Past Meetings: 2016–17

Sjoerd Beentjes — A Hall algebra approach to representations of quivers

Hall algebras are associative algebras that can be associated to certain abelian categories. The multiplication of two objects in this category encodes extensions between them. As such, the Hall algebra is a natural place to study filtrations of representations of algebras. Moreover, some well-known algebras popped up as Hall algebras such as Macdonald's ring of symmetric functions and quantised universal enveloping algebras of semi-simple Lie algebras.

The theory has found applications in such diverse topics as categorification of quantum groups, canonical bases, wall-crossing and stability conditions, and curve-counting invariants on Calabi-Yau threefolds.

In this talk I will define a class of Hall algebras and survey some of the above results in the explicit example of quiver representations. This yields a neat class of examples since much of the general theory can be illustrated whereas explicit computations are still feasible.

Reading guide: This talk is loosely based on the paper "The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli" by Marcus Reineke. For a refresher on representations of quivers, there is Michel Brion's set of lecture notes called "Representations of quivers". For a comprehensive introduction to the algebraic side of the topic there are two sets of lecture notes about Hall algebras by Olivier Schiffmann. 

Miguel Couto — The Drinfeld double of infinite dimensional Hopf algebras

In this talk I will attempt to introduce some basic concepts such as Hopf algebras and Hopf dual, I will present the construction of the Drinfeld double for finite dimensional Hopf algebras, enumerate some of its nice properties and then generalize it for the infinite dimensional case. 

Mel Chen — Persistent Homology and Quiver Representations

Persistent homology is a method which associates sequences of homology groups to filtered topological spaces. Due to its ease of computation and ability to capture topological structures, it has seen application in analysing various data sets from images, to viral genomes, and protein structure.

In this talk, I will give a brief introduction to persistent homology, and go through some of the theory behind its use in data analysis. In particular, if homology is taken over a field, then the persistent homology can be represented as a multiset of intervals, which provides a simpler object for further analysis.

This talk will be mostly based on material from the book "Persistence Theory: From Quiver Representations to Data Analysis" by Steve Y. Oudot. 

Matt Booth — Deformation Theory and DGLAs

Deformation theory studies the infinitesimal deformations of algebraic or geometric objects. We'll see what this means, and look at some examples of deformations. I'll then describe a general framework into which deformation problems fit, namely that of deformation functors. Every deformation problem gets assigned such a functor.

A differential graded Lie algebra (DGLA for short) is essentially a chain complex of vector spaces given a Lie algebra structure. Given a DGLA, we can construct a deformation functor using solutions of its Maurer-Cartan equation. Surprisingly, in some sense, we can obtain every deformation functor this way, hence the slogan that 'every deformation problem is controlled by a DGLA'.

Time permitting, I'll briefly mention derived or noncommutative deformations, and how they fit into the framework.

Angela Tabiri — Reducible and Compact Real Form Singular Curves which are Quantum Homogeneous Spaces 

In this talk, we will describe the construction a Hopf algebra A(f,g,a,b,p,q) which contains the coordinate ring of a decomposable plane curve (i.e. f(y)=g(x) ) as a right coideal subalgebra and is free over the coordinate ring of the curve. Concrete examples of A(f,g,a,b,p,q) enable us to show that a reducible singular curve (e.g. the coordinate crossing) or a singular curve with compact real form (e.g. the leminiscate) can be a quantum homogeneous space. Conditions are provided on when these Hopf algebras are domains. These Hopf algebras can be viewed as a generalisation of the Hopf algebras B(n,p_0, ... ,p_s,q) from Construction 1.2 in the paper by Goodearl and Zhang in the references below. http://web.math.ucsb.edu/~goodearl/GK2Hopf31oct09.pdf http://link.springer.com/article/10.1007/s10468-016-9658-8

Tim Weelinck — The Braided Geometry of Quantum Symmetric Pairs

Factorisation homology is a new bridge between algebra and the geometry of manifolds. It has recently been used by D. Jordan et al to 'integrate quantum groups over surfaces' which yields a quantization of character varieties. This established deep connections to amongst others quantum D-modules and Hecke algebras. In this talk we study quantum symmetric pairs ('quantum subgroups') and explain how to interpret the generalised reflection equation as coming from the geometry of coloured braids (we will see these are braids in the orbifold R^2/Z_2). We then explain how one can use that to 'integrate Q.S.Pairs over orbifold surfaces'. Time permitting we explain why we expect DAHAs should appear. 

Lucia Rotheray — Bialgebras and Real-world processes 

We will look at some examples of processes described by the multplication and comultiplication in a bialgebra, common themes and classification. 

Luke Jeffreys — Statistical Hyperbolicity in Groups

The Cayley graph of a group is a fundamental object in geometric group theory that allows us to study groups using the theory of metric spaces. Indeed, understanding the geometric properties of the Cayley graph can tell us important information about the group itself. The notion of the sprawl of a metric space, introduced by Duchin, Lelièvre and Mooney, is a highly sensitive measure of the large-scale geometry of a metric space and, when applied to the Cayley graphs of groups, reveals some very interesting behaviour.

In this talk, I will provide all of the definitions required to introduce the notion of sprawl and to say what it means for a space to be statistically hyperbolic; that is, having sprawl equal to two. Following this, I will give examples demonstrating the application of these ideas and discuss the interesting results that arise. 

Dr. Travis Schedler — Poisson traces, D-modules, and symplectic resolutions

Given a Poisson algebra A, the space of Poisson traces are those functionals annihilating {A,A}, i.e., invariant under Hamiltonian flow. I explain how to study this subtle invariant via D-modules (the algebraic study of differential equations), conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations.