Fall2015

Sep 4 Alexander Treyer (Omsk State University) "Canonical and existentially closed groups for universal classes of abelian groups"

Abstract. The talk is based on joint work with A. Mishenko and V.Remeslennikov and devoted to universal classes of abelian groups. In our work we classify universal classes of abelian groups in terms of f.g. groups closed under discriminating operator. Also we introduce the principal universal classes of abelian groups and canonical groups for them. For arbitrary universal class K we describe the class of existentially closed groups relatively universal theory of class K and show that this class is axiomatizable.

Oct 2 Bob Gilman (Stevens Institute) Title: How to Sample Hard Instances of the Word problem.

Abstract: It is well known that there are finitely presented groups with unsolvable word problem. Any (partial) algorithm for the word problem of such a group must fail on infinitely many instances. Nevertheless it can be hard to find these instances. In this talk we exhibit a finitely presented group whose hard instances can be sampled in linear time.

Oct 9 Lisa Carbone (Rutgers U.)

Title: Finite presentations of hyperbolic Kac-Moody groups

Abstract: Tits defined Kac--Moody groups over commutative rings, providing

infinite dimensional analogues of the Chevalley-Demazure group schemes. Tits' presentation can be simplified considerably when the Dynkin diagram is hyperbolic and simply laced. Over finitely generated rings R, we give finitely many generators and defining relations parametrized over R and we describe a further simplification for R=Z. We highlight the role of the group E10(R), conjectured to play a role in the unification of superstring theories.

Oct 16 I. Lysenok (Stevens and Steklov Institute) Towards optimization of the Novikov-Adian exponent

Abstract: The Novikov-Adian theorem states that a non-cyclic Burnside group B(m,n) of

odd exponent n greater or equal 665 is infinite. I will discuss ideas behind

different known proofs of infiniteness of groups B(m,n) as well as possibility

of a substantional reduction of the lower bound 665.

Oct 23 Jason Behrstock (Lehman College, CUNY)

Title: Random graphs and applications to Coxeter groups.

Abstract: Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then, explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.

Oct 30 Alex Gamburd (CUNY Graduate Center) Markoff Triples and Strong Approximation.

Abstract: Markoff triples are integer solutions to Markoff equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. We will review some of these (in particular connection to Product Replacement Graphs) and will then discuss recent joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo $p$ under the action of the group generated by Vieta involutions. We show in particular that for almost all prime the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.

Nov 6 Aner Shalev (Hebrew University of Jerusalem)

Words and generarion in linear groups

Abstract: We will discuss two recent results showing that finitely generated

linear groups with certain properties are virtually solvable.

The first one (joint with Kantor and Lubotzky) concerns invariable generation by a finite subset.

The second one (joint with Larsen) concerns probabilistic identities.

We will deduce the following probabilistic Tits alternative:

a finitely generated linear group is either virtually solvable

or almost all n-tuples in its profinite completion generate a

free subgroup of rank n.

Related open problems will also be discussed.

Nov 13 Ruth Charney (Brandeis University) Random Graph Products of Cyclic Groups

Abstract: For a finite graph $\Gamma$, the right-angled Artin group $A_\Gamma$ is the group generated by the vertices of $\Gamma$ with relations given by commutators of adjacent vertices. More generally, if we also allow (all or some) of the generators to have finite order, we get a collection of groups $G_\Gamma$ known as graph products of cyclic groups. In random group theory, one asks what the probability is that a randomly chosen group will satisfy a given property. While the notion of a ``random group" usually involves fixing a generating set and randomly choosing relators, in the case of graph products, the natural model to consider is a group $G_\Gamma$ associated to a random graph in the sense of Erdos-Renyi. In this talk I will review the basics of random graph theory and survey applications to graph products of cyclic groups and their automorphism groups, including my joint work with M. Farber, as well as subsequent work by various other authors.

Nov 20 (Canceled) Benson Farb (U. of Chicago) Title: Some problems about mapping class groups.

Abstract: The mapping class group Mod(S_g) of a genus g surface is the group of homotopy classes of homeomorphisms of S_g. It plays a central role in low-dimensional topology, combinatorial and geometric group theory, and algebraic geometry. In this talk I'll explain some open problems about mapping class groups that I find to be interesting and important.

Dec 4 Manhattan Algebra Day http://web.stevens.edu/algebraic/MADAY2015/