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|January 31/15:00/||Benjamin Horton (University of Leeds)||Manifolds, Cobordisms, Braids and Loop Braids|
|February 7/15:00/||Dionne Ibarra (Monash University)||Gram Determinants motivated by Knot Theory|
|February 14/15:00/||Joel Costa da Rocha (University of Leeds)||Parametrising positroid cells using bicolored tilings|
|February 21/15:00/||Catherine Meusburger (Friedrich-Alexander-Universität)||A Generalization of Kitaev’s quantum double models|
|March 7/15:00/||Nikita Nikolaev (University of Birmingham)||Spectral Clusters from the Abelianisation of Groupoid Representations|
|March 14/15:00/||Eugen Rogozinnikov (University of Strasbourg)||Noncommutative Coordinates for Maximal Symplectic Representations|
|March 28/15:00/||Hannah Dell (University of Edinburgh)||Stability conditions on free abelian quotients|
|April 18/15:00||Eric Hanson (Université du Québec à Montréal)||Infinitesimal wall-and-chamber structures|
|May 9/15:00/||Colin Krawchuk (University of Cambridge)||Higher Friezes and Higher AR Theory|
|May 16/15:00/||Dani Kaufman (University of Copenhagen)||Special Folding of Grassmannian Cluster Algebras|
|May 23/15:00/||İpek Tuvay (Mimar Sinan Fine Arts University)||Splendid Morita equivalence between block algebras and Brauer indecomposability of Scott modules|
Benjamin Horton (University of Leeds)
Manifolds, Cobordisms, Braids and Loop Braids.
We consider a categorifieed version of the category of manifolds and cobordisms, in the form of a pseudo-double category of manifolds, collared cobordisms, and equivalence classes of certain collar compatible diffeomorphisms between cobordisms. We discuss why this is the appropriate setting to formulate the monoidal categories of generalised braids and loop braids. We aim to give a pedagogical talk, and we will not assume previous knowledge on topological quantum field theory.
Dionne Ibarra (Monash University)
Gram Determinants motivated by Knot Theory.
In the 1990’s, a general formula for the Gram determinant of Type A was formulated in order to prove the existence and uniqueness of Lickorish’s construct of the Witten-Reshetikhin-Turaev invariants of 3-manifolds. Since then, various types of Gram determinants have been created from knot theory by using crossingless connections in surfaces with boundary. In this talk we will introduce the different types and present recent work on the progress to proving a closed formula for a particularly complicated conjectured closed formula for the Gram determinant of Type Mb. We will end with observations and questions about the potential connection between Gram determinants motivated by knot theory and statistical mechanics.
Joel Costa da Rocha (University of Leeds)
Parametrising positroid cells using bicolored tilings
Triangulations of an n-gon are in bijection with reduced Postnikov diagrams of type (2,n), with diagonal flips in the triangulation corresponding to geometric exchanges in the Postnikov diagram, and both corresponding to mutations of corresponding quivers. We introduce bicolored tilings to generalise this correspondence. We can then use these tilings to parametrise positroid cells of the Grassmannian, or generate Postnikov diagrams for any given permutation.
Catherine Meusburger (Friedrich-Alexander-Universität)
A Generalization of Kitaev’s quantum double models.
We describe a construction inspired by Kitaev’s quantum double models and related to factorization homology. It assigns to an embedded graph on an oriented surface S and a Hopf monoid in a (co)complete symmetric monoidal category C an object in C that depends only on S, equipped with an action of the mapping class group Map(S).
We determine this object explicitly for simplicial groups and crossed modules as Hopf monoids in SSet and in Cat. For the latter this yields a groupoid with representation varieties Hom(pi_1(S), B)/B as objects and certain equivalence classes of representation varieties as morphisms.
This is based on joint work with Thomas Voss and Anna-Katharina Hirmer.
Nikita Nikolaev (University of Birmingham)
Spectral Clusters from the Abelianisation of Groupoid Representations
Perhaps surprisingly, flat meromorphic connections on holomorphic bundles over complex manifolds — objects that you’d probably consign to a purely differential-geometric context — can be equivalently studied using the representation theory of naturally associated groups or, better yet, Lie groupoids. The moduli spaces of these representations are really remarkable geometric objects that carry many interesting structures, including a symplectic structure and a cluster algebra structure. I will talk about various aspects of my ongoing series of projects to study these spaces using a method called abelianisation which one can think of as a kind of spectral theory for connections.
Eugen Rogozinnikov (University of Strasbourg)
Noncommutative Coordinates for Maximal Symplectic Representations
Representations of the fundamental group of an orientable surface of finite type into a Hermitian Lie group with maximal Toledo invariant are of particular interest in higher Teichmüller theory. These representations have been studied by M. Burger, A. Iozzi, and A. Wienhard and generalize Fuchsian representations of the fundamental group of a surface into PSL(2, R). Moreover, maximal representations have particularly nice properties, e.g. they are injective with a discrete image.
In my talk I will introduce coordinates on the space of decorated maximal representations of the fundamental group of a punctured surface into Sp(2n, R). These coordinates generalize the Fock-Goncharov coordinates for representations into SL(2, R) and have a nice non-commutative cluster-like structure. If time permits, I will talk about how we can use these coordinates to understand the topology of the space of decorated maximal representations. This is a joint work with D. Alessandrini, O. Guichard and A. Wienhard.
Hannah Dell (University of Edinburgh)
Stability conditions on free abelian quotients
Stability conditions on triangulated categories were introduced by Tom Bridgeland in 2007. They generalise several notions of stability, such as slope-stability for vector bundles on curves and King stability for quiver representations. Bridgeland showed that the space of stability conditions on a given triangulated category can be viewed as a complex manifold, giving us a way to extract geometry from homological algebra. However, an explicit description of the stability manifold is only known in a few cases.
In this talk, I will discuss two approaches to studying stability conditions on derived categories of surfaces that are free quotients by finite abelian groups. One method is via Le Potier functions, which characterise the existence of slope-semistable sheaves. The second method uses Deligne’s notion of group actions on triangulated categories to describe a connected component of so-called geometric stability conditions inside the stability manifold of these finite free abelian quotients. A consequence of this is a disproof of the expectation that surfaces with irregularity 0 always admit a wall of the geometric chamber.
Eric Hanson (Université du Québec à Montréal)
Infinitesimal wall-and-chamber structures
Loosely speaking, a wall-and-chamber structure is determined by a set of convex codimension 1 cones (called “walls”) in R^n. Such structures in particular arise when considering stability conditions for finite-dimensional algebras over fields. In this talk, we explain how the “infinitesimal structure” near an accumulation point of walls can be used to define a new wall-and-chamber structure of smaller dimension. We then study these new structures over tame hereditary algebras. This talk will not assume prior knowledge of stability conditions or of finite-dimensional algebras.
Colin Krawchuk (University of Cambridge)
Higher Friezes and Higher AR Theory
Friezes are patterns of integers that may be viewed as specialisations of cluster variables belonging to an associated cluster algebra. More abstractly, one can associate a frieze to each cluster tilting object in a cluster category via the Caldero-Chapoton map. Recently McMahon introduced (k,n)-frieze patterns as a generalisation of friezes and showed that clusters in the homogeneous coordinate ring of the Grassmannian could similarly produce (k,n)-friezes. In this talk we discuss a representation theoretic connection between higher friezes and higher homological algebra.
Dani Kaufman (University of Copenhagen)
Special Folding of Grassmannian Cluster Algebras
In a recent preprint I defined a notion of a “special folding” of a cluster algebra. A usual folding of a cluster algebra is one that can be represented with a skew-symmetrizable exchange matrix, like the folding representing a type C_n algebra from a type A_(2n-1) algebra. Special folding, which cannot be represented by a skew symmetrizable matrix, can have surprising new properties like producing a finite exchange complex from an infinite one. In this talk I will define this notion and give several examples coming from Grassmannian cluster algebras.
İpek Tuvay (Mimar Sinan Fine Arts University)
Splendid Morita equivalence between block algebras and Brauer indecomposability of Scott modules
Let p be a prime number and P a finite p-group. In the representation theory of finite groups in characteristic p, understanding finite block algebras with defect groups isomorphic to P up to splendid Morita equivalence is important. There is a strong connection between the existence of such equivalence between principal block algebras and Brauer indecomposability of Scott modules. In this talk, after a short introduction we will present a recent result from a joint work with S. Koshitani which gives an equivalent condition for a Scott module to be Brauer indecomposable. Then we will discuss how the notion of Brauer indecomposability is used to show that Puig Conjecture holds for principal blocks in tame representation type.