Algebra

Abstract - An associahedron is a polytope arising from combinatorics of Catalan-type objects (for example, from a collection of all triangulations of a given polygon). Fomin and Zelevinsky found a way to construct the same combinatorial structure from considering the Coxeter group of type A_n. This allowed them to define a generalized associahedron for every finite reflection group. For generalized associahedra arising from crystallographic reflection groups, it was also shown that they can be realized as polytopes. We use the folding technique to construct polytopal realisations of generalized associahedra for all non-simply-laced root systems, including non-crystallographic ones. This is a joint work with Pavel Tumarkin and Emine Yildirim.

Abstract - The Graph Reconstruction Conjecture is a long-standing problem in Graph Theory formulated by Kelly (1957) and Ulam (1960). The conjecture states that every graph with at least three vertices can be uniquely reconstructed (up to isomorphism) from their deck one-vertex deleted subgraphs. It is well known that the graph isomorphism problem can be worded using Invariant Theory, although this is not particularly interesting in practice as the computations get quickly out of hand. In this talk, we explore how invariant theory can be used to approach the Graph Reconstruction conjecture. This naturally brings the focus to K-weighted graphs. We focus on the attempt by Thiéry (2000), which led to a disproof of a stronger statement using a computational argument. This also turns out not to be particularly practical, but we'll see how it still brings valuable insight. This talk is based on a survey paper joint with Gabriela Jerónimo, Jenny Kenkel, Haydee Lindo and Nelly Villamizar.

Abstract - Dehn’s famous decision problems for finitely presented groups have been studied for over a century by combinatorial and geometric group theorists. In recent years, a variant of one of these classical problems, namely the twisted conjugacy problem, has been studied. The motivation for this problem comes from Bogopolski, Martino and Ventura who, in 2009, proved an equivalence between conjugacy in group extensions and twisted conjugacy.
In this talk I will give a brief survey of this lesser-known decision problem, and discuss some of the latest results in this area. This includes a framework which can be applied to dihedral Artin groups.

Abstract: With (almost) every finite mutation class of quivers we associate two groups: an extended affine Weyl group of type A or D, and a certain quotient of a Coxeter group which behaves nicely with respect to mutations (in most cases, the latter can also be obtained as a quotient of certain surface braid group). I will discuss a connection between these groups and a (still conjectural) characterization of mutation-finite quivers in terms of positive semidefinite symmetric matrices. The talk is based on joint works (some still in progress) with Anna Felikson, John Lawson and Michael Shapiro.

Classical knot theory associates to a combinatorial knot diagram a knot embedding in the 3-sphere. The fundamental group (and peripheral system) of the complement give rise to powerful, combinatorially computable invariants. This talk explores an analogous construction for welded knots, a diagrammatic extension of classical knot theory corresponding to ribbon knotted tori in 4-space via the tube map.

We introduce a new topological invariant, the fundamental \pi-module (built using the first and second homotopy groups), and show how it can be computed combinatorially from a diagram. We then define a natural topological generalisation of the peripheral system, using the free loop space of the complement, and again show how this admits a simple combinatorial description.

Cluster ensembles were introduced by Fock and Goncharov to encode seed data for cluster algebras in terms of certain lattices, enabling the definition of cluster varieties by gluing the associated tori. We will explain aspects of two pieces of joint work, with Gratz and Pressland respectively, in which we show that categorifications of cluster algebras yield cluster ensembles, that these categorifications are often quantizable and that there is also a category governing interactions between cluster ensembles. Furthermore, these structures can help us understand mirror symmetry for cluster varieties.

Abstract: In this talk, I will review the basic notions of Fomin-Kirillov algebras and discuss some open problems. I will explain how these algebras arise in combinatorial Schubert calculus and in the theory of Hopf algebras. Finally, I will present some crazy numerology that seems to connect Fomin-Kirillov algebras with various topics in representation theory.

Abstract - The main aim of affinization (in the sense discussed here) is to formulate and analyse (additive or linear) algebraic structures in a way in which no selection of a specific element is necessary. In this talk I will explain the main motivation and various aspects of affinization and illustrate the process by affinization of familiar algebraic structures such as groups, vector spaces, and associative and Lie algebras.

Abstract:

Module categories have two types of generators: projective modules and simple modules. Abstractions of projective modules have led to (tau-)tilting theory and cluster-tilting theory. Cluster-tilting theory is well suited to positive Calabi-Yau categories. However, there are natural examples of negative Calabi-Yau categories, such as the stable module category of a symmetric algebra. Projective generators are less useful in this context because they are killed by stabilisation, making understanding the simple-like generators an important question. 

In this talk I will explain how simple-minded systems are negative Calabi-Yau analogues of cluster-tilting objects and give an overview of aspects of their theory for hereditary algebras.