The University of Leeds Algebra Seminar 2022


The Leeds Algebra Group .
DATE and TIME and PLACE  SPEAKER  TITLE 
January 25, 15:00
MALL/online 
Bethany Marsh
(University of Leeds) 
Tauexceptional sequences 
February 8, 15:00
Online 
Patricia Klein
(University of Minnesota) 
Geometric vertex decomposition and liaison 
February 15, 15:00
Online 
Gwyn Bellamy
(University of Glasgow) 
Invariant holonomic systems for symmetric spaces 
February 22, 15:00
Online 
Esther Banaian
(Aarhus University) 
Algebras from Orbifolds 
March 1, 15:00
MALL/Online 
Al Kasprzyk
(University of Nottingham) 
Maximally mutable Laurent polynomials 
March 8, 15:00
Online 
Giulio Belletti
(Universite Paris Saclay) 
Quantum invariants, the volume conjecture and representations of the mapping class group 
March 15, 15:00
Online 
Iva Halacheva
(Northeastern University) 
Crystals and cacti in representation theory 
March 22, 15:00
MALL 
Simon Crawford
(University of Manchester) 
Hopf actions, quivers, and superpotentials 
April 26, 15:00
Online 
Emily Cliff
(Universite de Sherbrooke) 
Moduli spaces of principal 2group bundles and a categorification of the Freed Quinn line bundle 
May 3, 15:00
Online 
Julia Plavnik
(Indiana University Bloomington) 
Classifying modular categories using Galois theory 
Bethany Marsh (University of Leeds): Tauexceptional sequences.
Joint work with Aslak Bakke Buan (NTNU).
We introduce the notion of a (signed) tauexceptional 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 nonhereditary and hereditary algebras. The work is motivated by the signed exceptional sequences introduced, in the hereditary case, by IgusaTorodov, and by tautilting theory. We show that there is a bijection between the set of complete signed exceptional sequences and ordered basic support tautilting 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 Gbiliaisons of height 1 to an ideal of indeterminates and, conversely, how every Gbiliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several wellknown 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 HottaKashiwara introduced a certain Dmodule HC on g called the HarishChandra Dmodule. This is precisely the module whose distributional solutions are the invariant eigendistributions appearing in character theory for real reductive Lie groups. A key result by HottaKashiwara is that HC is semisimple 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 Dmodule on the tangent space of a symmetric space. I'll explain the extent to which HottaKashiwara'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 Dmodules.
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 threedimensional classification of MoriMukai. But each ansatz has limitations. For example, the Minkowski ansatz is strictly threedimensional. 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 3manifolds 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 2group bundles and a categorification of the FreedQuinn line bundle. A 2group is a higher categorical analogue of a group, while a smooth 2group is a higher categorical analogue of a Lie group. An important example is the string 2group in the sense of SchommerPries. We study the notion of principal bundles for smooth 2groups, and investigate the moduli "space" of such objects. In particular in the case of flat principal bundles for a finite 2group over a Riemann surface, we prove that the moduli space gives a categorification of the Freedâ€“Quinn line bundle. This line bundle has as its global sections the state space of ChernSimons 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 StolzTeichner). This is based on joint work with Dan BerwickEvans, 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 .