Algebra

Organizer(s): 
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Upcoming Lectures

Uri Brezner (Hebrew University of Jerusalem)
22/11/2017 - 10:30 - 11:30

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.

Dr. Liran Shaul (Ben Gurion University)
29/11/2017 - 10:30 - 11:30

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.

Previous Lectures

Dr. Amiya Kumar Mondal (Bar-Ilan University)
15/11/2017 - 10:30 - 11:30

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.

Mark Shusterman (Tel Aviv University)
08/11/2017 - 10:30 - 11:30

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.

Prof. Louis Rowen (Bar-Ilan University)
01/11/2017 - 10:30 - 11:30

See attached file.

Prof. Mee Seong Im (United States Military Academy)
19/07/2017 - 10:30 - 11:30

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.

Peter Schneider, Stefano Morra, François Legrand, and Gabor Wiese
08/06/2017 - 10:30 - 16:10

See attached poster.

Dr. Eyal Subag (Pennsylvania State University)
17/05/2017 - 10:30 - 11:30
In 1975 George Mackey pointed out an analogy between certain unitary representations of a semisimple Lie group and its Cartan Motion group. 
Recently this analogy was proven to be a part of a bijection between the tempered dual of a real reductive group and the tempered dual of its Cartan Motion group. 
In this talk I will show, in the case of SL(2,R), how algebraic families of Harish-Chandra modules can be used to characterize the Mackey bijection and extend it to an algebraic isomorphism between the admissible duals.

 

Dr. Devika Sharma (Weizmann Institute of Science)
10/05/2017 - 10:30 - 11:30

See attached.

Dr. Inna Entova-Aizenbud (Ben-Gurion University)
03/05/2017 - 11:00 - 12:00

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.

Prof. Jasbir Chahal (Brigham Young University)
03/05/2017 - 10:00 - 11:00

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

Dr. Eli Matzri (Bar-Ilan University)
26/04/2017 - 10:30 - 11:30

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.

Dr. Uriya First (University of British Columbia)
19/04/2017 - 10:30 - 11:30

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.

Be'eri Greenfeld (Bar-Ilan University)
05/04/2017 - 10:30 - 11:30

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.

Prof. Stefan Gille (University of Alberta)
22/03/2017 - 10:30 - 11:30

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.

Prof. Shamgar Gurevich (University of Wisconsin and Yale University)
01/02/2017 - 10:30 - 11:30

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). 

Dr. Sergey Malev (Bar-Ilan University)
25/01/2017 - 10:30 - 11:30

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.

Shira Gilat (Bar-Ilan University)
18/01/2017 - 10:30 - 11:30

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.

Dr. Oren Ben-Bassat (Haifa University)
11/01/2017 - 10:30 - 11:30

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.

Dr. Uriya First (University of British Columbia)
28/12/2016 - 10:30 - 11:30

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).

David El-Chai Ben-Ezra (Hebrew University of Jerusalem)
21/12/2016 - 10:30 - 11:30

See attached file.

Dr. Eyal Kaplan (Bar-Ilan University)
14/12/2016 - 10:30 - 11:30

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.

Prof. George Glauberman (University of Chicago)
07/12/2016 - 10:30 - 11:30

 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.

Dr. Jianrong Li (Weizmann Institute of Science)
30/11/2016 - 10:30 - 11:30

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).

Yotam Hendel (Weizmann Institute of Science)
23/11/2016 - 10:30 - 11:30

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).

Dr. Rony Bitan (Bar-Ilan University)
16/11/2016 - 10:30 - 11:30

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).  

Dr. Andrew Dolphin (Universiteit Antwerpen)
27/10/2016 - 10:30 - 11:30

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.

Dr. Gili Golan (Vanderbilt University)
15/06/2016 - 10:30 - 11:30

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. 

Prof. Patrice Ntumba (University of Pretoria)
01/06/2016 - 10:30 - 11:30

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.

Prof. Daniel Wise (McGill University and Technion)
25/05/2016 - 10:30 - 11:30

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.

Dr. Pradeep Kumar Rai (Bar-Ilan University)
04/05/2016 - 11:00 - 12:00

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.

Dr. Rony Bitan (Université Claude Bernard Lyon I)
04/05/2016 - 10:00 - 11:00

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.

Dr. Laura Peskin (Weizmann Institute of Science)
06/04/2016 - 10:30 - 11:30

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. 

Dr. Inna Entova Aizenbud (Hebrew University of Jerusalem)
30/03/2016 - 10:30 - 11:30

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.

Dr. Benjamin Beeker (Hebrew University of Jerusalem)
16/03/2016 - 10:30 - 11:30
We show that the boundary of a one-ended hyperbolic group that has enough codimension-1 surface subgroups and is simply connected at infinity is homeomorphic to a 2-sphere. Together with a result of Markovic, it follows that these groups are Kleinian groups. 
 
In my talk, I will describe this result and give a sketch of the proof.
 
This is joint work with N. Lazarovich.
Dr. Rishi Vyas (Ben-Gurion University)
09/03/2016 - 10:30 - 11:30
The notion of a weakly proregular sequence in a commutative ring was first formally introduced by Alonso-Jeremias-Lipman (though the property that it formalizes was already known to Grothendieck), and further studied by Schenzel, and Porta-Shaul-Yekutieli. 
 
Roughly speaking, an element s in a commutative ring A is said to be weakly proregular if every module over A can be reconstructed from its localisation at s considered along with its local cohomology at the ideal generated by s. This notion extends naturally to finite sequences of elements: a precise definition will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set.  Every ideal in a commutative noetherian ring is weakly proregular.

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.

Dr. Shalini Bhattacharya (Bar-Ilan University)
02/03/2016 - 10:30 - 11:30

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.

Dr. Solomon Vishkautsan (Scuola Normale Superiore di Pisa)
30/12/2015 - 10:30 - 11:30

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).

Dr. Edva Roditty-Gershon (University of Bristol)
23/12/2015 - 10:30 - 11:30
One of the most famous conjectures in number theory is the Hardy-Littlewood conjecture, which gives an asymptotic for the number of integers n up to X such that for a given tuple of integers a_1,.., a_k all the numbers n+a_1,.., n+a_k  are prime. This quantifies and generalises the twin-prime conjecture.

 

Function field analogue of this problem has recently been resolved in the limit of large finite field size q by Lior Bary-Soroker. However, in this limit the correlations disappear: the arithmetic functions become uncorrelated. It is therefore important to understand the terms of lower order in q, which must account for the correlations. We compute averages of these terms which detect correlations. Our results show that there is considerable cancellation in the averaging and have implications for the rate at which correlations disappear when q tends to infinity. This is a joint work with Jon Keating
Prof. Leonid Makar-Limanov (Wayne State University)
16/12/2015 - 10:30 - 11:30

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. 

Dr. Shifra Reif (ORT Braude College)
09/12/2015 - 10:30 - 11:30

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.

Dr. Claudio Quadrelli (Ben-Gurion University)
02/12/2015 - 10:30 - 11:30

See attached file.

Dr. Eyal Kaplan (Ohio State University)
25/11/2015 - 10:30 - 11:30
One of the tools frequently used in the study of group representations and L-functions is called a model. Roughly speaking, a model is a unique realization of a representation in a convenient space of functions on the group. We will discuss examples of models on linear and covering groups. We will present a novel model: the metaplectic Shalika model. This is the analog of the Shalika model of GL(2n) of Jacquet and Shalika. One interesting representation having this model is the so-called exceptional representation of Kazhdan and Patterson, which is the analog for linear groups of the Weil representation. This representation is truly exceptional.  We will describe it and its role in the study of the symmetric square L-function, and related problems.
Dr. R. Venkatesh (Weizmann Institute of Science)
11/11/2015 - 10:30 - 11:30

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.

Prof. Nir Avni (Northwestern University)
04/11/2015 - 10:30 - 11:30
I will talk about the following questions:
 
 
1) 

Given a system of polynomial equations with integer coefficients, how many solutions does it have in the ring Z/N?

 
​2)​ Given a polynomial map f:R^a-->R^b and a smooth, compactly supported measure m on R^a, does the push-forward of m by f have bounded density?
 
3) Given a lattice in a higher rank Lie group (say, SL(n,Z) for n>2). How many d-dimensional representations does it ​have?
 
I will explain how these questions are related to the singularities of certain varieties. Along the way, I'll talk about canonical singularities, random commutators, and the moduli space of local systems.
 
 
This is a joint work with Rami Aizenbud
Prof. Volodymyr Mazorchuk (Uppsala University)
28/10/2015 - 10:30 - 11:30

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.

Ofir Gorodetsky (Tel Aviv University)
21/10/2015 - 11:00 - 12:00
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.
Prof. Marianne Akian (INRIA Saclay--Ile-de-France and CMAP, Ecole Polytechnique)
24/06/2015 - 10:30 - 11:30

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.

Dr. Adi Niv (INRIA Saclay Ile-de-France and Ecole Polytechnique)
17/06/2015 - 10:30 - 11:30

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

Dr. Hau-Wen Huang (Hebrew University of Jerusalem)
03/06/2015 - 10:30 - 11:30

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.

Dr. Sefi Ladkani (Ben-Gurion University)
27/05/2015 - 10:30 - 11:30

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.

Nicola Sambonet (Technion)
20/05/2015 - 10:30 - 11:30

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.

David Corwin (Massachusetts Institute of Technology)
13/05/2015 - 10:30 - 11:30
Let E be an elliptic curve over a number field K with algebraic closure K'. For an integer n, the set of n-torsion points in E(K') forms a group isomorphic to Z/n X Z/n, which carries an action of G_K=Gal(K'/K).
 
A result of Serre shows that if K=Q, then the associated homomorphism from G_K to GL_2(Z/n) cannot be surjective for all n. The result is false, however, over other number fields. The 2010 PhD thesis of Greicius found the first counterexample, over a very special non-Galois cubic extension of Q.
 
In this talk I will describe the above background and then describe more recent results of the speaker and others allowing one to find such elliptic curves over a more general class of number fields. As time allows, I may describe other results from our paper about finding elliptic curves with maximal (but not surjective) Galois action given certain constraints, or a forthcoming paper doing similar work for abelian surfaces.
Mark Shusterman (Tel Aviv University)
06/05/2015 - 11:15 - 12:15
The talk is going to be about the work carried out as part of my MSc thesis.
 
Motivated by recent arithmetic results, we will consider new and improved
results on the freeness of subgroups of free profinite groups:
 
1.The Intermediate Subgroup Theorem - A subgroup (of infinite index) in a
nonabelian finitely generated free profinite group, is contained in a free profinite group of infinite rank.
 
2. The Verbal Subgroup Theorem - A subgroup containing the normal closure of a (finite) word in the elements of a basis for a free profinite group, is free profinite.
 
These results shed light on several theorems in Field Arithmetic and may be combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a generalization of a result of Bary-Soroker, Fehm, and Wiese on the profinite freeness of subgroups arising from Galois representations.
Beeri Greenfeld (Bar-Ilan University)
06/05/2015 - 10:15 - 11:15
Graded algebras play a major role in many topics, including algebraic geometry, topology, and homological algebra, besides classical ring theory. These are algebras which admit a decomposition into a sum of homogeneous components which 'behave well' with respect to multiplication.
In this talk we present several structure-theoretic results concerning affine (that is, finitely generated) Z-graded algebras which grow 'not too fast'.
In particular, we bound the classical Krull dimension both for algebras with quadratic growth and for domains with cubic growth, which live in the heart of Artin's proposed classification of non-commutative projective surfaces. We also prove a dichotomy result between primitive and PI-algebras, relating a graded version of a question of Small.
From a radical-theoretic point of view, we prove that unless a graded affine algebra has infinitely many zero homogeneous components, its Jacobson radical vanishes. Under a suitable growth restriction, we prove a stability result for graded Brown-McCoy radicals of Koethe conjecture type: they remain Brown-McCoy even after being tensored with some arbitrary algebra.
Finally, we pose several open questions which could be seen as graded versions of the Kurosh and Koethe conjectures.
The talk is based on joint work with A. Leroy, A. Smoktunowicz and M. Ziembowski.
Dr. Rony Bitan (Bar-Ilan University)
29/04/2015 - 10:30 - 11:30
Let C be a smooth projective curve defined over a finite field F_q (q is odd), and let K=F_q(C) be its (global) function field.  This field is considered as the geometric analogue of a number field.
 
Removing one closed point from C results in an affine curve C^af.  The ring of regular functions over C^af is an integral domain, over which we consider a non-degenerate bilinear and symmetric form f of any rank n. 
 
We express the number c(f) of isomorphism classes in the genus of f in cohomological terms and use it to present a sufficient and necessary condition depending only on C^af, under which f admits the Hasse local-global principle.  We say that f admits the Hasse local-global principle if c(f)=1, namely, |Pic(C^\af)| is odd for any n other than 2 and equal to 1 for n=2.  
 
This result emphasizes the difference between Galois cohomology and etale cohomology. Examples are provided.
Dr. Uriya First (University of British Columbia)
15/04/2015 - 10:30 - 11:30
Let R be a discrete valuation ring with fraction field F. Two algebraic objects (say, quadratic forms) defined over R are said to be rationally isomorphic if they become isomorphic after extending scalars to F. In the case of unimodular quadratic forms, it is a classical result that rational isomorphism is equivalent to isomorphism. This has been recently extended to "almost umimodular" forms by Auel, Parimala and Suresh. I will present further generalizations to hermitian forms over (certain) involutary R-algebras and quadratic spaces equipped with a group action ("G-forms"). The results can be regarded as versions of the Grothendieck-Serre conjecture for certain non-reductive groups.
(Joint work with Eva Bayer-Fluckiger.)
Prof. Darrell Haile (Indiana University)
25/03/2015 - 11:15 - 12:15

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.

Prof. Gabriella D'Este (University of Milan)
25/03/2015 - 10:15 - 11:15

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

Moshe Newman
18/03/2015 - 10:30 - 11:30

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.

Prof. Victor Batyrev (Universität Tübingen)
11/03/2015 - 11:15 - 12:15

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.

Prof. Mikhail Borovoi (Tel Aviv University)
11/03/2015 - 10:15 - 11:15

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.

Dr. Mark Berman (ORT Braude College of Engineering)
28/01/2015 - 11:15 - 12:15

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.

Prof. Alessandra Cherubini (Politecnico di Milano)
28/01/2015 - 10:15 - 11:15

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.

Efrat Bank (Tel Aviv University)
21/01/2015 - 10:30 - 11:30

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.

Dr. Klim Efremenko (University of California, Berkeley)
31/12/2014 - 10:30 - 11:30

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.

Dr. Klim Efremenko (University of California, Berkeley)
31/12/2014 - 10:30 - 11:30

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.

Dr. Shaul Zemel (Technische Universität Darmstadt)
24/12/2014 - 11:00 - 12:00
We show how the simple property of 2-Henselianity suffices to reduce the classification of lattices over a general valuation ring in which 2 is invertible (with no restriction on the value group) to classifying quadratic spaces over the residue field. The case where 2 is not invertible is much more difficult. In this case we present the generalized Arf invariant of a unimodular rank 2 lattice, and show how in case the lattice contains a primitive vector with norm divisible by 2, a refinement of this invariant and a certain class suffice for classifying these lattices.
Dr. Adam Chapman (Michigan State University)
24/12/2014 - 10:00 - 11:00
We discuss the situation where two quaternion algebras over a field of characteristic 2 share the same genus, i.e. have the same set of isomorphism classes of quadratic field extension of the center. We provide examples of pairs of nonisomorphic quaternion algebras who satisfy this property, and show that over global fields and the fields of Laurents series over perfect fields the quaternion algebras are uniquely determined by their maximal subfields.
This talk is based on a joint work with Andrew Dolphin and Ahmed Laghribi.
Adina Cohen (Hebrew University of Jerusalem)
10/12/2014 - 10:30 - 11:30

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.

Prof. G. A. Soifer (Bar-Ilan University)
03/12/2014 - 10:30 - 11:30

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?

Prof. Ido Efrat (Ben Gurion University)
26/11/2014 - 10:30

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.
 

Dr. Eli Matzri (Ben Gurion University)
19/11/2014 - 10:30

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.
 

Arie Levit (Weizmann Institute of Science)
12/11/2014 - 10:30
Subgroup growth usually means the asymptotic behavior of the number of subgroups of index n of a given f.g. group as a function of n.
We generalize this to discrete (torsion-free) subgroups of the Lie group G=SO+(n,1) for which the quotient admits finite volume, as a function of the co-volume. Conjugacy classes of such discrete subgroups correspond geometrically to n-dimensional hyperbolic manifolds of finite volume.

 

By a classical result of Wang, for n >=4 there are only finitely many such conjugacy classes up to any given finite volume V. More recently, Burger, Gelander, Lubotzky and Mozes showed that this number grows like V^V.
 
In this talk we focus on counting commensurability classes. Two subgroups are commensurable if they admit a common finite index subgroup (in our context, up to taking conjugates). We show that surprisingly, for n >= 4 this number grows like V^V as well. Since the number of arithmetic commensurability classes grows ~polynomially (Belolipetsky), our result implies that non-arithmetic subgroups account for “most" commensurability classes.
 
Our proof uses a mixture of arithmetic, hyperbolic geometry and some combinatorics. In particular, recall that a quadratic form of signature (n,1) over a totally real number field, whose conjugates are positive definite, defines an arithmetic discrete subgroup of finite covolume in G. As in the classical construction of Gromov--Piatetski-Shapiro, several non-similar quadratic forms can be combined to construct amalgamated non-arithmetic subgroups.


This is a joint work with Tsachik Gelander.

Dr. Oren Ben-Bassat (University of Oxford and University of Haifa)
29/10/2014 - 10:30

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).

Prof. Miriam Cohen (Ben-Gurion University)
21/05/2014 - 10:30

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).

Prof. Christopher Voll (Bielefeld University)
14/05/2014 - 10:30

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).

Adi Niv (Bar-Ilan University)
07/05/2014 - 10:30
The Max-Plus (tropical) algebra, is the set of real numbers R, together with  -\infty, equipped with the operations maximum and the usual plus. We start by presenting some basic notation in this setting, and show how the lack of additive inverse causes failure of some classic algebraic properties. Then, we present the extended (supertropical) algebra, introduced and studied by Izhakian and Rowen, which adds a layer of singular elements to R. We show how this extension recovers these failed properties. In the last part we introduce definitions and theorems in supertropical linear algebra, and state the connection between the eigenvalues of a matrix to those of its powers, tropical-inverse and conjugates. If time allows, we will give some details of the proof.
 
*The results on characteristic polynomials are a part of the speaker's PhD thesis. 
 
Prof. Ido Efrat (Ben-Gurion University)
30/04/2014 - 10:30


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. 

 
Dr. Lior Bary-Soroker (Tel Aviv University)
02/04/2014 - 10:30
Many problems in algebra and number theory reduce to 
the problem of calculating Galois groups.
In this talk, I will focus on the proof of the following theorem:
 
Thm: Let x |--> f(x) be a polynomial map from the Riemann sphere to itself of degree n=deg f. 
Assume that f(x) is Morse (in the  sense that the critical points are non-degenerate and the critical values are distinct).
Then the Galois group is the full symmetric group.
 
The proof involves some geometry and some finite group theory.
Gili Golan (Bar-Ilan University)
26/03/2014 - 10:30
The Tarski number of a group G is the minimal number of pieces in a paradoxical decomposition of it. We investigate how Tarski numbers may change under various group-theoretic operations. Using these estimates and known properties of Golod-Shafarevich groups, we show that the there are 2-generated groups with property (T) and arbitrarily large Tarski numbers.  
We also prove that there exist groups with Tarski number 6. These provide the first examples of non-amenable groups without free subgroups whose Tarski number has been computed precisely.
 
Joint work with Mikhail Ershov and Mark Sapir. 
Prof. Darrell Haile (Indiana University)
19/03/2014 - 10:30

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.

Dr. Ivan Mitrofanov (Moscow State University)
29/01/2014 - 10:30

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 $V$ 

Dr. Shaul Zemel (Technische Universitaet Darmstadt)
27/11/2013 - 10:30

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.

Dr. Oren Ben-Bassat (University of Oxford and University of Haifa)
20/11/2013 - 10:30

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).

Sergey Malev (Bar-Ilan University)
13/11/2013 - 10:30
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 n-by-n matrices is either zero, or the set of
scalar matrices, or the set sl_n(K) of matrices of trace 0, or
all of M_n(K). I prove the conjecture when K is the field of real numbers and
n=2, and give a partial solution for
an arbitrary field K.
Uriya First
01/02/2012 - 10:30

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.