# Algebra Seminar[ Edit ]

## Moderator: Danny Neftin

*Abstract:*

**NOTICE THE SPECIAL DATE AND TIME!**

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 state a conjecture characterizing the Mackey bijection as an algebraic isomorphism between the admissible duals. This will be done in terms of certain algebraic families of Harish-Chandra modules. We shall see that the conjecture hold in the case of SL(2,R).

*Abstract:*

Sample 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 whether a given element is nilpotent is algorithmically unsolvable, for the second the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.

Joint work with Ilya Ivanov-Pogodaev.

*Abstract:*

Contramodules are module-like algebraic structures endowed with infinite summation or, occasionally, integration operations understood algebraically as infinitary linear operations subject to natural axioms.For about every abelian category of torsion, discrete, or smooth modules there is a no less interesting, but much less familiar, dual analogous abelian category of contramodules. So there are many kinds of contramodule categories, including contramodules over coalgebras and corings, associative rings with a fixed centrally generated ideal, topological rings, topological Lie algebras, topological groups, etc. The comodule-contramodule correspondence is a covariant equivalence between additive subcategories in or (conventional or exotic) derived categories of the abelian categories of comodules and contramodules. Several examples of contramodule categories will be defined in the talk, and various versions of the comodule-contramodule correspondence discussed.

*Abstract:*

We will prove that for any finite solvable group G, there exists a cyclic extension K/Q and a Galois extension M/Q such that the Galois group Gal(M/K) is isomorphic to G and M/K is unramified.

We will apply the theory of embedding problem of Galois extensions to this problem and gives a recursive procedure to construct such extensions.

*Abstract:*

The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0,1,2,4 or 8. The analogous question in the case of characteristic 2 remained open for a long time. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in a joint work by Andrew Dolphin and the speaker.

*Abstract:*

Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? This can be seen as a question about a property the equation XY=YX. Studying analogous problems for more general equations (or systems of equations) leads to the notion of "locally testable groups" (aka "stable groups").

We will take the opportunity to say something about "local testability" in general, which is an important subject in computer science. We will then describe some results and methods developed (in a work in progress), together with Alex Lubotzky, to decide whether various groups are locally testable or not.This will bring in some important notions in group theory, such as amenability, Kazhdan's Property (T) and sofic groups.

*Abstract:*

Milnor-Witt K-groups of fields have been 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 applications of this result is that the unramified Milnor-Witt K-groups are a birational invariant of smooth proper schemes over a field.

(joint work with Stephen Scully and Changlong Zhong)

*Abstract:*

Let $K$ be a commutative ring. Consider the groups $GL_n(K)$. Bernstein and Zelevinsky have studied the representations of the general linear groups in case the ring $K$ is a finite field. Instead of studying the representations of $GL_n(K)$ for each $n$ separately, they have studied all the representations of all the groups $GL_n(K)$ simultaneously. They considered on $R:=\oplus_n R(GL_n(K))$ structures called parabolic (or Harish-Chandra) induction and restriction, and showed that they enrich $R$ with a structure of a so called positive self adjoint Hopf algebra (or PSH algebra). They use this structure to reduce the study of representations of the groups $GL_n(K)$ to the following two tasks:

1. Study a special family of representations of $GL_n(K)$, called "cuspidal representations''. These are representations which do not arise as direct summands of parabolic induction of smaller representations.

2. Study representations of the symmetric groups. These representation also has a nice combinatorial description, using partitions.

In this talk I will discuss the study of representations of $GL_n(K)$ where $K$ is a finite quotient of a discrete valuation ring (such as $\Z/p^r$ or $k[x]/x^r$, where $k$ is a finite field). One reason to study such representation is that all continuous complex representations of the groups $GL_n(\Z_p)$ and $GL_n(k[[x]])$ (where $\Z_p$ denotes the $p$-adic integers) arise from these finite quotients. I will explain why the natural generalization of the Harish-Chandra functors do not furnish a PSH algebra in this case,and how is this related to the Bruhat decomposition and Gauss elimination.

In order to overcome this issue we have constructed a generalization of the Harish-Chandra functors. I will explain this generalization, describe some of the new functors properties, and explain how can they be applied to studying complex representations.

The talk will be based on a joint work with Tyrone Crisp and Uri Onn.

*Abstract:*

Given a finite group G, we consider in this talk ``parametric sets'' (over $\mathbb{Q}$), {\it{i.e.}}, sets $S$ of (regular) Galois extensions of $\mathbb{Q}(T)$ with Galois group $G$ whose specializations provide all the Galois extensions of $\mathbb{Q}$ with Galois group $G$. This relates to the Beckmann-Black Problem (which asks whether the strategy by specialization to solve the Inverse Galois Problem is optimal) which can be formulated as follows: does a given finite group $G$ have a parametric set over $\mathbb{Q}$?

We show that many finite groups $G$ have no finite parametric set over $\mathbb{Q}$. We also provide a similar conclusion for some infinite sets, under a conjectural ``uniform Faltings theorem''.

This is a joint work with Joachim K\"onig.

*Abstract:*

I will briefly recall the theory of Hurwitz spaces and their relevance to the Inverse Galois Problem. I will then describe techniques for explicit computation. Finally I will give a survey of problems to which these techniques can be applied. The focus will be on producing "nice" polynomials for nice groups, rather than providing exhaustive theoretical results.

*Abstract:*

We shall present structural results of the profinite completion $\widehat G$ of a 3-manifold group $G$ and its interrelation with the structure of $G$. Residual properties of $G$ also will be discussed.

*Abstract:*

The goal of this talk is to present the definition, the motivation and the main properties of (graded) Cohen-Macaulay rings. It will include the notions of homogeneous regular sequences and system of parameters, and a solution for the main problem -- under which conditions a ring is a free module over a polynomial subring generated by a system of parameters?

The talk assumes familiarity with basic Commutative Algebra results, which will be reminded during the talk.

*Abstract:*

What are the irreducible constituents of a smooth representation of a p-adic group that is constructed through parabolic induction?

In the case of GL_n the problem can be formulated as a study of the multiplicative behavior of irreducible representations in the so-called Bernstein-Zelevinski ring.

I will try to convey the idea that such problems are in fact universal in Lie theory. The theory of Kazhdan-Lusztig polynomials points on intriguing equivalences between several settings, such as representations of Lie algebras, affine Hecke algebras, canonical bases in quantum groups and more recently KLR algebras. All of which give different tools and points of view on similar phenomena.

*Abstract:*

The notion of a weakly proregular idea 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. The precise definition is quite technical, but will be given in the talk. 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.

At the beginning of this talk, these ideas will be motivated by studying what happens in a very particular case: power series in one variable over a field. In particular, a portion of this talk will be elementary and accessible to any one with a background in basic commutative and homological algebra.

Time permitting, after a brief survey of the general theory we will proceed to 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.

*Abstract:*

**NOTICE THE SPECIAL DAY AND PLACE!**

**The lecture designed for graduate students!**

Many properties of a finite group G can be approached using formulas involving sums over its characters. A serious obstacle in applying these formulas seemed to be lack of knowledge over the low dimensional representations of G. In fact, the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some conjectures which are currently out of reach.

This talk will discuss a joint project with Roger Howe (Yale), where we introduce a language to define, and a method for systematically construct, the small representations of finite classical groups.

I will demonstrate our theory with concrete motivations and numerical data obtained with John Cannon (Head of MAGMA, Sydney) and Steve Goldstein (Scientific Computing, Madison).

*Abstract:*

How many numbers between X and X+H are square-free, where X is large and H > X^ε? In how many ways can a large number N be given as a sum N = x^k + r of a positive k-th power and a positive square-free number r? In full generality, both questions are still mostly open. They can be seen as special cases of a more general question - how many values of a polynomial f(x) are square-free, where the coefficients of the polynomial are much larger than the values the argument assumes? We answer these questions in the function field setting, over a fixed finite field with degrees going to infinity, following the techniques of Poonen and Lando, who solved similar questions for polynomials with fixed coefficients.

*Abstract:*

The inverse Galois problem over a field E asks which finite groups occur as Galois groups over E. The most interesting case is E being the field of rationals numbers, wherethe problem is wide open. The question has been given a positive answer for many classes of function fields, via a method called Patching, invented by Harbater in the 1980s and refined by many researchers since. In this talk we'll describe this method and survey theorems achieved by it, leading up to recent results.

*Abstract:*

**NOTICE THE SPECIAL TIME!**

Hyperbolic polynomials are one of the central topics of study in real algebraic geometry. Though their study was initiated in the 50's of the previous century in connection with Cauchy problems for PDEs, since then they have found applications in various fields both theoretical and applied. Recently Markus, Spielman and Srivastava used stable polynomials (cousins of the hyperbolic polynomial) to prove the long standing Kadison-Singer conjecture.

In this talk we will define the notion of hyperbolicity for real subvarieties of $\mathbb{P}^d$ and show that this notion gives rise the notion of a real-fibered morphism. A real morphism $f \colon X \to Y$ between two real varieties is called real fibered if it is finite, flat, surjective and the preimage of real points of $Y$ is always real. We will show that this abstract definition tells us a great deal about the ramification of $f$ at real points. This data in turn tells us about the structure of real points of a smooth real hyperbolic variety. Time permitting I will discuss Ulrich sheaves and bilinear forms on such sheaves, that coorespond to definite determinantal representation of hyperbolic varieties.

The talk is based on a joint work with M. Kummer (Konstanz).

*Abstract:*

Suppose $\tilde{G}$ is a connected reductive group over a finite field $k$, and $\Gamma$ is a finite group acting on $\tilde{G}$, preserving a Borel-torus pair. Then the connected part $G$ of the group of $\Gamma$-fixed points of $\tilde{G}$ is reductive, and there is a natural map from (packets of) representations of $G(k)$ to those of $\tilde{G}(k)$. I will discuss this map, its motivation in the study of $p$-adic base change, prospects for refining it, and a generalization: the pair of groups $(\tilde{G},G)$ must satisfy some axioms, but $G$ need not be a fixed-point subgroup of $\tilde{G}$, nor even a subgroup at all.

*Abstract:*

I will present an elementary proof of the following theorem of Alexander Olshanskii:

Let F be a free group and let A,B be finitely generated subgroups of infinite index in F. Then there exists an infinite index subgroup C of F which contains both A and a finite index subgroup of B.

The proof is carried out by introducing a 'profinite' measure on the discrete group F, and is valid also for some groups which are not free.Some applications of this result will be discussed:

1. Group Theory - Construction of locally finite faithful actions of countable groups.

2. Number Theory - Discontinuity of intersections for large algebraic extensions of local fields.

3. Ergodic Theory - Establishing cost 1 for groups boundedly generated by subgroups of infinite index and finite cost.

*Abstract:*

When studying noncommutative f.d. algebras, the building blocks, in a sense, are the matrix algebras over division algebras (e.g. the real quaternions). This led to the idea of a generic division algebra such that all the division algebras are just specializations of it. In particular, many properties satisfied by the generic division algebra are inherited by all other division algebras. The generic crossed product arises in a similar manner, when we consider division algebras with a crossed product structure. In this lecture, I will talk about the place of division algebras and crossed products in the study of f.d. algebras, and how to construct their generic versions. Moreover, I will show why the center of these generic objects play such a central role, and how to compute it using field invariants

*Abstract:*

**NOTICE THE SPECIAL TIME AND DATE!**

Let $p$ be a multilinear polynomial in several non-commutingvariables with coefficients in an arbitrary field $K$. Kaplanskyconjectured that for any $n$, the image of $p$ evaluated on theset $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set ofscalar matrices, or the set $sl_n(K)$ of matrices of trace $0$, orall of $M_n(K)$. We prove the conjecture for $K=\mathbb{R}$ orfor quadratically closed field and $n=2$, and give a partial solution for an arbitrary field $K$.We also consider homogeneous and Lie polynomials and providethe classifications for the image sets in these cases.

*Abstract:*

The generalized circle problem asks for the number of lattice points of an n-dimensional lattice inside a large Euclidean ball centered at the origin. In this talk I will discuss the generalized circle problem for a random lattice of large dimension n. In particular, I will present a result that relates the error term in the generalized circle problem to one-dimensional Brownian motion. The key ingredient in the discussion will be a new mean value formula over the space of lattices generalizing a formula due to C. A. Rogers.

This is joint work with Andreas Strömbergsson.

*Abstract:*

Ramanujan graphs, constructed by Lubotzky, Phillips and Sarnak and known also as the LPS graphs, are certain quotients of the Bruhat-Tits building of PGL_2(Q_p). These graphs form a family of expander graphs, and serve as an explicit construction of graphs of high girth and large chromatic number. High dimensional counterparts of the LPS graphs are the Ramanujan Complexes, constructed by Lubotzky, Samuels and Vishne, as quotients of the Bruhat-Tits building of PGL_d over a non-archimedean field of finite characteristic. I'll talk about the mixing of these complexes, which implies that they have good expansion and large chromatic number.

Joint work with S.Evra, A.Lubotzky.

*Abstract:*

In "All p-adic reductive groups are tame" Bernstein proved that for a reductive group G over a local non-archimedean field F and a compact open subgroup K of G there exists a uniform bound C(G,K) such that every irreducible, smooth, and admissible representation V of G satisfies dim(V^K) < C(G,K). In the talk I will repeat the proof of Bernstein and give my proof to one of the two main lemmas. The new proof of this lemma will give a new, sharper bound for the constant C(G,K).

*Announcement:*

See here for details.

*Abstract:*

We study measures induced by free words on the unitary groups U (n): let w be a word in the free group F_r on r generators x_1,...,x_r. For every i=1,...,r substitute x_i with an independent, Haar-distributed random element of U(n) and evaluate the product defined by w to obtain a random element in U(n). The measure of this element is called the w-measure on U(n).Let Tr_w(n) denote the expected trace of a random unitary matrix sampled from U (n) according to the w-measure. It was shown by Voiculescu (91') that for w \ne 1, this expected trace is o(n) asymptotically in n. We relate the numbers Tr_w(n) to the theory of commutator length of words and obtain a much stronger statement. Our analysis also sheds new light on the solutions of the equation [u_1, v_1] . . . [u_g, v_g] = w in free groups. I will also present some interesting related open problems.

Joint work with Michael Magee.

*Abstract:*

Cluster algebras are commutative rings with a distinguished set of generators that are grouped into overlapping finite sets of the same cardinality. Among many other examples, cluster algebras appear in coordinate rings of various algebraic varieties.

Using the notion of compatibility between Poisson brackets and cluster algebras in the coordinate rings of simple complex Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang Baxter equation. For a simple complex Lie group G and a Belavin-Drinfeld class, one can define a corresponding Poisson bracket on the ring of regular functions on G. For certain types of classes in SLn, a compatible cluster structure can be constructed. Cluster algebras will be defined and explained, and a description of compatible structures and their properties will be given.

*Abstract:*

After the proof of the Bloch-Kato conjecture, we know that the $\mathbb{F}_p$-cohomology ring $H^\bullet(G,\mathbb{F}_p)$of a maximal pro-$p$ Galois group $G$ is a quadratic algebra.Recently L.~Positselsky conjectured that such ring is a quadratic Koszul algebra -- and he proved it is for localand global fields.We prove this conjecture for the class of pro-$p$ groups of elementary type, and for such pro-$p$ groups the quadratic (or Koszul)dual of $H^\bullet(G,\mathbb{F}_p)$ is a canonical graded algebra induced by thecomplete group algebra $\mathbb{F}_p[\![G]\!]$.Moreover, we prove that for any maximal pro-$p$ Galois group $G$ the quadratic dual of $H^\bullet(G,\mathbb{F}_p)$is the ``quadratic cover'' of such graded algebra, which carries also some arithmetic information.This is a joint work with J.~Min\'a\v{c} and N.D.~T\^an.

*Abstract:*

An intersective polynomial is a monic polynomial in one variable with rational integer coefficients, with no rational root and having a root modulo $m$ for all positive integers $m$. Let $G$ be a finite noncyclic group and let $r(G)$ be the smallest number of irreducible factors of an intersective polynomial with Galois group $G$ over $\dQ$. Let $s(G)$ be smallest number of proper subgroups of $G$ having the property that the union of their conjugates is $G$ and the intersection of all their conjugates is trivial. It is known that $s(G)\leq r(G).$ It is also known that if $G$ is realizable as a Galois group over the rationals, then it is also realizable as the Galois group of an intersective polynomial. However it is not known, in general, even for the symmetric groups $S_n$, whether there exists such a polynomial which is a product of the smallest feasible number $s(G)$ of irreducible factors.

**Theorem**: For every $n$, either $r(S_n)=s(S_n)$ or $r(S_n)=s(S_n)+1$, with the first equality holding for all odd $n$. When $n$ is the product of at most two odd primes, $r(S_n)$ is computed explicitly. General upper and lower bounds for $r(S_n).$ are also given. (Joint work with Daniela Bubboloni)

*Abstract:*

In this talk, I will present results on the number of ramified prime numbers in the specialization of a finite Galois extension of $\mathbb{Q}(T)$ at a positive integer. In particular, I will give a central limit theorem for this number. The talk will be partially based on a joint work with Lior Bary-Soroker.

*Abstract:*

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.

*Abstract:*

Let f be a square-free polynomial in Fq[t][x] where Fq is a field of qelements. We view f as a univariate polynomial in x with coefficientsin the ring Fq[t]. We study square-free values of f in sparse subsetsof Fq[t] which are given by a linear condition. The motivation for ourstudy is an analogue problem of representing square-free integers byinteger polynomials, where it is conjectured that setting aside somesimple exceptional cases, a square-free polynomial f in Z[x] takesinfinitely many square-free values. Let c(t) be an arbitrarypolynomial in Fq[t]. A consequence of the main result we show, is thatif q is sufficiently large with respect to the degree of c(t) and thedegrees of f in t and x, then there exist v,w in Fq such thatf(t,c(t)+vt+w) is square-free, i.e. a square-free value of f isobtained by varying the first two coefficients of c(t).

*Abstract:*

Kemer's representability theorem is one of the less understood gems of PI theory,whereas the PI exponent is, by now, a well known and incredibly handy tool in thearsenal of PI theory. It is remarkable however that these two topics have a lot incommon and one can greatly bene t the other. One such occasion occurs when onegeneralizes the representability theorem and Amitsur's PI exponent conjecture tothe framework of H-module F-algebras satisfying an ordinary polynomial identity.Here F is a characteristic zero eld and H is a semisimple nite dimensional Hopfalgebra over F. In particular, this includes ( nite) group graded and group actedalgebras.In this talk I will tell about recent and (less recent) results concerning the abovewith emphasis on the intersection of these two theories.

*Abstract:*

The Schur multiplier is a very interesting invariant, being the archetype of group cohomology. An explicit description of the multiplier is often a too difficult task, it is therefore of interest to obtain information about its numerical attributes, 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 a very interesting construction which has been the major contribution of my PhD research.