Spring 2017

All the talks are on Fridays, 4:00pm-5:00pm in room 5417 at the CUNY Graduate Center. 
Wine and cheese are available after the seminar in the Graduate Center math lounge, room 4214.

Feb 3. Alexei Miasnikov (Stevens Institute); Title:  Homogeneous and isotypic groups and algebras
The type of a tuple of elements of a group (or a ring) G  is the set of all formulas of the group (ring) language that hold on the tuple.  For example, elements in an algebraically closed field are either algebraic (in which case the type is determined by the minimal polynomial) or transcendental (the type is the correspondent set of all polynomial inequalities). Group elements may have much more interesting and complex types.   We say that G is defined by its types if any group (ring) isotypic to G, i.e., having the same types of elements as in G, is isomorphic to G.   I will discuss finitely generated groups defined by their types – there are a lot of such groups!  
A group (ring) G is homogeneous if any two isotypic tuples of elements of G are conjugated by an automorphism of G.  Homogeneous structures are well studied in model theory, usually they arise as completions or limits in some classes (fields of complex or real numbers, or Fraisse limits, or saturated structures).  My focus will be on homogeneous finitely generated groups and their algebras.  Finitely generated homogeneous groups (rings) are defined by their types, but the converse does not hold even for some hyperbolic groups.
The original motivation of this work goes back to model theory and universal algebraic geometry, but recent developments show that the topic is interesting in its own right.   The talk is based on joint results with Olga Kharlampovich and Nikolay Romanovskii.

Feb 10. Jason Behrstock (CUNY);  TitleAsymptotic dimension of mapping class groups
The goal of this talk will be to describe our recent result proving that the asymptotic dimension of the mapping class group of a closed surface is at most quadratic in the genus (building on and strengthening a prior result of  Bestvina-Bromberg giving an exponential estimate). We obtain this result as a special case of a result about the asymptotic dimension of a general class of spaces, which we call hierarchically hyperbolic; this class includes  hyperbolic spaces, mapping class groups, Teichmueller spaces endowed with either the Teichmuller or the Weil-Petersson metric, fundamental groups of non-geometric 3-manifolds, RAAGs, etc. We will discuss the general framework and a sketch of how this machinery provides new tools for studying special subclasses, such as mapping class groups. The results discussed are joint work with Mark Hagen and Alessandro Sisto.


Feb 24. Ilya Kapovich (UIUC), Title:  Dynamics on free-by-cyclic groups
We develop a counterpart of the Thurston-Fried-McMullen ``fibered face'' theory in the setting of free-by-cyclic groups, that is, mapping tori groups of automorphisms of finite rank free groups.  We obtain information about the BNS invariant of such groups, and construct a version of McMullen's ``Teichmuller polynomial'' in the free-by-cyclic context. The talk is based on joint work with Chris Leininger and Spencer Dowdall.

March 3. Harald Helfgott  University of Göttingen and CNRS, Title: The Diameter of the Symmetric Group.   
Given a finite group $G$ and a set $A$ of generators, the diameter $\diam(\Gamma(G,A))$ of the Cayley graph $\Gamma(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding $\diam(G):= \max_A\diam(\Gamma(G,A))$. It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$. In 2011, Helfgott and Seress gave a quasipolynomial bound (exp((log n)^(4+epsilon))). We will discuss a recent, much simplified version of the proof.

March 10. No Seminar;   51 Spring Topology and Dynamical Systems conference, Jersey City, March 8-11.

March 17. No Seminar; Spring break at the Stevens Institute

March 24. Paul Schupp, University of Illinois at Urbana-Champaign, Title: How the asymptotic-generic point of view of geometric group theory has invaded the theory of computability
  ``Worst-case'' measures of complexity may not give a good overall
picture of a particular algorithm or problem.  The classic example
is Dantzig's Simplex Algorithm for linear programming which runs 
thousands of time every day, always very quickly, although its
worst-case complexity is exponential.  The point is that hard
instances are very sparse.
The idea of ``generic-case'' complexity arose in geometric group
theory where simple ``quick check'' algorithm for classical problems
often work ``most of the time''.  In recent years asymptotic density,
generic computability and coarse computability have had a large effect
on the theory of computabilty.  I will explain a few examples of this.
This is a general talk for group theorists and  knowledge of computabilty
theory is not assumed.

March 31. Chris Leininger (UIUC) Word-hyperbolic surface bundles
The work of Farb-Mosher and Hamenstadt provides a necessary and sufficient condition for the fundamental group of a closed surface bundle over any compact space to be word-hyperbolic.  The condition is geometric in nature, involving the monodromy homomorphism and the action on Teichmuller space.  Gromov's hyperbolization question, in the special case of surface bundles, asks whether the condition on the action can be relaxed to an algebraic one.  In this talk I will discuss this problem, and some joint work with Bestvina, Bromberg, and Kent providing results in this direction

April 7. Robert Young (NYU) Embeddings of the Heisenberg group and the Sparsest Cut problem
 (joint work with Assaf Naor) The Heisenberg group $\mathbb{H}$ is a sub-Riemannian manifold that is hard to embed in $\mathbb{R}^n$.  Cheeger and Kleiner introduced a new notion of differentiation that they used to show that it does not embed nicely into $L_1$.  This notion is based on surfaces in $\mathbb{H}$, and in this talk, we will describe new techniques that let us quantify the "roughness" of such surfaces, find sharp bounds on the distortion of embeddings of $\mathbb{H}$, and estimate the accuracy of an approximate algorithm for the Sparsest Cut Problem.

April 14. Spring break at CUNY

April 21. Khalid Bou-Rabee (CCNY) The Primitive Burnside Problem
Let P(a,k) be the subgroup of the rank a free group generated by kth powers of primitive elements. We show that P(2,k) is finite index if and only if k=1 or 2 or 3. We frame this as a solution to the Primitive Burnside Problem and discuss applications to the Bounded Burnside Problem. This covers joint work with Patrick W. Hooper. 

April 28. Bob  Gilman (Stevens) Groups with multiple context free word problem.
One way of looking at the word problem of a finitely generated group is as the formal language of all words representing the identity. A seminal result in this area, proved by  David Muller and Paul Schupp, is that this language is context free if and only if the group is virtually free.  This talk concerns groups whose word problems are in the larger class of multiple context free languages. Work of Turbo Ho, Rob Kropholler,  Saul Schleimer and the speaker will be presented.

May 5. Phillip Wesolek  Elementary amenable groups, descriptive set theory, and the space of marked groups
(Joint with J. Williams)  The class of elementary amenable groups is the smallest class that contains the abelian groups and the finite groups and that is closed under group extension, taking subgroups, taking quotients, and taking directed unions. This is a natural subclass of the amenable groups which contain all ``obviously" amenable groups. In this talk, we first give a characterization of elementary amenable groups in terms of well-founded descriptive-set-theoretic trees. From this characterization, we extract  a chain condition which characterizes elementary amenability. Using Grigorchuk's space of marked groups and tools from descriptive set theory, we then show that the set of elementary amenable marked groups is not in the Borel sigma algebra of the space of marked groups. This yields a new, non-constructive proof of a theorem of Grigorchuk: There are finitely generated amenable groups that are not elementary amenable.

May 12.  E. Rips  11:00 AM - 12:00 PM, Room 3209
Title: "Towards a group-like small cancellation theory for rings", a joint work with Agata Atkarskaya, Alexei Kanel-Belov and Eugene Plotkin