Past Meetings: 2023-24

Speaker: Laura Ciobanu

Title: Solving equations in groups (but not teams!)

Abstract: For a group or semigroup or ring G, solving equations where the coefficients are elements in G and the solutions take values in G can be seen as akin to solving systems of linear equations in linear algebra, Diophantine equations in number theory, or more generally polynomial systems in algebraic geometry.
I will give a short, nontechnical, survey about solving equations in infinite nonabelian (semi)groups, with emphasis on the free and hyperbolic ones. In particular, I will explain that the solutions to equations can be beautifully described in terms of words and languages, and that this was made possible by results coming from both geometry and computer science. No background in computer science is required, only some basic group theory.

Speaker: Isambard Goodbody

Title: Global dimension and strong generation

Abstract: The global dimension of a ring is a measure of its complexity and in the commutative case it had a geometric interpretation relating to smoothness. It is closely related to another invariant: the Roquier dimension of the ring's derived category. This viewpoint can be used to study the behaviour of global dimension under direct limits of rings. I'll mention some fun examples which demonstrate that the relationship between these two notions can breakdown even in some relatively mild situations.

Speaker: Luke Naylor

Title: Conformal mapping of Bridgeland stabilities

Abstract: When transitioning the theory of stability condition to triangulated categories, one gets a natural action by auto equivalences of the category. In the case of smooth projective Picard rank 1 surfaces, this action can be linked to Mobius transforms allowing a nice picturesque explanation of some results in the area.

Speaker: Scott Warrander

Title: Coherent-Constructible Relationships in Geometric Representation Theory  

Abstract: Something geometric representation theorists like to do is find and study geometric constructions of interesting algebras and representations, i.e. finding some variety which gives us the desired algebra in its (co)homology, or as the Grothendieck group of some associated sheaf category. This talk will be about the sheaf theoretic approach, of which there are two kinds: firstly we have constructible (complexes of) sheaves, which are sheaves of modules in either the etale or analytic topology; and secondly we have coherent sheaves, which of course only see the Zariski topology. In this talk I will discuss some examples of relationships between constructible and coherent constructions, and how phenomena on one side can lead to interesting research on the other in both directions.

Speaker: Simone Castellan 

Title: Isomorphisms of deformations and quantizations of Kleinian singularities

Abstract: Given a non-commutative algebra Q and its semiclassical limit A, an intriguing question has always been “Do the properties of Q always reflect the (Poisson) properties of A?”. Of particular interest is the behaviour of automorphisms. The most famous example is the Belov Kanel-Kontsevich Conjecture, which predicts that the group of automorphisms of the nth-Weyl algebra A_n is isomorphic to the group of Poisson automorphisms of the polynomial algebra C[x_1,…,x_2n]. In this talk, I will present my work on a problem similar to the BKK Conjecture. Take a symplectic quotient singularity; the parameter spaces of filtered deformations and the parameter space of filtered quantizations coincide. Do the Poisson isomorphisms between the deformations coincide with the automorphisms between the quantizations? The answer is affirmative in the case of Kleinian singularities of type A and D.

Subrabalan Murugesan- Topological representations of the quantum group Uq(sl2).

Abstract: A certain class of 2d quantum field theories known as conformal field theories are interesting to physics because they are more often than not physical theories that are exactly solvable. If you know any physics, then you would know that this is very very rare. At the same time, it has also captured the imagination of mathematicians thanks to its relation to Teichmuller theory, isomonodromic deformations, quantum groups, vertex algebras, so on and so forth. In this talk, I will talk about the connection between 2d CFTs and quantum groups. In particular, I will restrict myself to showing that there is a one-one correspondence between certain CFT operators and a highest weight representation of the quantum group $U_q(\mathfrak{sl}_2)$.

Riccardo Giannini- Artin groups and translation surfaces.

Abstract: In this talk, we are going to talk about geometric homomorphisms. These homomorphisms map small-type Artin groups, generated by elements that either commute or share a braid-like relation, to the mapping class group of a given finite-type surface. In some cases, geometric homomorphisms are known to be injective. However, in most of the cases the kernels are huge and we do not know how to describe them. We will see how to use geometric homomorphisms to shed some lights on a conjecture due to Kontsevich-Zorich. The conjecture is related to the moduli spaces of translation surfaces, which are closed Riemann surfaces with a flat metric on the complement of some finite number of points.

Tuan Anh Pham- Local representations of the Witt algebra.

Abstract: The orbit method is a fundamental tool to study a finite dimensional solvable Lie algebra g.  It relates the annihilators of simple U(g)-module to the coadjoint orbits of the adjoint group on g^* . In my talk, I will extend this story to the Witt algebra – a simple (non-solvable) infinite dimensional Lie algebra which is important in physics and representation theory.  I will construct an induced module from an element of W^*, show that it is simple, and establish some nice properties of the category of these induced modules under tensor product. I will also construct an algebra homomorphism that allows one to relate the orbit method for W to that of a finite dimensional solvable algebra.

Philipp Bader- Representations of mapping class groups of surfaces.

Abstract: Let S be a surface and Mod(S) its mapping class group. A big open question in the theory of low dimensional topology is whether Mod(S) is a linear group. It is therefore natural to study representations of mapping class groups. A theorem by Dehn-Nielsen-Baer relates Mod(S) to the automorphism group of the fundamental group of S.

In this talk, we present a method to construct representations of automorphism groups of fundamental groups and focus on the case of the fundamental group of a surface. The definition of these representations is purely algebraic, there is however a close relation to covering space theory which we briefly explore. We end by presenting some known results about the constructed representations.

Samuel Lewis- Classifying hyperbolic intersection arrangements.

Abstract: Choose your favourite connected graph Δ and shade a subset J of its vertices. The intersection arrangement associated to the data (Δ, J) is a collection of hyperplanes in dimension |Jc|, first defined by Iyama and Weymss.  This construction involves taking the classical Coxeter arrangement of Δ and then setting all variables indexed by J to be zero. It turns out that for many choices of J the chambers (connected components) of the intersection arrangement admit a nice combinatorial description, and there exists a wall-crossing rule to pass between them. I will start by making all of this more precise, before discussing work in progress to classify intersection arrangements of “hyperbolic” graphs. If time permits I will also talk about some of the geometric motivations for intersection arrangements and their applications to contracted preprojective algebras.

Yan Yau Cheng- Arithmetic Chern Simons Theory

Abstract: Mazur first observed in the 60s a deep analogy between the embedding of a knot in a 3-manifold and primes in a number field. Using ideas from this analogy, Minhyong Kim and his collaborators developed the study of arithmetic field theories. This talk will be an introduction to Arithmetic Field Theories, in particular focusing on Arithmetic Chern-Simons Theory.

Robin Ammon- Arithmetic Statistics and the Cohen--Lenstra principle.

Arithmetic Statistics is an area of number theory that deals with statistical questions about various objects of number-theoretic relevance, ranging from century old questions about the distribution of prime numbers to more modern problems about number fields or elliptic curves, proofs of which often have far-reaching consequences. There has been a recent surge in work in arithmetic statistics following a seminal paper by H. Cohen and H. Lenstra, in which they observe that the previously mysterious seeming behaviour of ideal class groups of number fields appears to be governed by a fundamental principle about the distribution of random mathematical objects. Their paper demonstrated the power of a statistical approach in number theory and gave a new perspective on how to understand mathematical objects, and afterwards many other occurrences of their principle have been found, in number theory and beyond.

In my talk, I will give an introduction to arithmetic statistics and explain Cohen and Lenstra's principle with many examples. I will discuss their conjecture about the distribution of ideal class groups and briefly talk about my own work, which seeks to generalise the conjecture to ray class groups. I will not assume any knowledge in number theory!

Oli Jones- Fixed Points of Automorphisms of Torus Knot Groups.

Given an automorphism f of a group G, the set of fixed points Fix(f)= {g | f(g) = g} form a subgroup. Fixed point subgroups are a topic of long-term interest in geometric group theory, dating back to the Scott conjecture for free groups in the 1970s. In this talk we will explore how we can use an action of Aut(G) on a space X which extends an action of G on X to understand fixed points of automorphisms. We will illustrate this by framework by studying fixed points of the Torus Knot Groups <x,y | x^n = y^m>, via their actions on trees.

David Cueto Noval - Cluster Structures, Integrable Systems and Gauge Theories.

This talk is meant to be an introduction to the use of cluster structures in the context of integrable systems and gauge theories. We will start by reviewing the notion of cluster variety. We will focus on the relativistic Toda chain and its connections to 5d N =1 SU(N) gauge theory with no matter multiplets.