The Leeds Algebra Group .

Schedule:

Abstracts

Bethany Marsh (University of Leeds): Tau-exceptional sequences. Joint work with Aslak Bakke Buan (NTNU).
We introduce the notion of a (signed) tau-exceptional sequence for a finite dimensional algebra, which can be regarded as the generalisation of the notion of a classical exceptional sequence in the hereditary case. The new sequences behave well for both non-hereditary and hereditary algebras. The work is motivated by the signed exceptional sequences introduced, in the hereditary case, by Igusa-Torodov, and by tau-tilting theory. We show that there is a bijection between the set of complete signed exceptional sequences and ordered basic support tau-tilting objects.

Patricia Klein (University of Minnesota): Geometric vertex decomposition and liaison. Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, we will describe an explicit connection between these approaches. In particular, we describe how each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras. This connection also gives us a framework for implementing with relative ease Gorla, Migliore, and Nagel's strategy of using liaison to establish Gröbner bases. This talk is based on joint work with Jenna Rajchgot.

Gwyn Bellamy (University of Glasgow): Invariant holonomic systems for symmetric spaces. If G is a reductive (connected, complex) Lie group with Lie algebra g, then Hotta-Kashiwara introduced a certain D-module HC on g called the Harish-Chandra D-module. This is precisely the module whose distributional solutions are the invariant eigen-distributions appearing in character theory for real reductive Lie groups. A key result by Hotta-Kashiwara is that HC is semi-simple with simple summands in bijection with the irreducible representations of the Weyl group (the algebraic Springer correspondence). In this talk I'll describe a natural generalization of HC to a D-module on the tangent space of a symmetric space. I'll explain the extent to which Hotta-Kashiwara's results generalise to this setting. This is based on joint work with Levasseur, Nevins and Stafford; I won't assume any prior knowledge of D-modules.

Esther Banaian (Aarhus University): Algebras from Orbifolds. We give a construction for a gentle algebra which can be associated with a triangulated orbifold with all orbifold points having order three. First, we discuss how features of the module category can be viewed on the orbifold. Chekhov and Shapiro demonstrate how to associate a generalized cluster algebra from a triangulated orbifold. We then compare the modules over this gentle algebra with the elements of the generalized cluster algebra from the same orbifold with triangulation. This talk is based on joint work with Yadira Valdivieso.

Al Kasprzyk (University of Notthingham): Maximally mutable Laurent polynomials. Recent progress has been made classifying Fano manifolds via mirror symmetry: to each Fano manifold X we associate a Laurent polynomial mirror f, such that certain key invariants of X and f agree. If this method is to be applied systematically in high dimensions, a fundamental question which needs answering is "Which Laurent polynomials are mirrors to Fano manifolds?" Several successful ansatzs have been proposed over the past decade, giving a partial answer to this question. The most recent of these -- the Minkowski ansatz -- successfully recovered the three-dimensional classification of Mori--Mukai. But each ansatz has limitations. For example, the Minkowski ansatz is strictly three-dimensional. Here I will describe a new family of Laurent polynomials, called rigid maximally mutable, that conjecturally give a complete answer to this question.

Giulio Belletti (Universite Paris Saclay): Quantum invariants, the volume conjecture and representations of the mapping class group. In this talk I will give an introduction to quantum invariants, which are sequences of invariants of 3-manifolds arising from the representation theory of quantum groups. I will then discuss the most important open problem related to these invariants, the volume conjecture, and discuss how this conjecture has applications to the representation theory of the mapping class group.

Iva Halacheva (Northeastern University): Crystals and cacti in representation theory. One approach to studying the representation theory of Lie algebras and their associated quantum groups is through combinatorial shadows known as crystals. While the original representations carry an action of the braid group, their crystals carry an action of a closely related group known as the cactus group. I will describe how we can realize this combinatorial action both geometrically, as a monodromy action coming from a family of "shift of argument" algebras, as well as categorically through the structure of certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint work with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.

Simon Crawford (University of Manchester): Hopf actions, quivers, and superpotentials . There is currently a strong interest in extending results from commutative invariant theory to a noncommutative setting. One way to achieve this is by studying actions of Hopf algebras on "noncommutative polynomial rings". I will explain some of my recent results in this area and, in particular, show how to associate a quiver with relations to such a setup. I will also explain how some results in the literature follow quickly using these quivers.

Emily Cliff (Universite de Sherbrooke): Moduli spaces of principal 2-group bundles and a categorification of the Freed-Quinn line bundle. A 2-group is a higher categorical analogue of a group, while a smooth 2-group is a higher categorical analogue of a Lie group. An important example is the string 2-group in the sense of Schommer-Pries. We study the notion of principal bundles for smooth 2-groups, and investigate the moduli "space" of such objects. In particular in the case of flat principal bundles for a finite 2-group over a Riemann surface, we prove that the moduli space gives a categorification of the Freed–Quinn line bundle. This line bundle has as its global sections the state space of Chern-Simons theory for the underlying finite group. We can also use our results to better understand the notion of geometric string structures (as previously studied by Waldorf and Stolz-Teichner). This is based on joint work with Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.

Julia Plavnik (Indiana University Bloomington): Classifying modular categories using Galois theory. Modular categories arise naturally in many areas of mathematics, such as conformal field theory, representations of braid groups, quantum groups, and Hopf algebras, low dimensional topology. The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter.
A mixture of Galois theory and representation techniques has played a key role in the classification program of modular tensor categories.
In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories. We will give an overview of the current situation of the classification program and focus on some recent results of classification by the number of Galois orbits.

For past algebra seminars see: 2020/21 , 2019/20 , 2018/19 , 2017/18 .

This page is maintained by Eleonore Faber .

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