Tag - Group theory

Fabienne Chouraqui: Connections between the Yang-Baxter equation and Thompson’s group F

22nd June, 2023

The quantum Yang-Baxter equation is an equation in mathematical physics and it lies in the foundation of the theory of quantum groups. One of the fundamental problems is to find all the solutions of this equation. Drinfeld suggested the study of a particular class of solutions, derived from the so-called set-theoretic solutions. A set-theoretic solution of the Yang-Baxter equation is a pair (X,r), where X is a set and

r : X ⨯ X → X ⨯ X     r(x,y)=(σx(y),γy(x))

is a bijective map satisfying r12r23r12 = r23r12r23, where r12 = r ⨯ IdX and r23 = IdX ⨯ r. We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang-Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and the Thompson's group F. This raises the question of whether there are further connections between partial solutions and Thompson's groups in general.

Vladimir Vankov: Bestvina-Brady groups and generalizations

5th June, 2023

Right-angled Artin groups are perhaps the most ubiquitous manifestations of polyhedral products in geometric group theory and low-dimensional topology. The theory of their subgroups has been of great importance in the last couple of decades. This is especially true with regards to what are known as 'finiteness properties' - meaningful criteria for measuring ways in which infinite groups may behave like finite ones - as well as the theory of three-dimensional manifolds. We will visit some celebrated theorems and, if time allows, discuss problems arising from deck transformations of branched covering maps.

Yifan Jing: Measure Doubling for Small Sets in SO3(ℝ)

30th May, 2023

Let SO3(ℝ) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure μ. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO3(ℝ) with sufficiently small measure, then μ(A2) > 3.99 μ(A).

Michael Wibmer: Expansive endomorphisms of profinite groups

25th May, 2023

Étale algebraic groups over a field k are equivalent to finite groups with a continuous action of the absolute Galois group of k. The difference version of this well-known result asserts that étale difference algebraic groups over a difference field k (i.e., a field equipped with an endomorphism) are equivalent to profinite groups equipped with an expansive endomorphism and a certain compatible difference Galois action. In any case, understanding the structure of expansive endomorphisms of profinite groups seems a worthwhile endeavour and that's what this talk is about.

Michael Wibmer: Difference algebraic groups

25th May, 2023

Difference algebraic groups are a generalization of algebraic groups. Instead of just algebraic equations, one allows difference algebraic equations as the defining equations. Here one can think of a difference equation as a discrete version of a differential equation. Besides their intrinsic beauty, one of the main motivations for studying difference algebraic groups is that they occur as Galois groups in certain Galois theories. This talk will be an introduction to difference algebraic groups.

Richard Ng: Discovering modular data from congruence representations

14th May, 2023

Modular fusion categories (MFCs) arise naturally in many areas of mathematics and physics. Associated with an MFC is a pair of complex matrices, called modular data, which are arguably the most important invariants of an MFC. The modular data of an MFC generate some uncanonical congruence representations of SL2(ℤ). In this talk, we will discuss how modular data could be reconstructed or discovered from congruence representations of SL2(ℤ). The talk is based on a joint work with Eric Rowell, Zhenghan Wang and Xiao-Gang Wen.