Algebra Day on the Hudson

EVENT DETAILS

Saturday, December 7, 2024
9 a.m. – 5 p.m.
Gateway South 216 or via Zoom
For the complete agenda, please visit the Algebra on the Hudson's website.

ZOOM LINKS

Morning Session, 8:30 a.m. – 12:30 p.m.
https://stevens.zoom.us/j/97995694860

Afternoon Session, 1:30 p.m. – 5:30 p.m.
https://stevens.zoom.us/j/93410923708

MEET OUR SPEAKERS

Gil Goffer, Visiting Assistant Professor, University of California

Uncovering Group Laws via Random Walks
Abstract
The phenomenon of probability gaps suggests that if a property holds in a finite group with high probability, it often holds universally within the group. Extending this idea to an infinite group is challenging, in the absence of a natural probability measure. In the talk I’ll discuss probability gaps on infinite groups measured by random walks, and present recent results, joint with Greenfeld and Olshanskii, answering a few questions of Amir, Blachar, Gerasimova, and Kozma.

Thomas Koberda, Professor, University of Virginia

Locally Approximating Groups of Homeomorphisms
Abstract
I will survey the model theory of locally approximating groups of homeomorphisms of compact manifolds, which are groups of homeomorphisms which are ”sufficiently dense” in the full group of homeomorphisms, with the compact-open topology. These groups always interpret first-order arithmetic; using arithmetic, one can prove that all finitely generated subgroups of locally approximating groups are definable, with parameters. Under some further conditions, one can prove that these groups are prime models of their theories. I will also discuss action rigidity for these groups: if an arbitrary group G is elementarily equivalent to a locally approximating group of homeomorphisms of a compact manifold M, then for any locally approximating group action of G on a manifold N, we must have that M and N are homotopy equivalent to each other. In low dimensions, we may in fact conclude that M and N are homeomorphic to each other. This represents joint work with J. De la Nuez Gonzalez.

Daniel Studenmund, Assistant Professor, Binghamton University

Countable Unions of Finite Groups as Hidden Symmetries of the Free Group
Abstract
Symmetries of a group G are encoded in the automorphism group Aut(G). ”Hidden symmetries” are encoded in the abstract commensurator Comm(G). While many classes of finitely generated groups have reasonably well-understood commensurators — for example, when G is an arithmetic group, Comm(G) is typically a group of matrices with rational entries — the abstract commensurator of a free group, Comm(F2), is still somewhat mysterious. I will explain how Edgar A. Bering IV and I fleshed out a topological perspective of commensurations that allowed us to show that every countable locally finite group is a subgroup of Comm(F2).

Jennifer Taback, Isaac Henry Wing Professor of Mathematics, Bowdoin College

A Solution to the Conjugacy Problem for a Large Class of One-Relator Groups
Abstract
In this talk, I will introduce ongoing work (joint with Bob Gilman and Alden Walker) whose goal is to show that any one-relator group whose relator is a commutator is biautomatic and hence has a solvable conjugacy problem. This would answer a question of Baumslag asking whether groups of this form are necessarily automatic. Our arguments rely on a method of proving a group is biautomatic introduced by Gilman which is based on ideas from small cancellation theory. In applying Gilman’s techniques, we take advantage of the symmetry in the commutator relator. I will outline our results to date and provide an overview of the remaining steps in the proof of our theorem.

Filippo Calderoni, Assistant Professor, Rutgers University

Groups, Orders, and Classification
Abstract
In this talk, we will survey some recent results about left-orderable groups and their interplay with descriptive set theory. In particular, we use Borel classification theory to describe the complexity of spaces of left-orderings modulo the conjugacy action. Then we will discuss how our results are related to a central open problem in geometric topology, known as the L-space conjecture. Most of the results presented here are joint work with Adam Clay.

Alexander Grishkov, Associate Professor, University of Sao Paulo

An Algebraic Analog of the Exponential Map for Algebraic Groups
Abstract
In 1955 A. Malcev introduced the notion of analytic diassociative loops and its connection with binary-Lie algebras, using classical exponential map. Later, in 1963 Yu. Manin noted that cubic form admits a structure of an algebraic diassociative loop. Recall, that a loop is a diassociative if every two elements generate a subgroup. In this talk, we discuss the last progress in the theory of analytic and algebraic diassociative loops. In particular, we study the question of the existence of an algebraic exponential map between Lie algebra and correspondent algebraic group.

Koichi Oyakawa, PhD Student, Vanderbilt University

Classifying Group Actions on Hyperbolic Spaces
Abstract
Studying groups via their actions on Gromov hyperbolic spaces has been a recurrent theme in geometric group theory over the past three decades. Of particular interest in this approach are actions of general type, i.e., non-elementary actions without fixed points at infinity. For a given group G, it is natural to ask whether it is possible to classify all general-type actions of G on hyperbolic spaces. In a joint paper with D. Osin, we propose a formalization of this question based on the complexity theory of Borel equivalence relations. Our main result is the following dichotomy: for every countable group G, general type actions of G on hyperbolic spaces can either be classified by an explicit invariant ranging in the infinite-dimensional projective space or are unclassifiable by countable structures. Special linear groups over countable fields provide examples satisfying the former alternative, while every non-elementary hyperbolic group satisfies the latter.

SPONSORS:

National Science Foundation

Department of Mathematics at Stevens

Algebra and Cryptology Center at Stevens