Research

Geometry/Topology
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Monday, November 27, 2023

Title: Geometric Group Theory for Pedestrians
Speaker: Milé Krajčevski
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

I will review a few dominant (in my view) themes in Geometric Group Theory and my personal journey in their discovery.

Monday, November 20, 2023

Title: Singular limits in representation theory
Speaker: Razvan Teodorescu
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

Resolving the singular limit of \(\widehat{sl}(2)_{k \to -2}\) is a subject of wide interest in representation theory, knot invariants, and integrable system alike, due to its many implications, both theoretical and applied. We consider the special case of a sequence \(k_m+2=\frac{4}{8m+1}\to 0\) so that the level \(k_m\) is admissible in the sense of Kac and Kazhdan and realize the corresponding representations of the braid group through the homology of a hyperelliptic curve in \(\mathbb{\widehat{C}}^2\). The associated Hecke algebra is a \(q\)-deformation of the symmetric group with fixed \(q=e^{i\pi/4}\), which allows to identify the \(m\to\infty\) limit as a Dubrovin system of nonlinear differential equations, leading to explicit solutions in terms of hyperelliptic \(\theta\)-functions.

Monday, November 13, 2023

Title: Synergodic actions of product groups
Speaker: Peter Burton, University of Texas at Austin
Time: 2:00pm–3:00pm
Place: Online

Abstract

We will discuss a structural aspect of measure-preserving actions of products of countable discrete groups, involving a so-called ‘synergodic decomposition’ in terms of the ergodic components of the actions of the two factor groups. We show that this construction provides a canonical way to detect whether the action is built as a product of actions on independent measure spaces, and we use it to prove a result about convergence of ergodic averages on product groups which are highly imbalanced between the factors. Defining an action to be synergodic if it is isomorphic to its synergodic decomposition, we show that if a countable group \(G\) is amenable then every action of \(G \times G\) can be approximated by synergodic actions and that this statement fails if \(G\) is a nonabelian free group. The last result relies on the refutation of Connes’ embedding conjecture.

Monday, November 6, 2023

Title: Liftable self-similar groups and scale groups
Speaker: Dima Savchuk
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the \((d+1)\)-regular tree \(T_{d+1}\). Then we introduce the class of liftable self-similar groups acting on a \(d\)-regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on \(T_{d+1}\) fixing one of the ends. We show that their closures in \(\mathrm{Aut}(T_{d+1})\) are totally disconnected locally compact (TDLC) group that belongs to the class of scale groups. We give multiple examples of liftable groups coming from various sources and expose their connections to the classes of scale invariant groups, groups admitting finite \(L\)-presentations, amenable but not elementary or subexponentially amenable groups, groups of intermediate growth, etc. In particular, we show that the finitely presented group constructed by Grigorchuk embeds into the group of dilations of the field 2-adic numbers, and that its closure fixes an end of \(T_3\) and is a scale group that acts 2-transitively on the punctured boundary of \(T_3\). This is a joint work with Rostislav Grigorchuk./p>

Monday, October 30, 2023

Title: Free products of bi-ordered groups
Speaker: Kyrylo Muliarchyk
Time: 2:00pm–3:00pm
Place: Online

Abstract

Rivas demonstrated that there is no isolated left-ordering on a free product of left-orderable groups. We modify his technique to show that no bi-ordering on a free product of bi-orderable groups is isolated. In particular, we prove that the bi-orderings space of a free product of countable groups is homeomorphic to the Cantor set. Furthermore, we demonstrate that the natural action of the automorphism group of a free group on the bi-orderings space of that group does not have dense orbits. This contrasts with Clay’s discovery that the action of the inner automorphism group on the space of left-orderings has a dense orbit.

Monday, October 23, 2023

Title: A Geometric Tour of Grushin-type Spaces
Speaker: Zack Forrest
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

In recent years, Grushin-type spaces have attracted considerable attention due to their interesting geometry, which results from the lack of an underlying algebraic group law.

In this talk, we will explore the geometric properties of Grushin-type spaces and discuss calculus and PDEs on these spaces.

Monday, October 16, 2023

Title: Multivariate growth and cogrowth and related topics
Speaker: Rostislav Grigorchuk
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

I will begin with recalling a basic facts about cogrowth, i.e., a relative growth of a subgroup in a free group introduced by the speaker in the mid-`70s. This will include the cogrowth criterion of amenability, the relation to the random walk on a group, the rationality of cogrowth series in the case of finitely generated subgroups, etc. Then I will mention some results about Hopf decomposition for the boundary action obtained in a joint work with V. Kaimanovich and T. Nagnibeda (2011).

After that I will switch to the multivariate version of the growth and cogrowth paying the most attention to the case of regular languages. It will be explained how convexity theory and large deviation theory are the right tools to compute the defined objects. The result of computation will be shown for the language of freely reduced words and Fibonacci language.

If time permits I will finish with more advanced results concerning a fine multivariate asymptotic. The latter part of the talk is based on the recent joint results of the speaker with J-F. Quint and A. Saikh.

Monday, October 9, 2023

Title: Hyperlinear approximations to amenable groups are induced by sofic approximations
Speaker: Maksym Chaudkhari
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

A discrete group is called sofic or hyperlinear if it admits metric approximations by finite permutation groups or by unitary groups of finite dimensional Hilbert spaces, respectively. It is well-known that every sofic group is also hyperlinear, and moreover, the class of sofic groups includes all amenable and all residually finite groups. In this talk we will describe the relationship between sofic and hyperlinear approximations for torsion-free amenable groups. The talk is based on a joint work with Peter Burton, Kate Juschenko and Kyrylo Muliarchyk.

Monday, October 2, 2023

Title: Instanton Floer homology and surface knot detection
Speaker: Zhenkun Li
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

Given a knot \(K\) inside a thickened surface \([0,1]*S\), a natural question to ask is how can we determine whether the knot \(K\) can be isotopic onto the surface \(S\) or not? In this talk I will introduce the Asaeda-Przytycki-Sikora homology, which generalizes the Khovanov homology for knots in \(S^3\), and which can be combinatorially defined for links inside thickened surfaces. I will show that if \(S\) has genus zero, then a knot can be isotoped onto the surface if and only if its APS homology has rank two. This is a joint work with Yi Xie and Boyu Zhang.

Monday, September 25, 2023

Title: Instanton Floer homology and Dehn surgery
Speaker: Zhenkun Li
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

Instanton Floer homology is introduced by Floer in 1980s. It is a powerful invariant for 3-manifolds and knots and links inside them. In this talk, I will present a surgery formula for instanton theory, which describes the instanton Floer homology of 3-manifolds coming from Dehn surgeries along knots. As applications I will discuss how this technique can be applied in establishing the existence of irreducible \(SU(2)\) representations for the fundamental group of 3-surgery of any non-trivial knots in \(S^3\), answering a question proposed by Kronheimer and Mrowka back to 2004. This is a joint work with John Baldwin, Steven Sivek, and Fan Ye.

Monday, September 18, 2023

Title: Residual finiteness and orderability of link quandles
Speaker: Manpreet Singh
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

In the talk, I will introduce the residual finiteness of quandles and prove that all link quandles are residually finite. Using the preceding result, we will see that the word problem is solvable for link quandles. I will talk about the orderability properties of link quandles. Since all link groups are left-orderable, it is reasonable to speculate that link quandles are left (right)-orderable. In contrast, we will see that the orderability of link quandles behaves quite differently than that of the corresponding link groups.

Monday, September 11, 2023

Title: Low dimensional topology and orderability of groups and quandles
Speaker: Mohamed Elhamdadi
Time: 2:00pm–3:00pm
Place: NES 102

Abstract

In recent decades, there has been a remarkable interplay between the theory of orderable groups and topology (orderability of knot groups and fundamental groups of 3-manifolds). We will review left orderability and bi-orderability of groups and quandles. Some applications to knot groups and fundamental groups of 3-manifolds will be discussed. The talk will be self-contained.