# Tel Aviv Algebra Seminar

## 2020-2021, Semester B

**Time: June 6, 2021, 14:10**

Speaker: Uri Bader (Weizmann Institute)

Title: TBA

Abstract: TBA

**Time: May 30, 2021, 14:10 **

Speaker: Ido Efrat (Ben Gurion University)

Title: Filtrations of Profinite Groups as Intersections and Absolute Galois Groups

Abstract: Let $F$ be a field with absolute Galois group $G=G_F$.
By a 25-year old result by Minac and Spira, the third term in the lower 2-central filtration of $G$ is the intersection of all closed normal subgroups $N$ of $G$ such that $G/N$ is either the group of order $2$, the cyclic group of order $4$, or the dihedral group $D_4$. Similar "Intersection Theorems" were subsequently proved for odd primes, and for other filtrations of $G$, such as the Zassenhaus filtration. All these results are based on deep cohomological properties of absolute Galois groups.
In our talk we will describe a general "Transfer Theorem", which underlies all these intersection results.

**Time: May 23, 2021, 14:10 **

Speaker: Danny Neftin (Technion)

Title: Reducibility of polynomials under specializations

Abstract:
Given an algebraic family of irreducible polynomials f(t,x) in Q(t)[x] that depends on a parameter t, how large is the set of rational values t=t_0 for which f(t_0,x) in Q[x] is reducible? When is it infinite?
We shall discuss this problem, its role in arithmetic geometry, applications to arithmetic dynamics, and, as time permits, its relation to the question: "can you hear the shape of the drum?"

**Time: May 9, 2021, 14:10 **

Speaker: Philip Dittmann (TU Dresden)

Title: Galois groups of iterated polynomials

Abstract: I will discuss some results, both old and recent, on the Galois groups of the iterates f(X), f(f(X)), f(f(f(X))), ... of a polynomial f. These Galois groups were originally studied by Odoni in the 1980s, and are related to elementary problems on integer sequences as well as current work in arithmetic dynamics. Among other things I will discuss recent joint work with Borys Kadets concerning behaviour over hilbertian fields, using tools from field arithmetic.

**Time: May 2, 2021, 14:10 **

Speaker: Elad Sayag (Tel Aviv University)

Title: Hilbert's Irreducibility Theorem

Abstract: In this lecture I will describe the elementary proof of Hilbert Irreducibility theorem as in the book "Groups as Galois groups"

**Time: April 18, 2021, 14:10**

Speaker: Tamar Bar-On (Bar-Ilan University)

Title: Segal&Nikolov Theorem and applications

Abstract: A profinite group is called strongly complete if every subgroup of finite index is open. Strongly complete groups are very useful, since in such groups the algebra determines the topology. For example, every homomorphism from a strongly complete group to any profinite group is continuous, and thus a homomorphism in the category of profinite groups.
For many years it was an open question, whether every finitely generated profinite group is strongly complete. In 2000 Segal and Nikolov published a positive proof for this conjecture.
In the talk we present the general idea of the proof, and show some nice results relying on this theorem.

**Time: April 11, 2021, 14:10**

Speaker: Oren Becker (Cambridge)

Title: Stability of approximate group actions

Abstract: An approximate unitary representation of a group G is a function f from G to U(n) such that f(gh) is close to f(g)f(h) for all g,h. Is every approximate unitary representation just a slight deformation of a unitary representation? The answer depends on G and on the norm on U(n). If G is amenable, the answer is positive for the operator norm on U(n) (Kazhdan '82). The answer remains positive if we use the normalized Hilbert-Schmidt norm and allow a slight change in the dimension n (Gowers-Hatami '15, De Chiffre-Ozawa-Thom '17). For both norms, the answer is negative if G is a nonabelian free group (or a nonelementary word-hyperbolic group). In this talk we shall discuss a similar notion where U(n) is replaced by Sym(n) with the normalized Hamming metric. We study the cases where G is either free, amenable or equal to SL_r(Z), r>=3. When G is finite, a slight variation of our main theorem provides an efficient probabilistic algorithm to determine whether a function f from G to Sym(n) is close to a homomorphism when |G| and n are both large. Based on a joint work with Michael Chapman.

**Time: March 21, 2021, 14:10**

Speaker: Uriya First (Haifa University)

Title: The Grothendieck--Serre conjecture for classical groups in low dimensions

Abstract: Let R be a regular local ring and K its field of fractions. Let q and q' be two quadratic forms which become isomorphic K. Are q and q' already isomorphic over R? This and similar statements of this flavor are special cases of an open conjecture of Grothendieck and Serre about principal homogeneous bundles of reductive group schemes over R. The conjecture has been studied extensively over the years and has seen significant progress in the past decade, but it remains open in general. I will survey the conjecture and discuss a recent work with Eva Bayer-Fluckiger and Raman Parimala in which we establish the conjecture for low-dimensional R and G a classical group; this can be restated in terms of hermitian forms and Witt groups.

**Time: March 7, 2021, 14:10 **

Speaker: Mark Shusterman (Harvard)

Title: Finite presentation for fundamental groups in characteristic p

Abstract: Let X be a smooth projective variety over an algebraically closed field k. If k has characteristic zero, for instance the complex numbers, homotopy theory can be used to show that the fundamental group of X is finitely presented. In a joint work with Esnault and Srinivas we show that in case the characteristic of k is positive, even though the corresponding homotopical tools are lacking, it is still possible to show that the fundamental group of X is finitely presented.

## 2020-2021, Semester A

**Time: January 17, 2021, 14:10 **

Speaker: Umberto Zannier (Scuola Normale Superiore)

Title: Abelian varieties not isogenous to any Jacobian - **dedicated to Rudnick for his 60th birthday**

Abstract: It is well known that in dimension g\ge 4
there exist complex abelian varieties not isogenous to
any Jacobian. A question of Katz and Oort asked whether
one can find such examples over the field of algebraic numbers.
This was answered affirmatively by Oort-Chai under the
Andre'-Oort conjecture, and by Tsimerman unconditionally.
They gave examples within Complex Multiplication.
In joint work with Masser, by means of a completely
different method, we proved that in a sense the "general
abelian variety over \overline\Q is indeed not isogenous
to any Jacobian. I shall illustrate the basic principles
of the proofs.

**Time: January 10, 2021, 14:10 **

Speaker: Pierre Debes (Univeristy of Lille)

Title: The Hilbert-Schinzel specialization property

Abstract: Hilbert's Irreducibility Theorem shows that irreducibility over the field of rationals is “often” preserved
when one specializes a variable in some irreducible polynomial in several variables. I will present a version "over
the ring'' for which the specialized polynomial remains irreducible over the ring of integers. The result also relates
to the Schinzel Hypothesis about primes in value sets of polynomials: I will discuss a weaker “relative” version for
the integers and the full version for polynomials. The results extend to other base rings than the ring of integers;
the general context is that of rings with a product formula. (Joint work with A. Bodin, J. Koenig and S. Najib).

**Time: January 3, 2021, 14:10 **

Speaker: Arno Fehm (Dresden University of Technology)

Title: Existential rank of diophantine sets in fields

Abstract: A diophantine subset of a field K is the projection of the zero set of a polynomial over K. After plenty of motivation and examples I will report on joint work with Nicolas Daans and Philip Dittmann in which we introduce and study several complexity measures of diophantine sets. One result that I will explain in more detail requires the construction of a polynomial f in two variables over certain fields of characteristic p (e.g. rational function fields over finite field) for which f(x,y) is a p-th power if and only if both x and y are.

**Time: December 27, 2020, 14:10**

Speaker: Brad Rodgers (Queen's University)

Title: The distribution of random polynomials with multiplicative coefficients

Abstract: A classic paper of Salem and Zygmund investigates the distribution of trigonometric polynomials whose coefficients are chosen randomly (say +1 or -1 with equal probability) and independently. Salem and Zygmund characterized the typical distribution of such polynomials (gaussian) and the typical magnitude of their sup-norms (a degree N polynomial typically has sup-norm of size $\sqrt{N \log N}$ for large N). In this talk we will explore what happens when a weak dependence is introduced between coefficients of the polynomials; namely we consider polynomials with coefficients given by random multiplicative functions. We consider analogues of Salem and Zygmund's results, exploring similarities and some differences.
Special attention will be given to a beautiful point-counting argument introduced by Vaughan and Wooley which ends up being useful.
This is joint work with Jacques Benatar and Alon Nishry.

**Time: December 20, 2020, 14:10**

Speaker: Asaf Yekutieli (Tel Aviv University)

Title: Triple product of Maass Forms

Abstract: Maass forms are a particular class of smooth functions defined on a hyperbolic Riemann surface.
In the special case of Riemann surfaces associated with a congruence subgroup, it is often the case that results concerning Maass forms bear witness to the existence of profound arithmetic relations.
Our main goal is to describe the problem of estimating the triple product functional, explain its significance, and illustrate the representation theoretical techniques employed by Bernstein and Reznikov to make progress.
If time permits, we shall discuss non-Archimedean instances of the above theory.
I will not be assuming familiarity with any of the abovementioned notions.

**Time: December 6, 2020, 14:10**

Speaker: Chen Meiri (Technion)

Title: Verbal width and the model theory of higher rank arithmetic groups

Abstract: In this talk we will present joint work with Nir Avni and Alex Lubotzky concerning the model theory of higher rank arithmetic groups. We will show that many of these groups are determined by their first order theory as individual groups and also as a collection of groups. These results follow from a model theoretic property called bi-interpretation which, in the context of higher rank arithmetic groups, is related to the verbal width.

**Time: November 29, 2020, 14:10**

Speaker: Uriel Sinichkin (Tel Aviv Univeristy)

Title: Enumeration of algebraic and tropical singular hypersurfaces

Abstract: It is classically known that there exist (n+1)(d-1)^n singular hypersurfaces of degree d in complex projective n-space passing through a prescribed set of points (of the correct size). In this talk we will deal with the analogous problem over the real numbers and construct, using tropical geometry, Omega(d^n) real singular hypersurfaces through a collection of points in RP^n. We will also consider the enumeration of hypersurfaces with more than one singular point.

No prior knowledge of tropical geometry will be assumed.

**Time (Note it is on Tuesday!): November 24, 2020, 11:00**

Speaker: Henry Wilton (Cambridge)

Title: On stable primitivity rank

Abstract: The commutator length of an element w of the commutator subgroup of a group G is the minimal number of commutators needed to express w as a product of commutators. A more fruitful definition is obtained by stabilising the definition, yielding the notion of *stable* commutator length.

In the context of free groups, Puder has recently introduced the notion of *primitivity rank*, which can be thought of as a homotopical version of commutator length. In this talk, I’ll propose a stable version of primitivity rank, and state some of its properties.

**Time: November 22, 2020, 14:10 **

Speaker: Sean Eberhard (Cambridge)

Title: Diameter of high-rank classical groups with random generators

Abstract: Babai's conjecture asserts that the diameter of the Cayley graph of any finite simple group G is bounded by (log |G|)^O(1). This conjecture has been resolved for groups of bounded rank, but for groups of unbounded rank such as SL_n(2) it is wide open. Even for random generators, only the case of alternating groups is resolved. In this talk we sketch the proof of Babai's conjecture for SL_n(p), p = O(1), with at least *three* random generators. The proof extends to other classical groups over F_q if we have at least q^{100} random generators. The heart of the proof consists of showing that the Schreier graph of SL_n(q) acting on F_q^n with respect to q^{100} random generators is an expander graph.

**Time: November 15, 2020, 14:10**

Speaker: Roy Shmueli (Tel Aviv University)

Title: The expected number of roots of random polynomials over the field of p-adic numbers

Abstract: There is a classical problem of estimating the expected number of real roots of a random polynomial, going back at least to Littlewood-Offord.
In this talk we shall present a p-adic version of this problem and a result estimating of the expected number of roots for a general family of distributions,
and several corollaries of the theorem for specific distributions such as a random Littlewood polynomial (polynomial whose coefficients are +1 or -1).

**Time: November 8, 2020, 14:10**

Speaker: Galyna Dobrovolska (PITP, Waterloo)

Title: Lattice gauge theory, combinatorics, and surfaces

Abstract: I will speculate about connections between partition functions in lattice gauge theory, the approach to representation theory of the symmetric group due to Okounkov-Vershik, and geometry of surfaces a la Magee-Puder (in progress, joint with V. Chelnokov).

**Time: November 1, 2020, 14:10**

Speaker: Benjamin Zack Kutuzov (Weizmann Institute)

Title: Tame Geometry and Applications in Diophantine Geometry

Abstract: We review the famous Yomdin-Gromov Lemma about smooth reparametrizations of semialgebraic sets, and state a version of this lemma for holomorphic reparametrizations and semialgebraic sets defined over $\bb{Q}$. We then introduce o-minimal theory and illustrate how it can deeply impact diophantine geometry.

**Time: October 25, 2020, 14:10**

Speaker: Daniele Garzoni (University of Padova)

Title: Invariable generation of finite simple groups

Abstract: A subset X of a group G invariably generates G if we are free to replace each element of X by an arbitrary conjugate, and we must always obtain a generating set of G. This concept was introduced by Dixon in the early nineties with motivations from computational Galois theory. We will review these motivations and their intimate connections with permutation groups.

We will then present some new results concerning the probability of generating invariably a finite simple group. For instance, we will see that two random elements of a finite simple group of Lie type of bounded rank invariably generate with probability bounded away from zero. Joint work with Eilidh McKemmie.

**Time: October 18, 2020, 14:10**

Speaker: Vlad Matei (Tel Aviv University)

Title: Average size of the automorphism group of smooth projective hypersurfaces

Abstract: We show that the average size of the automorphism group over $\mathbb{F}_q$ of a smooth degree $d$ hypersurface in $\mathbb{P}^{n}_{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow \infty$. We also discuss some consequences of this result for the moduli space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$.