have since been proven, particularly Weil's conjecture on Tamagawa numbers over number fields. One easily deduces that this same formula holds for all linear algebraic groups (not just reductive) over number fields. Sansuc's method still works to treat reductive groups in the function field setting, thanks to the recent resolution of Weil's conjecture in the function field setting by Lurie and Gaitsgory. However, due to the imperfection of function fields, the reductive case is very far from the general one; indeed, Sansuc's formula does not hold for all linear algebraic groups over function fields. We give a modification of Sansuc's formula that recaptures it in the number field case and also gives a correct answer for pseudo-reductive groups over function fields. The commutative case (which is essential even for the general pseudo-reductive case) is a corollary of a vast generalization of the Poitou-Tate nine-term exact sequence, from finite group schemes to arbitrary affine commutative group schemes of finite type. Unfortunately, there appears to be no simple formula in general for Tamagawa numbers of linear algebraic groups over function fields beyond the commutative and pseudo-reductive cases. Time permitting, we may discuss some examples of non-commutative unipotent groups over function fields whose Tamagawa numbers (and relatedly, Tate-Shafarevich sets) exhibit various types of pathological behavior.

This is an overview talk about my paper with Matt Baker with the same title. I will introduce matroids with coefficients, give a rough idea of how ordered blueprints enter the picture and how the moduli space of matroids is constructed, and then 
finally turn to an outline of a new proof of Tutte's theorem that a matroid is regular if and only if it is binary and orientable.

Understanding algebraic structures such as Galois extensions, quadratic forms, division algebras and torsors for linear algebraic groups, can give important insights into the arithmetic of fields. In this talk, I will discuss recent work and work in progress showing ways in which parts of the arithmetic of function fields over complete fields may be encoded in the combinatorial topology of the special fiber. I will then describe how these observations lead to algebraic versions of Meyer-Vietoris sequences, the Seifert–van Kampen theorem, and examples and counterexamples to local-global principles.

We will discuss important examples of infinite dimensional Lie superalgebras and their representations. 

Powering the adjacency matrix of an expander graph results in a better expander of higher degree. High dimensional expanders are simplicial complexes which generalize the notion of expanders. In these settings, we look for an analogue of the powering operation. We show that the naive approach to powering does not yield high dimensional expanders in general, but that for quotients of Bruhat Tits buildings a powering operation arises from so-called "geodesic walks". The analysis of the expansion in the power-complex boils down to intricate combinatorial relations between special flags in a free module over the ring Z/(p^r). Based on joint work with Tali Kaufman.

The talk will report on joint work with Angela Carnevale and Christopher Voll and on work in progress with Tomer Bauer.

Let K be a number field with ring of integers O.  We explicitly determine the local factors, at all primes unramified in K, of the normal subgroup zeta functions of a large class of finitely generated class-2-nilpotent torsion-free groups over O.  This class includes the free class-2-nilpotent groups, various amalgamations of the Heisenberg group, and direct products of any these with abelian groups.  We study the analytic properties of these functions and also give some indication of what happens at the ramified primes.  In particular, these results unify and generalize work of many previous authors and prove a conjecture of Grunewald, Segal, and Smith from 1988 on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.

We will discuss the field of definition of a rational function and in what ways it can change under iteration, in particular when the degree over the base field drops. We present two families of rational functions with the property above, and prove that in the special case of polynomials, only one of these families is possible. We also explain how this relates to Ritt's decomposition theorem on polynomials. Joint work with Francesco Veneziano (SNS Pisa). 

We say that a group G is wide if it contains an element which is not representable 

as a single commutator of elements of G. Recently it was proven that a finite simple 

group cannot be wide, thus confirming a conjecture of Ore of 1950's. On the other hand, 

during the past decades there were discovered several examples of wide infinite simple 

groups. 

In a similar vein, we say that a Lie algebra is wide if it contains an element which is not 

representable as a single Lie bracket. A natural question to ask is whether there exist 

wide simple Lie algebras. Our goal is to present first examples of such Lie algebras.

The simplest example relies on a recent work of Billig and Futorny on Lie algebras of vector 

fields on smooth affine varieties.

This talk is based on a work in progress, joint with Andriy Regeta.

This work provides an example of a finitely presented semigroup H with zero containing an infinite ideal of the form I = LH, where L is a generator of H, such that every word in generators representing an element of I is square free (i.e. any word of the type LXYYZ, for non-empty Y, equals zero in H).

This is a joint work with Ilya Ivanov-Pogodaev and Olga Sapir.

Keywords: Finitely presented semigroups; nil ideals; nil semigroups; nil rings.

In the construction of Hamiltonian reductions in symplectic geometry, interesting and rich connections to Hilbert schemes, Calogero-Moser spaces, and rational spherical Cherednik algebras have emerged over the last two decades. A Borel analogue of the classical general linear group construction (realized after a reduction from the cotangent bundle of enhanced Grothendieck-Springer resolutions) potentially opens doors for its connections to isospectral Hilbert schemes, flag Hilbert schemes, and other algebraic varieties, that are important to geometric representation theory, algebraic combinatorics, and quantum topology. 

Our construction can also be realized by certain quiver flag varieties, appearing in the geometric interplay in quiver Hecke algebras that categorify quantum groups. 

I will discuss a Borel analogue of the cotangent bundle of the extended general linear Lie algebra, discussing the complete intersection of the zero fiber of a moment map (as conjectured by Thomas Nevins), an enumeration of the irreducible components, and a Borel analog of an almost-commuting scheme appearing in the study of Calogero-Moser systems. No background is necessary and I will give plenty of examples throughout my talk. 

This is joint with Travis Scrimshaw. 

I shall describe some open problems, relating to representation
of integers by integral binary quadratic forms: new things are just 
well forgotten old ones !

It has been known for almost a hundred years that most polynomials with integral coefficients are irreducible and have a big Galois group. 

For a few dozen years, people have been interested in whether the same holds when one considers sparse families of polynomials—notably, polynomials with plus-minus 1 coefficients. 

In particular, “some guy on the street” conjectures that the probability for a random plus-minus 1 coefficient polynomial to be irreducible tends to 1 as the degree tends to infinity 

(a much earlier conjecture of Odlyzko-Poonen is about the 0-1 coefficients model) .  In this talk, I will discuss these types of problems, their connection with analytic number theory.

If G is a finite group and M is a G-module, there is a norm map from the homology of G with coefficients in M to the cohomology. This map arises from a morphism in the derived category from the derived co-invariants to the derived invariants of G.

The resulting map is always an isomorphism over the rational numbers but rarely an isomorphism in mod p representation. In stable homotopy theory, there are many "intermediate" characteristics (p.n) associated with the so called "Morava K-theories". It turns out that the norm map is an isomorphism in all those intermediate characteristics and a vast generalization to this fact was discovered by Hopkins and Lurie. They call this generalization Ambidexterity. In my talk I will explain the notion of ambidexterity in stable infinity categories,  present Hopkins and Lurie's result of the ambidexterity in characteristic (p,n) and discuss a recent work on the subject by Tomer Schlank, Lior Yanovski and myself. 

According to the Langlands philosophy, there should be a close relationship between automorphic representations and Galois representations. When are these Galois representations irreducible?

In the 1970s, Ribet proved that the p-adic Galois representation attached to a modular form f is irreducible if and only if f is cuspidal. More generally, it is conjectured that the p-adic Galois representation associated to any cuspidal automorphic representation of GL(n) is irreducible. 

The goal of this talk is to provide an overview of this conjecture, focusing on the special case of Galois representations attached to low weight, genus 2 Siegel modular forms. These two-dimensional analogues of weight 1 modular forms are, conjecturally, the automorphic objects that correspond to abelian surfaces.

In this talk I provide a graph-theoretic proof of the tropical Jacobi identity, alternative 

to the matrix-theoretic proof recently obtained jointly with Akian and Gaubert. The latter was inspired by the classical identity: 

The (J^c,I^c)-minor of a matrix A corresponds, in some way to be defined, to the (I,J)-minor of A^{-1}). 

That is, the compound of order k of A corresponds to the compound of order (n-k) of its inverse.

We provide an application of this theorem to optimal assignments with supervisions.
More precisely, we consider the problem of assigning multiple tasks to one team, or daily tasks to 
multiple teams, where each team has a supervisor-task or a supervised task. 
This is a joint work with Marie Maccaig and Sergey Sergeev.

A foliation subdivides a manifold $M$ into a union of "leafs". It is usually defined by specifying a "distribution": a choice of tangent space at every point of $M$. Going from a distribution to a leaf tangent to it amounts to solving a differential equation, and the leafs are therefore usually transcendental objects.

Consider the case that $M$ and the distribution are algeraic over a number field. We give bounds for the number of intersections between a leaf of the foliation and an algebraic subvariety of complementary dimension by using a combination of ideas from differential equations, value distribution theory and algebraic geometry. I will explain this result and how it leads to estimates for the number of algebraic points of specified degree and height on a leaf. These estimates significantly sharpen the Pila-Wilkie counting theorem in this context.

I will also indicate how this result, applied to foliations arising naturally in the study of abelian varieties and their moduli spaces (modular curves, Siegel varieties, Shimura varieties), can lead to significant information on classical problems in diophantine geometry following ideas of Pila, Zannier, Masser and others.

The reducibility and structure of parabolic inductions is a basic problem in the representation theory of p-adic groups.  Of particular interest are principal series and degenerate principal series representations, that is parabolic induction of 1-dimensional representations of Levi subgroups.

In this talk, I will start by describing the functor of normalized induction and its left adjoint, the Jacquet functor, and by going through several examples in the group SL_4(Q_p) will describe an algorithm which can be used to determine reducibility of such representations.

This algorithm is the core of a joint project with Hezi Halawi, in which we study the structure of degenerate principal series of exceptional groups of type En (see https://arxiv.org/abs/1811.02974).

Let a group G be given by generators and defining relations. It is known that we cannot, in general, extract specific information about the structure of G using the defining relations. However, if the defining relations satisfy small cancellation conditions, then we possess a great deal of knowledge about G. In particular, such groups are hyperbolic, that is, we can express the multiplication in the group by means of thin triangles. It seems of interest to develop a similar theory for rings.

Let kF be the group algebra of the free group F over some field k. Let F have a fixed system of generators. Then its elements are reduced words in these generators that we call monomials. Let I be an ideal of kF generated by a set of polynomials, and let kF / I be the corresponding quotient algebra. In the present work we state conditions on these polynomials that will enable a combinatorial description of the quotient algebra similar to small cancellation quotients of the free group. In particular, we construct a linear basis of kF / I and describe a special system of linear generators of kF / I for which the multiplication table amounts to a linear combination of thin triangles.

Constructions of groups with exotic properties make extensive use of small cancellation theory and its generalizations. In a similar way, generalizations of our approach allow one to construct various examples of algebras with exotic properties.

This is a joint work with A. Kanel-Belov, E. Plotkin and E. Rips.

One of the natural questions of Universal Algebraic Geometry is the following one: when do two algebras from a variety of algebras have the same algebraic geometry?

This question can be interpreted in various ways. For instance, one can say that algebraic geometries of the algebras are the same if the categories of algebraic sets over the given algebras are isomorphic.

An important role in the study of the categories of algebraic sets is played by investigations of automorphisms of the category of free finitely generated  algebras in a given variety.

We will present the method of verbal operations for the study of automorphisms of the category of free finitely generated algebras, consider some results in this area, and discuss open problems.

A classical theorem of Rajan states that a tensor product of simple finite-dimensional modules over a simple Lie algebra admits unique factorization. In this talk, we shall discuss various types of infinite dimensional Lie algebras and the factorization of tensor products for these algebras. Joint with Venkatesh. 

Let S be an integral domain with field of fractions F, and let A be an F-algebra having an S-stable basis. We prove the existence of an S-subalgebra R of A lying over S whose localization with respect to S is A (we call such R an S-nice subalgebra of A). We also show that there is no such minimal S-nice subalgebra of A. Given a valuation v on F with a corresponding valuation domain Ov, and an Ov-stable basis of A over F, we prove the existence of a quasi-valuation on A extending v on F. Moreover, we prove the existence of an infinite decreasing chain of quasi-valuations on A, all of which extend v. Finally, we present applications for the above existence theorems; for example, we show that if A is commutative and C is any chain of prime ideals of S, then there exists an S-nice subalgebra of A having a chain of prime ideals covering C.

Consider a prime number p and a free profinite group S on basis X.

We describe the quotients of S by the lower p-central filtration in terms of the shuffle algebra on X.  This description is obtained by combining tools from the combinatorics of words with Galois cohomology methods.

In the context of absolute Galois groups, this machinery gives a new general perspective on recent arithmetical results on Massey products and other cohomological operations. 

The study of Whittaker models for representations of reductive groups over

local and global fields has become a central tool in representation theory and the theory of automorphic forms, though their Fourier coefficients.  We will start by recalling the classical results on the existence and uniqueness of such models.

In order to encompass representations that do not have Whittaker models, one attaches a degenerate (or a generalized) Whittaker model WO, or a Fourier coefficient in the global case, to any nilpotent orbit.  We will discuss the relation between different kinds of degenerate Whittaker models, and applications to the existence of these models.

We will give several examples for GLn, and discuss the relation to the Bernstein – Zelevinsky derivatives.

We use the language of formal power series to construct finite state automata generating groups of the form A \wr Z, where A is the additive group of a finite commutative ring and Z is the integers. We then provide conditions on the ring and the power series which make automata bireversible. 

This is a joint work with Benjamin Steinberg.

We say that I^n(F) is m-linked if any m bilinear n-fold Pfsiter forms have a common (n-1)-fold factor. In a recent publication, Karim Becher pointed out that when F is a global field, I^n(F) is m-linked for every positive integer m, and raised the question of whether I^n(F) being 3-linked implies that it is m-linked for every positive integer m. In the special case of characteristic 2, this question can be phrased in two versions - one for bilinear forms and another for quadratic forms. We will provide negative answers to both versions of the question in characteristic 2 and discuss some open problems.

Various algebraic and topological situations give rise to compatible sequences of representations of different groups, such as the symmetric groups, with stable asymptotic behavior. Representation stability is a recent approach to studying such sequences, which has proved effective for extracting important invariants. Coming from this point of view, I will introduce the associated character theory, which explains many of the approach's strengths (in char 0). Central examples are simultaneous characters of all symmetric groups, or of all Gl(n) over some finite field. Their mere existence gives applications to statistics of random matrices over finite fields, and raises many combinatorial questions.

Let X be a curve over a p-adic field K with semi-stable reduction and let ω be a meromorphic differential on X. There are two p-adic integrals one may associate to this data. One is the Vologodsky (abelian, Zarhin, Colmez) integral, which is a global function on the K-points of X defined up to a constant. The other is the collection of Coleman integrals on the subdomains reducing to the various components of the smooth locus. In this talk I will prove the following Theorem, joint with Sarah Zerbes:  The Vologodsky integral is given on each subdomain by a Coleman integral, and these integrals are related by the condition that their differences on the connecting annuli form a harmonic 1-cocyle on the edges of the dual graph of the special fiber. I will further explain the implications to the behavior of the Vologodsky integral on the connecting annuli, which has been observed independently and used, by Stoll and Katz-Rabinoff-Zureick-Brown, in works on global bounds on the number of rational points on curves, and an interesting product on 1-forms used in the proof of the Theorem as well as in work on p-adic height pairings. Time permitting I will explain the motivation for this result, which is relevant for the interesting question of generalizing the result to iterated integrals.

A group is said to be invariably generated (IG) by a set S if any conjugation of elements of S still generates G, and topologically invariably generated (TIG) by S if every such conjugation generates G topologically.
I will give a short review of this notion and present new results from joint work with Gennady Noskov.

An octonion algebra is a unital, non-associative algebra endowed with a non-degenerate, multiplicative quadratic form. Such algebras are crucial in the construction of exceptional groups. Over fields, it is known that the quadratic form determines the algebra structure completely. Remarkably, this is not true over commutative rings in general, as was shown by P. Gille in 2014 using cohomological arguments.

I will talk about a recent joint work with Gille, where we give an explicit construction of all octonion algebras having the same quadratic form. I will explain the point of view of torsors and cohomology, and how the phenomenon of triality plays a key role in relating this to a classical construction of alternative algebras.

Given a finite group G and two unitary G-representations V and W, possible restrictions on Brouwer degrees of equivariant maps between the representation spheres S(V) and S(W) are usually expressed in terms of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted α(V)).  Effective application of these congruences is limited by answers to the following questions:

(i) Under which conditions is α(V)>1?

(ii) Does there exist an equivariant map whose degree is easy to calculate?

In my talk I'll address mainly the first question. It will be shown that α(V)>1 for every irreducible non-trivial C[G]-module if and only if G is solvable. So this result provides a new solvability criterion for finite groups.

This is a joint work with Z. Balanov and Haopin Wu.

The recent work of Abe--Henniart--Herzig--Vigneras gives a classification of irreducible admissible mod-p representations of a p-adic reductive group in terms of supersingular/supercuspidal representations. However, supersingular representations remain mysterious largely, and in general we know them very little. So far, there are only classifications of them for the group GL_2 (Q_p) and a few other closely related cases. 

In this talk,  we will present some work on the unramified unitary group G=U(2, 1) defined over a non-archimedean local field of odd residue characteristic p, in which via a local method we show the pro-p-Iwahori invariants of certain supersingular representations of G, as right modules over the pro-p-Iwahori--Hecke algebra of G, are not simple.  This gives a large amount of examples which unveils a possible new feature of supersingular representations in general (note that such a phenomenon never happens in complex representations).  

A finite group is called semi-rational if the distribution induced on it by any word map is a virtual character. Amit and Vishne give a sufficient condition for a group to be semi-rational, and ask whether it is also necessary. We answer this in the negative, by exhibiting two new criteria for semi-rationality, each giving rise to an infinite family of semi-rational groups which do not satisfy the Amit-Vishne condition. On the other hand, we use recent work of Lubotzky to show that for finite simple groups the Amit-Vishne condition is indeed necessary, and we use this to construct the first known example of an infinite family of non-semi-rational groups.

Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? Arzhantseva and Paunescu (2015) formalized this question and answered affirmatively. This can be viewed as a property of the equation XY=YX, and turns out to be equivalent to the following property of the group Z^2 = < X,Y | XY=YX >: Every "almost action" of Z^2 on a finite set is close to a genuine action of Z^2. This leads to the notion of stable groups. Another point of view is that of property testing: The result of [AP] implies that for a pair of permutations, “being (close to) a solution for XY=YX” is a locally testable property, and one may ask which other equations, or systems of equations, are locally testable in this sense.

We will describe a relationship between stability, invariant random subgroups and sofic groups, giving, in particular, a characterization of stability among amenable groups. We will then show how to apply the above in concrete cases to prove and refute stability of some classes of groups. Finally, we will discuss stability of groups with Kazhdan's property (T), and some results on the quantitative aspect of stability.

Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.

Let G be a group and H_1,...,H_s be subgroups of G of  indices d_1,...,d_s respectively. In 1974, M. Herzog and J. Schönheim conjectured that if \{H_i a_i\}_{i=1}^{i=s} is a coset partition of G, then d_1,..,d_s cannot be distinct. We consider the  Herzog-Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied. Furthermore, under a certain assumption, we show there is a finite number of cases to study in order to show the conjecture is true for every coset partition. Since every finitely generated  group is a quotient of a free group of finite rank, we show these results  extend to finitely generated groups.

In analysis, the convolution of two functions results in a smoother, better behaved function. A natural question is then whether this phenomenon has an analogue in the setting of algebraic geometry.

Let f,g be two morphisms from algebraic varieties X,Y to an algebraic group G. We define their convolution to be a morphism f*g from X x Y to G by first applying each morphism to its respective coordinate and then multiplying using the group structure of G.

In this talk, we will present some properties of this convolution operation, as well as a recent result which states that, under mild conditions, after sufficiently many self convolutions every morphism f:X->G becomes flat, with reduced fibers of rational singularities (abbrevieted FRS). This gives a possible answer to the question above.

In addition, the FRS property is of particular interest since, by works of Aizenbud and Avni and of Mustata, it has close ties to the asymptotic point count of the fibers of f over Z/p^kZ. This connection allows us to draw interesting conclusions on algebraic families of random walks on finite groups.

Joint work with Itay Glazer.

The classification of finite subgroups of SL(2,C) has many applications from automorphy of Galois representations to singularity theory and to string theory. I shall describe the classification of finite subgroups of SL(3,C), and a few applications to questions of rationality and linearly reductive groups.

Following the natural instinct that when a group operates on a number field k, every term in the class number formula factorizes “compatibly” according to the representation theory (both complex and modular) of the group, we are led to some questions about the p-part of the class group of k.  The case when k is the cyclotomic extension Q(\mu_p) is the famous Herbrand-Ribet theorem.  We generalize these questions to k = Q(E[p]), where E[p] is the group of p-torsion points on an elliptic curve E over Q.  We answer these questions in a special case.

Various connections between the u-invariant of a field and the symbol length in Milnor K-theory and Kato-Milne cohomology have been proven in recent years.

Karshen and Saltman have each proven independently that when the characteristic is different from 2, the finiteness of the u-invariant implies the finiteness of the symbol length in all Milnor K-groups.

We present the analogous result in the case of characteristic two.

Unlike the previous case, in this case we are able to provide an explicit upper bound for the symbol length.

The talk is based on joint work with Kelly McKinnie.

The determination of which finite abelian groups can occur as the torsion subgroup of an elliptic curve over a number field has a long history starting with Barry Mazur who proved that there are exactly 15 groups that can occur as the torsion subgroup of an elliptic curve over the rational numbers. It is a theorem due to Loïc Merel that for every integer d the set of isomorphism classes of groups occurring as the torsion subgroup of a number field of degree d is finite. If a torsion subgroup occurs for a certain degree, then one can also ask for how many distinct pairwise non-isomorphic elliptic curves this happens. The question which torsion groups can occur for infinitely many non-isomorphic elliptic curves of a fixed degree is studied during this talk. The main result is a complete classification of the torsion subgroups that occur infinitely often for degree 5 and 6. This is joint work with Andrew Sutherland and heavily builds on previous joint work with Mark van Hoeij.

Given a  number field K, the unramified Inverse Galois problem Is concerned with the question of which finite groups $G$ can be realized as Galois groups of Galois unramified extensions $L/K$. The two main ways to attack the problem is by using class field theory (to analyze solvable extensions) and discriminant bounds (to analyze fields $K$ of small discriminant).  The goal of this talk is to show how using homotopical methods one can get results in the non-solvable case with no bound on the discriminant.  We will begin by describing a general method to obtain homotopy theoretical obstructions to problems in Galois theory called "Embedding problems". Then we will explain how to employ these obstructions to study the unramified inverse Galois problem. Specifically, using these obstructions on embedding problems with a non-solvable kernel, we'll give an example of an  infinite family of groups {G_i}i together with an infinite family of quadratic number fields such that for any number field K in this family, the maximal solvable quotient of G_i is realizable as an unramified Galois group over K; but G_i itself is not.

This is a joint work with Magnus Carlson.

We study rational points on two families of hypersurfaces in toric
varieties which are spherical threefolds when equipped with a suitable
action of the reductive group SL_2 x G_m. We are interested in the
asymptotic behavior of the number of rational points of bounded height,
which is predicted by a conjecture of Manin, refined by Batyrev,
Tschinkel, and Peyre. This is joint work with Ulrich Derenthal.

We consider a constant length step random walk on a hyperbolic surface, and deduce that the walker eventually gets lost (i.e., converges to the uniform distribution), and under the assumption of optimality of the non-trivial Laplace spectrum on the surface, the walker gets lost suddenly (i.e., the walk exhibits cut-off). We also prove that under the assumption of optimality the distances between pair of points of the surface are highly concentrated.

Analogous results were proved for graphs by Lubetzky and Peres, and for simplicial complexes by Lubetzky, Lubotzky and Parzanchevski. We show that conceptually the results in all three settings are closely related to the temperedness of representations of corresponding algebraic groups.

Joint work with Amitay Kamber [https://arxiv.org/abs/1712.10149]

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. 

I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting. 

I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, a Chebotarev density type theorem. 

This is a joint work with Tyler Foster.

A classical problem in number theory is to evaluate the number of primes in an arithmetic progression. This problem can be formulated in terms of the von Mangoldt function. I will introduce some conjectures concerning the fluctuations of the von Mangoldt function in arithmetic progressions. I will also introduce an analogous problem in the function field setting and discuss its generalization to arithmetic functions associated with higher degree L-functions (in the limit of large field size). The main example we will discuss is an elliptic curve L-function and statistics associated with its coefficients. This is a joint work with Chris Hall and Jon Keating.

The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:G^->H^ between their dual complex groups should naturally give rise to a map r*:Rep(G)->Rep(H) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to L-functions and recent work on weak functorial lifts to the exceptional group of type G_2.

The abelianization is a functor from groups to abelian groups, which is left adjoint to the inclusion functor. Being a left adjoint, the abelianization functor commutes with all direct limits. It is thus natural to wonder about the behavior of the abelianization functor under inverse limits. There is always a natural map from the abelianization of an inverse limit of groups to the inverse limit of their abelianizations. In this lecture I will present results giving restrictions on the kernel and cokernel of this natural map, in certain cases. These cases include countable directed inverse limits of finite groups, and can thus help in the calculation of the abelianization of certain profinite groups. If time permits I will also consider other families of functors into abelian groups. 

This is a joint work with Saharon Shelah.

 The Birch and Swinnerton-Dyer conjecture predicts that the group of rational points on an elliptic curve E over Q has rank equal to the order of vanishing of the L-function of E. A generalization of this conjecture to motives M was formulated by Bloch and Kato. I will explain a proof of a version of the Bloch-Kato conjecture in p-adic coefficients, when M is attached to a p-ordinary Hilbert modular form of any weight and the order of vanishing is 1.

The case of elliptic curves corresponds to classical modular forms of weight two, and was treated by Perrin-Riou in 1987 using the modular points on E(Q) constructed by Heegner. The proof in the general case is based on the universal p-adic deformation of Heegner points, via a formula for its height. 

The notion of an injective module is one of the most fundamental notions in homological algebra over rings. In this talk, we explain how to generalize this notion to higher algebra. 

The Bass-Papp theorem states that a ring is left noetherian if and only if an arbitrary direct sum of left injective modules is injective. 

We will explain a version of this result in higher algebra, which will lead us to the notion of a left noetherian derived ring. 

In the final part of the talk, we will specialize to commutative noetherian rings in higher algebra, 

show that the Matlis structure theorem of injective modules generalize to this setting, 

and explain how to deduce from it a version of Grothendieck’s local duality theorem over commutative noetherian local DG rings.

Let k be an algebraically closed, complete non-Archimedean field.
I will briefly describe Berkovich spaces, skeletons of Berkovich curves, skeletons of morphisms of curves and various enhancements of skeletons.

The semistable reduction theorem asserts that every nice curve possesses a skeleton.
Furthermore, the simultaneous semistable reduction theorem asserts that any finite generically etale morphism of nice compact curves possesses a skeleton. We are interested in the inverse direction: given a finite morphism of graphs that can arise as skeletons, can it be lifted to a morphism of nice compact curves?

In general, the answer is no. However, enhancing the graphs to metric graphs with reduction k-curves attached to the vertices changes the answer: a lifting theorem of Amini-Baker-Brugallé-Rabinoff (2015) shows that any suitable morphism of such graphs lifts to a finite (generically etale) residually tame morphism of Berkovich curves.

In a joint work with Temkin, we introduce a new enhancement of the skeleton and provide it with new invariants that are trivial in the residually tame case. In this setting, we were able to generalize the lifting result to minimally residually wild morphisms.

Typical representations appear in the Bushnell-Kutzko theory of types for the study of smooth representations of p-adic reductive groups. In this talk, we will present an overview of typical representations associated to level-zero Bernstein blocks of split classical groups.

Howson's theorem says that the intersection of two finitely generated subgroups of a free group is finitely generated.

Hanna Neumann conjectured a bound on the number of generators of the intersection, that after many years of works, has been established independently by Friedman and Mineyev.

I will discuss the history of this problem, surveying the proof techniques. I will then report on a new proof of the stengthened Hanna Neumann conjecture by Jaikin-Zapirain, and show that it generalizes to Demushkin groups (a class of pro-p groups that is of great importance in Galois theory).

No preliminaries are assumed beyond basic familiarity with the free group.

This is a joint work with Andrei Jaikin-Zapirain.

See attached file.

Considering a vector superspace with nondegenerate odd symmetric bilinear form, we define periplectic Lie superalgebras as a subalgebra satisfying this form in a certain way. I will discuss periplectic Lie superalgebras and their representation theory by discussing the action by the Temperley-Lieb algebra associated to the infinite symmetric group on the category of finite-dimensional representations of the periplectic Lie superalgebra as translation functors, the combinatorics behind these translation functors, and the blocks of this category. 

This is joint with I. Entova-Aizenbud, M. Balagovic, Z. Daugherty, I. Halacheva, J. Hennig , G. Letzter, E. Norton, V. Serganova, and C. Stroppel.

See attached poster.

See attached.

In the finite-dimensional representation theory of the symmetric groups
$$S_n$$ over the base field $$\mathbb{C}$$, there is an an interesting
phenomena of "stabilization" as $$n \to \infty$$: some representations
of $$S_n$$ appear in sequences $$(V_n)_{n \geq 0}$$, where each $$V_n$$
is a finite-dimensional representation of $$S_n$$, where $$V_n$$ become
"the same" in a certain sense for $$n >> 0$$.

One manifestation of this phenomena are sequences $$(V_n)_{n \geq 0}$$
such that the characters of $$S_n$$ on $$V_n$$ are "polynomial in $n$".
More precisely, these sequences satisfy the condition: for $$n>>0$$, the
trace (character) of the automorphism $$\sigma \in S_n$$ of $$V_n$$ is
given by a polynomial in the variables $$x_i$$, where $$x_i(\sigma)$$ is
the number of cycles of length $$i$$ in the permutation $$\sigma$$.

In particular, such sequences $$(V_n)_{n \geq 0}$$ satisfy the agreeable
property that $$\dim(V_n)$$ is polynomial in $$n$$.

Such "polynomial sequences" are encountered in many contexts:
cohomologies of configuration spaces of $$n$$ distinct ordered points on
a connected oriented manifold, spaces of polynomials on rank varieties
of $$n \times n$$ matrices, and more. These sequences are called
$$FI$$-modules, and have been studied extensively by Church, Ellenberg,
Farb and others, yielding many interesting results on polynomiality in
$$n$$ of dimensions of these spaces.

A stronger version of the stability phenomena is described by the
following two settings:
 

- The algebraic representations of the infinite symmetric group
$$S_{\infty} = \bigcup_{n} S_n,$$ where each representation of
$$S_{\infty}$$ corresponds to a ``polynomial sequence'' $$(V_n)_{n \geq
0}$$.

- The "polynomial" family of Deligne categories $$Rep(S_t), ~t \in
\mathbb{C}$$, where the objects of the category $$Rep(S_t)$$ can be
thought of as "continuations of sequences $$(V_n)_{n \geq 0}$$" to
complex values of $$t=n$$. 

I will describe both settings, show that they are connected, and
explain some applications in the representation theory of the symmetric
groups.

We will show how the arithmetic of three elliptic curves answers three old questions in the Euclidean geometry.

Fix an arbitrary prime p. Let F be a field containing a primitive p-th root of unity, with absolute Galois group G_F, and let H^n denote its mod p cohomology group, H^n(G_F,\Z/p\Z).
The triple Massey product (abbreviated 3MP) of weight (n,k,m) \in N^3, is a partially defined, multi-valued function 
< , , >: H^n x H^k x H^m \to  H^{n+k+m-1}.

The recently proved 3MP conjecture states that every defined 3MP of weight (1,1,1) contains the zero element.
In this talk I will present the idea of a new proof of the 3MP conjecture for odd primes, inspired by the idea of linearization. The nice thing is that it actually works for 3MP of weight (1,n,1) for arbitrary n.

Albert showed that a central simple algebra A over a field F admits an involution of the first kind, i.e. an F-antiautomorphism of order 2, if and only if the order of the Brauer class of A in the Brauer group of F divides 2.

Azumaya algebras are generalizations of central simple algebras, defined over an arbitrary commutative base ring (or scheme), and can be used to define the Brauer group of a commutative ring. They play an important role in the study of classical groups over schemes.

Albert's theorem fails in the more general setting where A is an Azumaya algebra over a commutative ring R. However, Saltman showed that in this case there is an Azumaya algebra B that is Brauer equivalent to A and admits an involution of the first kind. Knus, Parimala and Srinivas later showed that one can in fact choose B such that deg(B) = 2*deg(A).

I will discuss a joint work with Ben Williams and Asher Auel where we use topological obstructions to show that deg(B) = 2*deg(A) is optimal when deg(A)=4. More precisely, we construct a regular commutative ring R and an Azumaya R-algebra A of degree 4 and period 2 such that the degree of any Brauer equivalent algebra B admitting an involution of the first kind divides 8.

If time permits, I will also discuss examples of Azumaya algebras admitting only symplectic involutions and no orthogonal involutions. This stands in contrast to the situation in central simple algebras where the existence of a symplectic involution implies the existence of an orthogonal involution, and vice versa if the degree is even.

The famous Koethe conjecture asserts that the sum of two nil left ideals is always nil. This still open problem, which is sometimes considered the central open problem in ring theory, has attracted many researchers and inspired a flurry of results toward a better understanding of its validity.

Its most popular equivalent formulation nowadays is, that the polynomial ring R[x] over a nil ring R is equal to its own Jacobson radical.

The observation that R[x] is naturally graded, and every homogeneous element is nilpotent (i.e. R[x] is "graded nil") motivated L. Small and E. Zelmanov to ask ('06) whether a graded nil algebra is always Jaocbson radical.

This was disproved by A. Smoktunowicz a few years ago, and should be mentioned together with another result by Smoktunowicz, disproving a conjecture of L. Makar-Limanov: she proved that there exists a nil ring R such that after tensoring with central variables (specifically: R[x_1,...,x_6]) it contains a free subalgebra. Such ring can exist only over countable base fields.

In this talk we present a new construction, which provides a monomial, graded nilpotent ring (a stronger property than graded nil) which contains a free subalgebra. Our methods involve combinatorics of infinite words, and gluing together sequences of letters which arise from appropriate morphisms of free monoids. In particular, this resolves Small-Zelmanov's question and can be thought of as a continuation of Smoktunowicz's counterexample to Makar-Limanov's conjecture (as in our construction the base field can be arbitrary).

We also construct finitely generated graded Golod-Shafarevich algebras in which all homogeneous elements are nilpotent of bounded index, and prove that such phenomenon cannot appear in monomial algebras. This example also indicates the lack of a graded version for the Shirshov height theorem.

The talk is based on joint work with Jason P. Bell.

Milnor-Witt K-groups of fields were discovered by Morel and Hopkins within the framework of A^1 homotopy theory. These groups play a role in the classification of vector bundles over smooth schemes via Euler classes and oriented Chow groups. Together with Stephen Scully and Changlong Zhong we have generalized these groups to (semi-)local rings and shown that they have the same relation to quadratic forms and Milnor K-groups as in the field case. An application of this result is that the unramified Milnor-Witt K-groups are a birational invariant of smooth proper schemes over a field.  This is joint work with Stephen Scully and Changlong Zhong.

Suppose you have a finite group G and you want to study certain related structures (e.g., random walks, Cayley graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these formulas is lack of knowledge on the low dimensional representations of G. In fact, numerics shows that the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might assist in the solution of important problems. 

In this talk I will discuss a joint project (see arXiv:1609.01276) with Roger Howe (Yale). We introduce a language to speak about “size” of a representation, and we develop a method for systematically construct (conjecturally all the) “small" representations of finite classical groups.

I will illustrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison). 

Constructions of two algebras, both with the ideal of relations defined by a finite Groebner basis, will be presented. For the first algebra the question of whether a given element is nilpotent is algorithmically unsolvable, for the second the question of whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.

Quadratic Pfister forms are a special class of quadratic forms that arise naturally as norm forms of composition algebras.  The Witt group I_q F of quadratic forms (modulo hyperbolic forms) over a field F is a module over the Witt ring of bilinear forms.  This gives a most important filtration { I_q^n F }.  The n-fold Pfister forms, which are tensor products of n Pfister forms, generate I_q^n F.

We call a set of quadratic n-fold Pfister forms linked if they all share a common (n-1)-fold Pfister factor.  Since we wish to develop a characteristic-free theory, we need to consider the characteristic 2 case, where one has to distinguish between right linkage and left linkage.

To a certain type of set of s n-fold Pfister forms, we associate an invariant in I_q^{n+1} F which lives in I_q^{n+s-1} F when the set is linked.  We study the properties of this invariant and compute necessary conditions for a set to be linked.

We also consider the related notion of linkage for quaternion algebras via linkage of the associated norm forms.

I will talk about two topics which give support to a unified theory of archimedean and non-archimedean analytic geometry. In both examples I will review a topic in complex analytic geometry (results from the 1970's) and, after reinterpreting it, show that the same thing happens in non-archimedean geometry (giving new results). The first topic is a non-archimedean version of Ishimura's theorem. This theorem states that on a complex manifold, the continuous linear endomorphisms of the structure sheaf agrees with the sheaf of formal differential operators whose symbol is holomorphic on the cotangent bundle. The second topic is about acyclicity. On a complex analytic space, this is about "quasi-coherent sheaves" not having higher cohomology on Stein spaces. I explain a similar result in the non-archimedean context. The tools used involve an interesting mix of homological algebra and functional analysis. I will explain some potential applications of both of these topics related to number theory. No knowledge about cohomology, differential operators, Stein spaces, or any sort of analytic geometry will be assumed.

Let K/F be a quadratic Galois field extension and let s be the nontrivial F-automorphism of K. A celebrated theorem of Albert characterizes the kernel of the corestriction map Br(K)-->Br(F) as those Brauer classes containing a central simple K-algebra that admits an s-involution, i.e. an involution whose restriction to K is s.

Saltman generalized this result from quadratic Galois extensions of fields to quadratic Galois extension of commutative rings. A later proof given by Knus, Parimala and Srinivas applies in the greater generality of unramified double covers of schemes.

I will discuss a recent work with B. Williams in which we extend the aforementioned results to ramified double covers of schemes (and more generally of locally ringed topoi). Some fascinating phenomena that can occur only in the ramified case will also be discussed. For example, the classical construction of the corestriction of an Azumaya algebra does produce an Azumaya algebra when the corestriction is taken relative to a ramified double cover (so one cannot use it in proving our result).

See attached file.

I will present a joint work with Cai, Friedberg and Ginzburg. 

In a series of constructions, we apply the ``doubling method"

from the theory of automorphic forms to covering groups. 

We obtain partial tensor product L-functions attached to generalized Shimura lifts, 

which may be defined in a natural way since at almost all places the representations 

are unramified principal series.

 Let p be a prime.  To every finite group is associated a topological
space known as the p-completion of its classifying space.  The
Martino-Priddy conjecture states that for two groups G and H, these
spaces are homotopically equivalent if and only if there is a special
type of isomorphism between the Sylow p-subgroups of G and H
(an isomorphism of fusion systems, e.g., elements conjugate in G
are mapped to elements conjugate in H).

  The combined work of several authors has proved this conjecture
and some extensions, partly by assuming the classification of
finite simple groups.  Recently, J. Lynd and I removed this assumption.
I plan to discuss the main ideas of these results.

In this talk, I will discuss finite dimensional representations of quantum affine algebras. The main topics are Chari and Presslay's classification of finite-dimensional simple modules over quantum affine algebras, Frenkel and Reshetikhin's theory of q-characters of finite dimensional modules, Frenkel-Mukhin algorithm to compute q-characters, T-systems, Hernandez-Leclerc's conjecture about the cluster algebra structure on the ring of a subcategory of the category of all finite dimensional representations of a quantum affine algebra. I will also talk about how to obtain a class of simple modules called minimal affinizations of types A, B using mutations (joint work with Bing Duan, Yanfeng Luo, Qianqian Zhang).

For a finite group G and a subgroup H, we say that (G,H) is a Gelfand pair if the decomposition of C[G/H], the G-representation of complex-valued functions on G/H, into irreducible components has multiplicity one. In this case, the Gelfand property is equivalent to the commutativity of the Hecke algebra C[H\G/H] of bi-H-invariant functions on G. 

Given a reductive group G and a closed subgroup H, there are three standard ways to generalize the notion of a Gelfand pair, and a result of Gelfand and Kazhdan gives a sufficient condition under which two of these properties hold. Unfortunately, in contrast to the finite case, here the Gelfand property is not known to be equivalent to the commutativity of a Hecke algebra. In this talk we define a Hecke algebra for the pair (G,H) in the non-Archimedean case and show that if the Gelfand-Kazhdan conditions hold then it is commutative. We then explore the connection between the commutativity of this algebra and the Gelfand property of (G,H).

Let C be a smooth projective curve defined over the finite field F_q (q is odd)

and let K=F_q(C) be its (global) function field. 

Any finite set S of closed points of C gives rise to a Dedekind domain O_S:=F_q[C-S] in K.  

We show that given an O_S-regular quadratic space (V,q) of rank n >= 3,  

the group Br(O_S)[2]  is bijective to the set of genera in the proper classification of quadratic O_S-spaces  

isomorphic to V,q for the \'etale topology, thus there are 2^{|S|-1} such.   

If (V,q) is isotropic, then Pic(O_S)/2 properly classifies the forms in the genus of (V,q). 

This is described concretely when V is split by an hyperbolic plane, 

including an explicit algorithm in case C is an elliptic curve.   

For n >= 5 this is true for all genera hence the full classification is via the abelian group H^2_et(O_S,\mu_2).  

Determining whether a central simple algebra is isomorphic to the tensor product of quaternion algebras is a classical question. One can also ask similar decomposability questions when there is additional structure defined on the central simple algebra, for example an involution. We may ask whether an involution on a central simple algebra is isomorphic to the tensor product of involutions defined on quaternion algebras, i.e. whether the involution is totally decomposable. 

Algebras with involution can be viewed as twisted symmetric bilinear forms up to similarity, and hence also as twisted quadratic forms up to similarity if the characteristic of the underlying field is different from 2. In a paper of Bayer, Parimala and Quéguiner it was suggested that totally decomposable involutions could be a natural generalisation of Pfister forms, a type of quadratic form of central importance to the modern theory of quadratic forms. In this talk we will discuss recent progress on the connection between totally decomposable involutions and Pfister forms. 

We will also discuss fields of characteristic 2, where, since symmetric bilinear forms and quadratic forms are no longer equivalent, involutions are not twisted quadratic forms. Instead, if one wants a notion of a twisted quadratic form with analogous properties to involutions, one works with objects introduced in the Book of Involutions, known as a quadratic pairs. One can define an analogous notion of total decomposability for quadratic pairs, and there is a connection to Pfister forms very similar to that found between involutions and Pfister forms in characteristic different from 2.

We show that the generation problem in the Thompson group F is decidable, i.e., there is an algorithm which decides whether a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2-core of subgroups of F, which can be defined in an analogous way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary amenable subgroup B. The group B is a copy of a subgroup of F constructed by Brin. 

In the classical theory of quadratic forms and Clifford algebras, it is a well-known result that, given a finitely generated projective module P, if H[P] denotes the associated hyperbolic space of P, then the (graded) algebras Cl(H[P]) and End(^(P)) are isomorphic.  We investigate the conditions under which a counterpart of this result holds in the sheaf-theoretic context.  Next, we introduce standard involutions for O_X-algebras associated with K-algebras, where K is a unital commutative ring with no zero-divisors for the purpose of defining graded quadratic extensions of the ringed space (X, O_X), where X = Spec K.

This is joint work with C. Ndipingwi.

Also see the attached file.

The "nonpositive immersion" property is a condition on a 2-complex X

 that generalizes being a surface. When X has this property,  its

 fundamental group appears to have has some very nice properties which

 I will discuss. I will spend the remainder of the talk outlining a

 proof that the nonpositive immersion property holds for a 2-complex

 obtained by attaching a single 2-cell to a graph. This was proven

 recently with Joseph Helfer and also independently by Lars Louder and Henry Wilton.

See attached file.

Let G be a group.  An automorphism of G is called class-preserving if it maps each group element to a conjugate of it.  The obvious examples of class-preserving automorphisms are inner automorphisms. The first example of a group having non-inner class-preserving automorphisms was given by Burnside in 1913.  In this talk we shall present a brief survey of the topic and discuss the nilpotency of the outer class-preserving automorphism group, i.e. the factor group Aut_c(G) / Inn(G), where Aut_c(G) is the group of class-preserving automorphisms of G.

In this talk, we will interpret some classical results of Gauss in the language of flat cohomology and extend them.  Given a quadratic number field k = Q(\sqrt{d}) with narrow class number h_d^+, let O_d be the orthogonal Z-group of the associated norm form q_k.  We will describe the structure of the pointed set H^1_fl(Z, O_d), which classifies quadratic forms isomorphic to q_k in the flat topology, and express its cardinality via h_d^+ and h_{-d}^+.  Furthermore, if N_d is the connected component of O_d, we show that any N_d - torsor tensored with itself belongs to the principal genus.

Let F be a p-adic field. The irreducible admissible mod-p representations of a connected reductive group over F have recently been classified up to supercuspidals by Abe-Henniart-Herzig-Vigneras, building on a method introduced by Herzig in 2011. Their classification is part of an effort to formulate mod-p local Langlands correspondences. The complex representations of certain nonlinear covers of p-adic reductive groups play an interesting role in the classical LLC, and it is natural to ask whether this is also true in the mod-p setting. As a first step, I’ll explain how to modify Herzig’s method in order to classify irreducible admissible genuine mod-p representations of the metaplectic double cover of Sp_{2n}(F). The main consequence of the classification is that parabolically induced genuine mod-p representations are irreducible in the metaplectic case more often than in the reductive case; in particular, all parabolically induced genuine representations of the metaplectic cover of SL_{2}(F) are irreducible. This is joint work with Karol Koziol. 

Deligne categories Rep(GL_t) (for a complex parameter t) have been constructed by Deligne and Milne in 1982 as a polynomial extrapolation of the categories of algebraic representations of the general linear groups GL_n(C). 
In this talk, we will show how to construct a "free abelian tensor category generated by one object of dimension t", which will be, in a sense, the smallest abelian tensor category which contains the respective Deligne's category Rep(GL_t). 
The construction is based on an interesting stabilization phenomenon occurring in categories of representations of supergroups GL(m|n) when t is an integer and m-n=t. 
This is based on a joint work with V. Seganova and V. Hinich.

It turns out that weak proregularity is the appropriate context for the Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A.

In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.

I will describe the problem of mod p reduction of p-adic Galois representations.  For crystalline representations, the reduction can be computed using the compatibility of p-adic and mod p Local Langlands Correspondences; this method was first introduced by Breuil in 2003.  After giving a brief sketch of the history of the problem, I will discuss how the reductions behave for representations with slopes in the half-open interval [1,2).  This is based on joint works with Eknath Ghate, and also with Sandra Rozensztajn for slope 1.

A rational function defined over the rationals has only finitely many rational preperiodic points by Northcott's classical theorem. These points describe a finite directed graph (with arrows connecting between each preperiodic point and its image under the function). We give a classification, up to a conjecture, of all possible graphs of quadratic rational functions with a rational periodic critical point. This generalizes the classification of such graphs for quadratic polynomials over the rationals by Poonen (1998). This is a joint work with Jung Kyu Canci (Universität Basel).

Let z be an algebraic function of n variables and A(z) the algebra generated by all variables and all  partial derivatives of z (of all orders). If z is a polynomial then A(z) is just a polynomial algebra,  but when z is not a polynomial then it is not clear what is the structure of this algebra. I'll report on known cases and formulate a conjecture. 

We shall discuss the notion of superdimension and methods to compute it for simple modules of basic Lie superalgebras. We give a superdimension formula for modules over the general linear Lie superalgebra and propose ideas on how one should approach the general case. Joint with Chmutov and Karpman.

See attached file.

Suppose V is a finite dimensional representation of a complex finite dimensional simple Lie algebra that can be written as a tensor product of irreducible representations. A theorem of C.S. Rajan states that the non-trivial irreducible factors that occur in the tensor product factorization of V are uniquely determined, up to reordering, by the isomorphism class of V. I will present an elementary proof of Rajan's theorem. This is a joint work with S.Viswanath.

In this talk I will give a survey of that part of higher representation theory which studies finitary 2-categories and their 2-representations.  The plan is to present basic definitions, constructions, and results, and then describe some external applications.

Fermat was the first to characterize which integer numbers are sums of two perfect squares. A natural question of analytical number theory is: How many integers up to x are of that form? Landau settled this question using Dirichlet series and complex analysis.

We'll discuss Landau's proof and present recent results on the corresponding problem over the rational function field over a finite field, which requires new ideas.

We consider a matrix with entries over the field of Puiseux series,
equipped with its non-archimedean valuation (the leading exponent).
We establish majorization inequalities relating the
sequence of the valuations of the eigenvalues of a matrix
with the tropical eigenvalues of its valuation matrix
(the latter is obtained by taking the valuation entrywise).
We also show that, generically in the leading coefficients of the
Puiseux series, the precise asymptotics of eigenvalues, eigenvectors
and condition numbers can be determined.
For this, we apply diagonal scalings constructed from
the dual variables of a parametric optimal assignment constructed from
the valuation matrix.

Next, we establish an archimedean analogue of the above inequalities,
which applies to matrix polynomials with coefficients in
the field of complex numbers, equipped with the modulus as its valuation.
In particular, we obtain log-majorization inequalities for the eigenvalues
which involve combinatorial constants depending on the pattern of the matrices.

This talk covers joint works with Ravindra Bapat, Stéphane Gaubert,
Andrea Marchesini, and Meisam Sharify.

We start by presenting Gaubert's symmetrized tropical semiring, which defines a tropical additive-inverse and uses it to resolve tropical singularity. Then, we recall properties of totally positive matrices over rings, define tropical total positivity and total non-negativity of matrices using the symmetrized structure, and state combinatorial and algebraic properties of these matrices. By studying the tropical semiring via valuation on the field of Puiseux series, we relate the tropical properties to the classical ones.
Joint work with Stephane Gaubert

Motivated by the Racah coefficients, the Askey-Wilson algebra was introduced by the theoretical physicist Zhedanov. The algebra is named after Richard Askey and James Wilson because this algebra also presents the hidden symmetry between the three-term recurrence relation and $q$-difference equation of the Askey-Wilson polynomials. In this talk, I will present the progression on the finite-dimensional irreducible modules for Askey-Wilson algebra.

I will present a new combinatorial construction of finite-dimensional algebras with some interesting representation-theoretic properties: they are of tame representation type, symmetric and have periodic modules. The quivers we consider are dual to ribbon graphs and they naturally arise from triangulations of oriented surfaces with marked points.

The class of algebras that we get contains in particular the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin-Shapiro-Thurston and Labardini to triangulations of closed surfaces with punctures. Hence our construction may serve as a bridge between modular representation theory of finite groups and cluster algebras.

All notions will be explained during the talk.

The Schur multiplier is a very interesting invariant, being the archetype of group cohomology.
An explicit description of the multiplier is often too difficult a task.  Therefore it is of interest to obtain information about its arithmetical features, such as the order, the rank, and the exponent.
I will present the problem of bounding the exponent of the multiplier of a finite group, introducing the new concept of unitary cover.

This is joint work with Yuval Ginosar.  Let K/F be a finite Galois extension with Galois group G.  The Teichmüller map is a function that associates to every central simple K-algebra B normal over F an element of H^3(G, K*).  The value of the function is trivial precisely when the class of B is restricted from F.  The classical definition of this map involves the use of a crossed-product algebra over B.  The associativity of this algebra is also equivalent to the class of B being restricted from F.  The aim of this lecture is to elucidate the nature of the nonassociative algebras that arise when B is normal but not restricted.  It turns out that the resulting theory is remarkably similar to the theory of associative algebras arising from the noninvertible cohomology of a Galois extension L/F such that L contains K, and I want to explain that relationship.

In the first part of my talk I will describe with few words and many pictures some more or less ‘combinatorial’ results on tilting modules, bimodules and complexes, almost always obtained by means of elementary tools of two types:

- Linear Algebra arguments (that is, comparison of the dimensions of the underlying vector spaces of certain 

Hom  and  Ext  groups);

- Representation Theory arguments (that is, analysis of the Auslander - Reiten quivers of suitable finite dimensional algebras, almost always admitting only finitely many indecomposable modules up to isomorphism). 

In the second part of my talk I will describe other results (suggested by quivers) concerning ‘reflexive’ modules (not necessarity belonging to the tilting and cotilting worlds) and multiplicities of simple modules in the socle of certain injective cogenerators.  Almost all the results and examples are illustrated in two preprints available at  

http://arxiv.org/abs/1401.2085  and  

http://arxiv.org/abs/1411.4418 . 

The Gieseking group is a one-relator group defined by the
equation aab=bba. It is also the fundamental group of a certain
3-dimensional manifold. As a non-topologist trying to make use of the
latter fact, I learned some things the hard way, which I will share
with the audience.

Stringy Chern classes of singular projective algebraic varieties can be
defined by some explicit formulas using a resolution of singularities. It is important that the output of these formulas does not depend on the choice of a resolution.
The proof of this independence is based on nonarchimedean motivic integration.
The purpose of the talk is to explain a combinatorial computation of stringy Chern
classes for singular toric varieties. As an application one obtains
combinatorial formulas for the intersection numbers of stringy Chern classes
with toric Cartier divisors and some interesting combinatorial identities for convex lattice polytopes.

By the celebrated Hasse principle of Kneser, Harder and Chernousov,
calculating the Galois cohomology  H^1(K,G)  of a simply connected simple
K-group over a number field K reduces to calculating H^1(R,G) over the
field of real numbers R.  For some cases, in particular, for the split
simply connected R-group G of type E_7, the first calculations of
H^1(R,G)  appeared only in 2013 and 2014 in preprints of Jeffry Adams,
of Brian Conrad, and of the speaker and Zachi Evenor. All these
calculations used the speaker's note of 1988.

In the talk I will explain the method of Kac diagrams of calculating
H^1(R,G)  for a simply connected simple R-group G by the examples of
groups of type E_7. The talk is based on a work in progress with
Dmitry A. Timashev. No preliminary knowledge of Galois cohomology or
of groups of type E_7 is assumed.

Zeta functions of groups were introduced by Grunewald, Segal and Smith in 1988. They have proved to be a powerful tool for studying the subgroup structure and growth of certain groups, especially finitely generated nilpotent groups. Three types of zeta function have received special attention: those enumerating all subgroups, normal subgroups or "pro-isomorphic" subgroups: subgroups isomorphic to the original group after taking profinite completions. Of particular interest is a striking symmetry observed in many explicit computations, of a functional equation for local factors of the zeta functions. Inspired by wide-reaching results, due to Voll, for the first two types of zeta function, I will talk about recent progress on the functional equation for local pro-isomorphic zeta functions. Thanks to work of Igusa and of du Sautoy and Lubotzky, these local zeta functions can be analysed by translating them into integrals over certain points of an automorphism group of a Lie algebra associated to the nilpotent group and then applying a p-adic Bruhat decomposition due to Iwahori and Matsumoto. While this technique proves a functional equation for certain classes of such integrals, it is difficult to relate these results back to the nilpotent groups they arise from. In particular, it is not known whether the local pro-isomorphic zeta functions of all finitely generated groups of nilpotency class 2 enjoy local functional equations. I will discuss recent explicit calculations of pro-isomorphic zeta functions for specific nilpotent groups. Interesting new features include an example of a group whose local zeta functions do not satisfy functional equations, a family of groups whose global zeta functions have non-integer abscissae of convergence of arbitrary denominator, and an example whose calculation requires solving congruence equations modulo p^n for a prime p. The latter sheds new light on the types of automorphism groups that can be expected to arise. This is joint work with Benjamin Klopsch and Uri Onn.

In 2012 J. Meakin posed the following question: under what conditions is the word problem for amalgamated free products of inverse semigroups decidable?

Some positive results were interrupted by a result of Radaro and Silva showing that the problem is  undecidable even under some nice conditions.  Revisiting the proofs of decidability, we discuss  whether positive results can be achieved for wider classes of inverse semigroups and show how small the distance is between decidability and undecidability.

In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of $n$ linear functions, in the limit of a large finite field.
A key role is played by the computation of some Galois groups.

See attached file.

The goal of this talk is to show that natural questions in complexity theory raise very natural questions in algebraic geometry. 

More precisely,  we will show how to adapt an approach introduced by Landsberg and  Ottaviani, called Young Flattening, to questions about arithmetic circuits. We will show that this approach generalizes the method of shifted partial derivatives introduced by Kayal to show lower bounds for shallow circuits. 

We will also show how one can calculate shifted partial derivatives of the permanent using methods from homological algebra, namely by calculating a minimal free resolution of an ideal generated by partial derivatives.

I will not assume any previous knowledge about arithmetic circuits.  

Joint work with J.M. Landsberg, H Schenck, J Weyman.

The goal of this talk is to show that natural questions in complexity theory raise very natural questions in algebraic geometry. 

More precisely,  we will show how to adapt an approach introduced by Landsberg and  Ottaviani, called Young Flattening, to questions about arithmetic circuits. We will show that this approach generalizes the method of shifted partial derivatives introduced by Kayal to show lower bounds for shallow circuits. 

We will also show how one can calculate shifted partial derivatives of the permanent using methods from homological algebra, namely by calculating a minimal free resolution of an ideal generated by partial derivatives.

I will not assume any previous knowledge about arithmetic circuits.  

Joint work with J.M. Landsberg, H Schenck, J Weyman.

In this talk we will study the topological ramification locus of a generically étale morphism f : Y --> X between quasi-smooth Berkovich curves.  We define a different function \delta f : Y --> [0,1] which measures the wildness of the morphism.  It turns out to be a piecewise monomial function on the curve, satisfying a balancing condition at type 2 points analogous to the classical Riemann-Hurwitz formula.  We also explain how \delta can be used to explicitly construct the simultaneous skeletons of X and Y.

Joint work with Prof. M. Temkin and Dr. D. Trushin.

The talk will begin with a quick background on Berkovich curves.  All terms will be defined.

In a celebrated paper, J. Tits proved the following fundamental dichotomy for a finitely generated linear group:

Let G be a finitely generated linear group over an arbitrary field.  Then either G is virtually solvable, or G contains a free non-abelian subgroup.

Let G be a non-virtually solvable subgroup of a linear group.  We will discuss the following problem(s): is it possible to find a free subgroup of G that fulfills additional (topological, algebraic, and dynamical) conditions?

We will report on several recent works on Massey products in Galois cohomology,
and explain how they reveal new information on the structure of absolute Galois groups of fields.
 

We give explicit linear bounds on the p-cohomological dimension
of a field in terms of its Diophantine dimension. In particular,
we show that for a field of Diophantine dimension at most 4, the
3-cohomological dimension is less than or equal to the Diophantine dimension.
 

This is a joint work with Tsachik Gelander.

See attached file.

I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry. Our approach works for both complex analytic geometry and p-adic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in these various areas of math and how it is characterised categorically. In order to do this, we need to study algebras and their modules in the category of Banach spaces.  The categorical characterization that we need uses homological algebra in these 'quasi-abelian' categories which is work of Schneiders and Prosmans.  In fact, we work with the larger category of  Ind-Banach spaces for reasons I will explain. This gives us a way to establish foundations of  analytic geometry and to compare with the standard notions such as the theory of affinoid algebras, Grosse-Klonne's theory of dagger algebras (over-convergent functions) and others.  If time remains I will explain how this extends to a formulation of derived analytic geometry following the relative algebraic geometry approach of Toen, Vaquie and Vezzosi.

This is joint work with Federico Bambozzi (Regensburg) and Kobi Kremnizer (Oxford).

Generalizing the notion of nilpotency of groups to nilpotency of semisimple Hopf

algebras H we  give several criteria for  H to be nilpotent in terms

of various sequences of "commutators" and canonical matrices associated to H.  We also initiate the study of probabilistical methods for Hopf algebras and prove that quasi-triangular H  are

“probabilistically nilpotent” ( If G is a finite group then its group algebra kG is an example of such H).

The representation zeta function of a finitely generated nilpotent group is the Dirichlet generating series enumerating the group's irreducible finite-dimensional complex characters up to twists by one-dimensional characters. A simple example is the Heisenberg group over the integers: here the relevant arithmetic function is just Euler's totient function. In general, these zeta functions have natural Euler product decompositions, indexed by the places of a number field. The Euler factors are rational functions with interesting arithmetic properties, such as palindromic symmetries.

In my talk -- which reports on joint work with Alexander Stasinski -- I will (A) explain some general facts about representation zeta functions of finitely generated nilpotent groups and (B) discuss in detail some specific classes of examples, including groups generalizing the free class-2-nilpotent groups. One reason for interest in these classes of groups is the fact that their representation growth exhibits intriguing connections with some statistics on the hyperoctahedral groups (Weyl groups of type B).

A profinite group is equipped with various standard filtrations by closed normal subgroup,
such as the lower central series, the lower p-central series, and the p-Zassenhaus filtration.
In the case of an absolute Galois group of a field, these filtrations are related to the arithmetic 
structure of the field, as well as with its Galois cohomology.  We will describe some recent 
results on these connections, in particular with the Massy product in Galois cohomology. 

Noninvertible cohomology refers to Galois cohomology in which the values of the cocycles are allowed to be noninvertible.  In this talk I will describe an application of this theory to the following problem: Given L/F, a finite separable extension of fields, and an L-central simple algebra B, classify those F-algebras A containing B that are "tightly connected to B" in a sense I will make precise.  The answer uses the Teichmüller cocycle.  This is a three-cocycle that is the obstruction, when L/F is Galois, to a normal L/F central simple algebra (i.e. a central simple L-algebra B with the property that every element of Gal(L/F) extends to an automorphism of B) having the property that its Brauer class in Br(L) is restricted from B(F).  This is mostly work of two of my students, Holly Attenborough and Kevin Foster.

Let A=$\{a_1,\dots,a_n\}$  be a finite alphabet. Consider a substitution $S: a_i\to v_i; i=1,\dots, n$, where $v_i$ are some words.  
 
A DOL-system is an infinite word (superword) $W$ obtained by iteration of $S$. An HDOL-system is $V$ an image of $W$ under some other substitution $a_i\to u_i; i=1,\dots, n$.  
 
The general problem is: suppose we have 2 HDOL-systems. Do they have the same set of finite subwords? This problem is open so far, but the author proved a positive solution of the periodicity problem (is $U$ periodic?) and uniformly recurrence problem http://arxiv.org/abs/1110.4780. This result was obtained independently by Fabien Durand  http://arxiv.org/abs/1111.3268 using different method. see also http://arxiv.org/abs/1107.0185
 
 
We discuss algorithmical problems of periodicity of $V$ 

The multiplicative Borcherds singular theta lift is a well-known
tool for obtaining automorphic forms with known zeros and poles on
quotients of orthogonal symmetric spaces. This has been used by Borcherds
in order to prove a generalization of the Gross-Kohnen-Zagier Theorem,
stating that certain combinations of Heegner points behave, in an
appropriate quotient of the Jacobian variety of the modular curve, like
the coeffcients of a modular form of weight 3/2. The same holds for
certain CM (or Heegner) divisors on Shimura curves.

The moduli interpretation of Shimura and modular curves yields universal
families (Kuga-Sato varieties) over them, as well as variations of Hodge
structures coming from these universal families. In these universal
families one defines the CM cycles, which are vertical cycles of
codimension larger than 1 in the Kuga-Sato variety. We will show how a
variant of the additive lift, which was used by Borcherds in order to
extend the Shimura correspondence, can be used in order to prove that the
(fundamental cohomology classes of) higher codimensional Heegner cycles
become, in certain quotient groups, coefficients of modular forms as well.
Explicitly, by taking the $m$th symmetric power of the universal family,
we obtain a modular form of the desired weight $3/2+m$. Along the way we
obtain a new singular Shimura-type lift, from weakly holomorphic modular
forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.

I will review symmetric monoidal categories and explain how one can work with "algebras and modules" in such a category. Toen, Vaquie, and Vezzosi promoted the study of algebraic geometry relative to a closed symmetric monoidal category. By considering the closed symmetric monoidal category of Banach spaces, we recover various aspects of Berkovich analytic geometry. The opposite category to commutative algebra objects in a closed symmetric monoidal category has a few different notions of a Zariski toplogy. We show that one of these notions agrees with the G-topology of Berkovich theory and embed Berkovich analytic geometry into these abstract versions of algebraic geometry. We will describe the basic open sets in this topology and what algebras they correspond to.  These algebras play the same role as the basic localizations which you get from a ring by inverting a single element. In our context, the quasi-abelian categories of Banach spaces or modules as developed by Schneiders and Prosmans are very helpful. This is joint work with Kobi Kremnizer (Oxford).

The classical scenario in the algebraic theory of invariants is where a group G of automorphisms acts on a ring R. Working in a more general setting, where G need not be a group, I will discuss properties of R which are inherited by the ring of invariants R^G, focusing on cases when R is "almost" semisimple Artinian. 
In particular, if R is semiprimary (resp. left/right perfect; semilocal complete) then so is the invariant ring R^G for any set G of endomorphisms of R. However, that R is artinian or semiperfect need not imply this property for R^G, even when G is a finite group with an inner action. (Examples will be presented if time permits.) The former result actually holds in a more general context: Let S be a ring containing R and let G be a set of endomorphisms of S, then the ring R^G of G-invariant elements inside R inherits from R the properties: being semiprimary, being left (resp. right) perfect.
As easy corollaries, we get that if R is a subring of a ring S, then the centralizer in R of any subset of S inherits the property of being semiprimary or left perfect from R. Better still, the centralizer in R of a set of invertible elements in R inherits the property of being semilocal-complete.
Similarly, assume S is a ring containing R and let M be a right S-module. Then, that End_R(M) is semiprimary (resp. left/right perfect) implies that End_S(M) is. 
All ring-theoretic notions will be defined.