### Fall2018

All the talks are 4:00pm-5:00pm in Room 5417 at the CUNY Graduate Center.

Wine and cheese are served afterwards in the math lounge on the 4th floor.

Sep. 7

### Tullio Ceccherini-Silberstein (Universita degli Study del Sannio)

TITLE: Garden of Eden Theorems: from Symbolic Dynamics to Algebraic
Dynamical Systems.

ABSTRACT: The Garden of Eden Theorem is a central result in the theory of
cellular automata. Given a finite alphabet set A and a group G, a
continuous G-equivariant (w.r. to the G-shift) map $\tau \colon A^G \to A^G$ is called a cellular automaton. The GOE theorem states that a
cellular automaton is surjective if and only if it is pre-injective (a
weaker condition than injectivity).
It was proved by Moore and Myhill in 1963 with G = Z^d the free abelian
group of rank d, and it was extended to all amenable groups
(Ceccherini-Silberstein, Machi, and Scarabotti 1999; Gromov 1999).
It was later shown that it fails to hold for any nonamaneble group
(Bartholdi, 2010), thus yielding a new characterization of amenability.
Following a suggestiion by Gromov, namely that the Garden of Eden theorem
could be extended to dynamical systems with a suitable hyperbolic flavor,
a Garden of Eden type theorem was proved for Anosov diffeomorphisms on
tori (Ceccherini-Silberstein and Coornaert, 2015) and for principal algebraic
dynamical systems satisfying a weak form of expansivity
(Ceccherini-Silberstein, Coornaert, and Li, 2018).
The talk will be completely self-contained.

Sep. 14 Martin Kreuzer
"Computing the Canonical Decomposition of a Finite Z-Algebra".
Abstract: In this joint work with Alexei Miasnikov and Florian Walsh,
we present an efficient method for computing the canonical decomposition
of a finite Z-algebra R into irreducible factors. Although the existence
of this decomposition has been known for along time, its equational
definability
is a more recent result. The first step is to compute the maximal ring
of scalars S(R). This is a finite commutative Z-algebra with an
explicitly computable
presentation. Then we use strong Groebner bases to calculate the fundamental
idempotents of S(R),. They allow us to compute the desired canoncial
decomposition in the final step. All steps of the procedure have a
polynomial
complexity, except for the strong Groebner basis calculation which
has a singly exponential bound.

Sep.28 Vladimir Shpilrain, Complexity in SL_2(Z) and SL_2(Q)
Abstract. We reflect on how to define complexity of a matrix and how to sample a random invertible matrix. We also discuss a related issue of complexity of algorithms in matrix groups, focusing on  computational complexity of the subgroup membership problem for some important special subgroups of SL_2(Z) and SL_2(Q).
The talk is based on joint work with Anastasiia Chorna, Katherine Geller, Lisa Bromberg, and Alina Vdovina.

Oct.5 Alexei Miasnikov
Malcev's Problems and First-Order Paradise in Groups

Oct.12 Ilya Kapovich (Hunter College, CUNY)
Title: Counting conjugacy classes of fully irreducibles: double exponential growth
Abstract: A 2011 result of Eskin and Mirzakhani shows that for a closed hyperbolic surface S of genus $g\ge 2$, the number $N(L)$ of closed Teichmuller geodesics of length $\le L$ in the moduli space of $S$ grows as $e^{hL}/(hL)$ where $h=6g-6$. The number $N(L)$ is also equal to the number of conjugacy classes of pseudo-Anosov elements $\phi\in MCG(S)$ with $\log\lambda(\phi)\le L$, where $\lambda(\phi)$ is the dilatation" or stretch factor" of $\phi$. We consider an analogous problem in the $Out(F_r)$ setting for the number  $N_r(L)$ of fully irreducible elements $\phi\in Out(F_r)$ with $\log\lambda(\phi)\le L$. We prove, for $r\ge 3$, that  $N_r(L)$ grows doubly exponentially in $L$ as $L\to\infty$, in terms of both lower and upper bounds. These bounds reveal behavior not present in classic hyperbolic dynamical systems. The talk is based on a joint paper with Catherine Pfaff.

Oct.19 Daniel Studenmund (U of Notre-Dame)
Title: Commensurability growth of nilpotent groups
Abstract: A classical area of study in geometric group theory is subgroup growth, which counts the number of subgroups of a given group Gamma as a function their index. We will study a richer function, the commensurability growth, associated to a subgroup Gamma in an ambient group G. The main results of this talk concern arithmetic subgroups Gamma of unipotent groups G, following subgroup growth results by Grunewald, Segal, and Smith. We start with the simplest example of the integers in the real line. This is joint work with Khalid Bou-Rabee.

Oct.26 Lam Pham (Yale University)
Title: On Uniform Kazhdan Constants for Finitely Generated Linear Groups.
Abstract: If $G$ is a finitely generated group and $(\pi,\mathcal{H})$ is a unitary representation of $G$ on a Hilbert space $\mathcal{H}$ without $G$-invariant vectors, it is of interest to know if  $\pi$ has a spectral gap; when all such representations $(\pi,\mathcal{H})$ have a spectral gap, $G$ is said to have Kazhdan's Property $(T)$. in general these spectral gaps depend on the choice of the generating set $S$ of $G$, and an important question is whether this dependence on $S$ can be removed. It is an open problem to determine if $\mathrm{SL}(3,\mathbb{Z})$ is uniform Kazhdan (i.e., the Kazhdan constant is independent of the choice of generators of $\mathrm{SL}(3,\mathbb{Z})$). In this talk, I will: (1) give an overview of the literature on explicit Kazhdan constants of finitely generated groups since the first explicit computation due to Burger (1991), and (2) present some new results on uniform spectral gaps for actions of the affine group over the integers.

Nov.2  Ben Fine's 70 conference

Nov.9  Catherine Pfaff (Queen's University at Kingston)

Title: Random automorphisms of free groups and what happens when you iterate them.

Abstract:
Two of the most natural and interesting questions one can ask about an automorphism group is what a random element of the automorphism group looks like and what happens as one repeatedly applies the automorphism to an element of the group (the asymptotic conjugacy class invariants). In the mapping class group circumstance, these questions (and their intersection) have been thoroughly studied with results dating back to Nielsen and Thurston, and then more recently with Dahmani, Horbez, Maher, Rivin, Sisto, Tiozzo, etc. While some is known in the outer automorphism group of the free group setting, little to nothing has been known about the most basic questions in the intersection of the main classes of questions, i.e. understanding the asymptotic conjugacy class invariants of random (outer) automorphisms of free groups. Together with Ilya Kapovich, Joseph Maher, and Samuel Taylor, we give a fairly detailed answer to this question.

Nov.16 Ben Steinberg

Nov.23  Thanksgiving

Nov.30

Dec.7  Henry Bradford,  (Georg-August Universität Göttingen).