Algebra

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 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. 

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. 

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.

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.

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.

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.

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.

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.

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.

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). 

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.

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

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. 

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.

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).

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.

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.

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 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.

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.

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.

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).  

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.

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.

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.

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.

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.

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.

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 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. 

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.

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.

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.

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.

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.

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.

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). 

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.

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).

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.

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).

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).  

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. 

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.

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.

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. 

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.

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.

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 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

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.

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 . 

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.

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 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.

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.

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.
 

This is a joint work with Tsachik Gelander.

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).

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. 

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 $V$ 

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).