Seminars take place on Monday afternoons 14:00–15:00, unless specified otherwise. Everyone is welcome.
September 16, (EFRY 01.10), Benjamin Steinberg (The City College of New York), An Invitation to the representation theory of monoids.
30 September, (EFRY 01.05), Sam Gunningham (Kings College London), The finiteness condition for skein modules.
7 October, (EFRY 01.05), Neil Saunders (University of Greenwich), The Exotic Nilpotent Cone and Type C Combinatorics
14 October, (EFRY 01.05), Andrea Appel (University of Edinburgh), Meromorphically braided categories
21 October, (LT4), Andreea Mocanu (University of Nottingham), On the relation between Jacobi forms and elliptic modular forms.
28 October, (EFRY 01.05), Nadav Meir (Imperial College, London), O-minimality, pseudo-o-minimality: on first-order properties semialgebraic set
4 November, (LT3), Rachel Newton, (University of Reading), Counting failures of a local-global principle
11 November, (EFRY 01.05), Vahagn Aslanyan (University of East Anglia), A Remark on Atypical Intersections
18 November, (LT3), Nick Williams (University of Leicester), An algebraic interpretation of the higher Stasheff-Tamari orders
25 November, (EFRY 01.05), Lewis Topley (University of Birmingham), The dimensions of modules for Lie algebras in positive characteristics
2 December, (EFRY 01.05), Dimitris Michaelidis (University of Kent), On bases and BGG resolutions of simple modules of Temperley-Lieb algebras of type B
9 December, (EFRY 01.05), Dan Ciubotaru (University of Oxford), Dirac operators and Hecke algebras
Benjamin Steinberg (16 September)
An Invitation to the representation theory of monoids
We provide an overview of the representation theory of finite monoids. Our focus will be on examples (like the monoid of nxn matrices over a finite field and the monoid of all mappings on an n element set) and applications (to Markov chains and perhaps symbolic dynamics).
Sam Gunningham (30 September)
The finiteness condition for skein modules
Given a 3-manifold M, its Kaufmann bracket skein module is a certain family of vector spaces generated by embedded links in M modulo certain relations (called skein relations). It turns out that these modules naturally appear as the values of a topological quantum field theory (TQFT); more generally, there is such a TQFT attached to any suitable (ribbon) braided monoidal category. We use this structure to derive an expression for the skein module of a 3-manifold via a Heegard decomposition, and prove that it is generically finite dimensional using the theory of deformation quantization, confirming a conjecture of Witten.
I will give an overview of skein theory from a categorical point of view, and outline some of the ideas that go in to the proof. No prior knowledge of the subject will be assumed! This is joint work with David Jordan and Pavel Safronov.
Neil Saunders (7 October)
The Exotic Nilpotent Cone and Type C Combinatorics
The exotic nilpotent cone as defined by Kato gives a 'Type A-like' Springer correspondence for Type C. In particular, there is a bijection between the symplectic group orbits on the exotic nilpotent cone and the irreducible representations of the Weyl group of Type C. In this talk, I will outline the various geometric and combinatorial results that follow from this. These results are joint work with Vinoth Nanadakumar and Daniele Rosso, and Arik Wilbert.
Andrea Appel (14 October)
Meromorphically braided categories
The category Mod(H) of finite-dimensional modules over a quasi-triangular Hopf algebra H (that is, a Hopf algebra with an R-matrix R) is the prototype of a braided monoidal category. Similarly, the category of finite-dimensional modules over a coideal subalgebra in H is automatically endowed with a categorical action of Mod(H) and is the prototype of a “module" category.
The compatibility of this action with the braiding in Mod(H) is encoded by an additional datum, called a K-matrix, which provides, together with R, a universal solution of the reflection equation. Coideal subalgebras with a K-matrix are also called quasi-triangular and their finite-dimensional modules form a braided module category. While braided monoidal categories are associated with braid groups (Artin groups of type A), braided module categories corresponds to cylindrical braid groups (Artin groups of type B).
In this talk, I will review the construction of K-matrices for quantum groups of finite type due to Balagovic and Kolb. I will then describe alternative approaches to this construction, which are more suitable to generalizations and apply in particular to quantum affine algebras, where they produce examples of “meromorphically” braided module category. This is based on joint ongoing works with David Jordan and Bart Vlaar.
Andreea Mocanu (21 October)
On the connection between Jacobi forms and elliptic modular forms
Jacobi forms arise naturally in number theory, for example as functions of lattices or as Fourier-Jacobi coefficients of other types of modular forms. They have applications in algebraic geometry, string theory and the theory of vertex operator algebras, among other areas. We are interested in establishing a precise connection between Jacobi forms of lattice index and elliptic modular forms, in order to transfer information from one side to the other. In this talk, we illustrate this connection via an example, namely that of Jacobi forms whose indices are the root lattices of type D_n.
Nadav Meir (28 October)
O-minimality, pseudo-o-minimality: on first-order properties of semialgebraic sets
A semialgebraic set is a set given by the real solutions to a finite set of polynomial equations and inequalities. In Grothendieckʼs Esquisse dʼun Programme he suggested the following challenge: Investigate classes of sets with the tame topological properties of semialgebraic sets. O-minimality is model theory's response to Grothendieck's challenge; it is a property of ordered structures exhibiting the "tame" topological properties of semialgebraic sets, such as cell decompositions and stratifications. With applications both within and outside of model theory, the notion of an o-minimal structure has proven to be increasingly useful, with applications varying from real algebraic and real analytic geometry to economics and machine learning.
An elementary result on o-minimal structures states that any structure satisfying the same first-order sentences as an o-minimal structure is itself o-minimal. Despite that fact, there is no axiomatization of o-minimality by a set of first-order sentences; this can be seen by taking ultraproducts, as we will see.
In this talk, we will review the definition and key results of o-minimality. We will then survey a few first-order properties of semialgebraic sets, each property "tame" in its own way, generalizing the tameness of o-minimal structures. We will end by discussing structures satisfying all first-order properties which hold in every o-minimal structure, what tameness properties these structures satisfy and how they can be axiomatized.
Rachel Newton (4 November)
Counting failures of a local-global principle
Methods for solving polynomial equations in the integers and rationals have been sought and studied for more than 4000 years. Modern approaches try to piece together 'local' (meaning real and p-adic) information to decide whether a polynomial equation has a 'global' (meaning rational) solution. I will describe this approach and its limitations, with the aim of quantifying how often the local-global method fails within families of polynomial equations arising from the norm map between fields, as seen in Galois theory. This is joint work with Tim Browning.
Vahagn Aslanyan (11 November)
A Remark on Atypical Intersections
I will define atypical intersections of algebraic varieties and state the Conjecture on Intersections with Tori (CIT), which is a Diophantine conjecture generalising Mordell-Lang (in the appropriate setting). Although CIT is open, many special cases and weak versions have been proven. I will discuss a well-known weak version of CIT and explain how it can be generalised using the Mordell-Lang conjecture (which is a theorm).
Nick Williams (18 November)
An algebraic interpretation of the higher Stasheff–Tamari orders
The two higher Stasheff–Tamari orders generalise the well-studied Tamari lattice of triangulations of a convex n-gon to higher dimensions by considering instead the set of triangulations of a cyclic polytope. The orders were implicit in the work of Kapranov and Voevodsky, but were first defined explicitly by Edelman and Reiner, who conjectured them to be equal. Edelman and Reiner showed this to hold in low dimensions, but the general result is still unknown. Meanwhile, on the algebraic side, Oppermann and Thomas connected triangulations of even-dimensional cyclic polytopes with the tilting modules over Iyama’s higher Auslander algebras of type A. In this talk I outline recent work in which I show how the higher Stasheff–Tamari orders fit into the algebraic picture of Oppermann and Thomas. Indeed, it turns out that they coincide with higher-dimensional versions of orders on tilting modules studied by Happel, Unger, Riedtmann, and Schofield.
Lewis Topley (25 November)
The dimensions of modules for Lie algebras in positive characteristics
One of the main goals in representation theory is to understand simple modules for a chosen class of algebraic objects. Lie algebras are one of the most classical algebraic structures: they arise naturally as the infinitesimal analogues of (continuous) groups. Over the complex numbers representations of Lie algebras have been studied extensively and the situation is quite well-understood, although many mysteries still remain. Over fields of positive characteristics however, the situation is more complicated. I will begin this talk by giving a gentle introduction to this field, comparing the ordinary and the modular.
In the second half of the talk I will describe a joint work with Ben Martin and David Stewart in which we apply the Leftschetz principle, along with classical techniques from Lie theory, to prove (a generic version of) a conjecture of Kac and Weisfeiler from 1971, describing the maximal dimension of simple modules over a Lie algebra in positive characteristic.
Dimitris Michailidis (2 December)
On bases and BGG resolutions of Temperley-Lieb algebras of type B
Inspired from the study of certain models in physics, Martin and Saleur defined the Temperley-Lieb algebra of type B or blob algebra as the diagrammatic two parameter generalisation of the Temperley-Lieb algebra of type A. The blob algebra can also be viewed as quotient of the Hecke algebra of type B, hence it is isomorphic to a quotient of the (graded) KLR algebra via the Brundan-Kleshchev isomorphism. In this talk we shall construct bases of the simple representations for the blob algebra, indexed by paths in the Euclidean space with respect to the alcove geometry of affine type A1. We also prove that to each simple representation we attach a resolution of cell modules, called BGG resolution, which gives homological construction of simple representations.
Dan Ciubotaru (9 December)
Dirac operators and Hecke algebras
I will explain the construction and main properties of Dirac operators for representations of various Hecke-type algebras (e.g., Lusztig's graded Hecke algebra for p-adic groups, Drinfeld's Hecke algebras, rational Cherednik algebras). The approach is motivated by the classical Dirac operator which acts on sections of spinor bundles over Riemannian symmetric spaces, and by its algebraic version for Harish-Chandra modules of real reductive groups. The algebraic Dirac theory developed for these Hecke algebras turns out to lead to interesting applications: e.g., a Springer parameterisation of projective representations of finite Weyl groups (in terms of the geometry of the nilpotent cone of complex semisimple Lie algebras), spectral gaps for unitary representations of reductive p-adic groups, connections between the Calogero-Moser space and Kazhdan-Lusztig double cells. I will present some of these applications in the talk.