Algebra and Number Theory seminar

DateRoomSpeakerTitle

29/09/2014 5:30 PM103Jonathan Elmer (Aberdeen)Symmetric powers of modular representations of elementary abelian pgroups
In the 1970s, Almkvist and Fossum gave formulae which describe completely the decomposition of symmetric powers of modular representations of cyclic groups into indecomposable summands. We show how (in spite of the wildness of the representation type) some of their results can be generalized to representations of elementary abelian pgroups. Some applications to invariant theory will also be given.

06/10/2014 5:30 PM103Alonso Castillo Ramirez (Zaragoza)Majorana algebras
A Majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some properties of the 2Aaxes of the 196884dimensional Monster Griess algebra. The term was introduced by A. A. Ivanov in 2009 inspired by Sakuma's and Miyamoto's work on vertex operator algebras. In this talk, we are going to present some elementary examples of Majorana algebras, and we will sketch how to obtain the automorphism groups and maximal associative subalgebras of the twogenerated Majorana algebras.

13/10/2014 5:30 PM103Melanie de Boeck (Kent)Studying Foulkes modules using semistandard homomorphisms
The action of the symmetric group S_{mn} on set partitions of a set of size mn into n sets of size m gives rise to a permutation module called the Foulkes module. Structurally, very little is known about Foulkes modules, even in characteristic zero. In this talk, we will see that semistandard homomorphisms may be used as a tool for studying the module structure and, in particular, for establishing relationships between irreducible constituents of Foulkes modules.

20/10/2014 6:00 PM103Anton Evseev (Birmingham)Graded RoCK blocks and wreath products
The socalled RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic p, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block B_{0} of the wreath product S_{p}wr S_{w}, where w is the "weight" of the block. More precisely (and more simply), the conjecture states that the idempotent truncation in question is isomorphic to a tensor product of B_{0} and a certain matrix algebra. The talk will outline a proof of this conjecture, which uses an isomorphism between the group algebra of a symmetric group and a cyclotomic KhovanovLaudaRouquier algebra and the resulting grading on the group algebra of the symmetric group. This result generalizes a theorem of ChuangKessar, which applies to the case w

27/10/2014 4:30 PM103Maria Calvo Cervera (Granada)Cohomological classification of monoidal groupoids
Strongly inspired by Schreier’s analysis of group extensions and its extension to fibrations of categories by Grothendieck, we analyse the structure of monoidal categories in which every arrow is invertible. In particular, we state precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, by means of Leech’s cohomology groups of monoids.

03/11/2014 4:30 PMNo seminar  School colloquium

10/11/2014 4:30 PM103Emilio Pierro (Birkbeck)The Möbius function of the small Ree groups
In 1936 Hall showed that Möbius inversion could be applied to the lattice of subgroups of a finite group G in order to determine the number of nbases of G, that is, generating sets of G of size n. The question can be modified and nbases subject to certain relations can also be enumerated with applications to the theory of Riemann surfaces, Hurwitz groups, dessins d'enfants and various other algebraic, topological and combinatorial enumerations. In order to determine the Möbius function of a group it is necessary to understand the subgroup structure of a group and so we also give a description of the simple small Ree groups R(q)=^{2}G_{2}(q), in particular their maximal subgroups, in terms of their 2transitive permutation representations of degree q^{3}+1.

17/11/2014 4:30 PMNo seminar  School colloquium

24/11/2014 4:30 PM103Keith Brown (UEA)Properly stratified quotients of KhovanovLaudaRouquier algebras
Introduced in 2008 by Khovanov and Lauda, and independently by Rouquier, the KLR algebras are a family of infinitedimensional graded algebras which categorify the negative part of the quantum group associated to a graph. In finite types these algebras are known to have nice homological properties, in particular they are affine quasihereditary. In this talk I'll explain what it means to be affine quasihereditary and how this relates to properties of finite dimensional algebras. I'll then introduce a finite dimensional quotient of the KLR algebra which preserves some of the homological structure of the original algebra and provide a bound on its finitistic dimension. This work will form part of my PhD thesis, supervised by Dr Vanessa Miemietz.

01/12/2014 4:30 PMNo seminar  School colloquium

08/12/2013 4:30 PM103Adam Boocher (Edinburgh)Ideals of maximal minors and their resolutions
The set of k by n matrices with rank at most r naturally forms an algebraic variety. Its defining equations are given by determinants and it enjoys many beautiful properties. In this talk I'll discuss some recent work that describes how this variety behaves upon specialization with some applications to matroids and free resolutions.

09/01/2015 4:30 PMEngineering 371Xin Li (QMUL)On subsemigroups of groups
We discuss algebraic properties of subsemigroups of groups. These properties first arose in the study of operator algebras attached to semigroups. In this talk, however, the focus will not be on operator algebras. On the one hand, this means that no operator algebraic prerequisites are required, and on the other hand, it allows us to focus on purely algebraic aspects which are hopefully interesting on their own right. Our main concepts will be illustrated by natural examples like Braid groups, Artin groups or the Thompson group.

02/02/2015 4:30 PM103Thomas Kahle (OvGU Magdeburg)How primary decomposition of monoid congruences and binomial ideals is wrong
In a polynomial ring, a binomial says that two monomials are scalar multiples of each other. Forgetting about the scalars, a binomial ideal describes an equivalence relation on the monoid of exponents. Ideally one would want to carry out algebraic computations, such as primary decomposition of binomial ideals, entirely in this combinatorial language. We will present such a calculus, enabling one to compute by looking at pictures of monoids.

06/02/2015 4:30 PM103Noah White (Edinburgh)Schubert calculus and the cactus groupSeminar series:The RSK correspondence assigns a pair of standard tableaux to every element of the symmetric group. This describes a partitioning of the group into “cells”. More generally these cells can be defined for any Coxeter group. Recently Henriques and Kamnitzer defined an action of the “cactus group” on crystals for semisimple Lie algebras. I will explain, in type A, the connection between this action and a conjectural method of Bonnafe and Rouquier of defining cells for the symmetric group. I will show how this action appears using Schubert calculus or alternatively using the representation theory of the symmetric group and certain generalisations of the JucysMurphy elements called the Gaudin Hamiltonians.

23/02/2015 4:30 PM103Louise SuttonGraded Specht modulesSeminar series:
Specht modules play an integral role in the representation theory of the symmetric groups. Recent work by Brundan and Kleshchev and Khovanov, Lauda and Rouquier has added a wealth of structure to the Specht modules in positive characteristic. One ambitiously hopes to obtain a graded analogue of the hook length formula, introduced by Frame, Robinson and Thrall in 1954, which calculates the dimension of the Specht modules.
I will begin with the combinatorial construction of the Specht modules over a field of characteristic 0, as first developed by G. D. James in the 1970s. I will then give a review of the recent developments in modular representation theory of the symmetric groups, together with my progress in attaining a graded dimension formula for the Specht modules.

09/03/2015 4:30 PM103Daniel Schäppi (Sheffield)Tannaka duality for stacksSeminar series:
Classical Tannaka duality is a duality between groups and their categories of representations. It answers two basic questions: can we recover the group from its category of representations, and can we characterize categories of representations abstractly? These are often called the reconstruction problem and the recognition problem. In the context of affine group schemes over a field, the recognition problem was solved by Saavedra and Deligne using the notion of a (neutral) Tannakian category.
In this talk I will explain how this theory can be generalized to the context of certain algebraic stacks and their categories of coherent sheaves (using the notion of a weakly Tannakian category). On Tuesday [in the Quantum Algebras seminar] I will talk about work in progress to construct universal weakly Tannakian categories and some of their applications. The aim is to interpret various constructions on stacks (for example fiber products) in terms of the corresponding weakly Tannakian categories.

16/03/2015 4:30 PM103Arkady Vaintrob (Oregon)Cohomological field theories related to singularities and matrix factorizationsSeminar series:
I will discuss a cohomological field theory associated to a quasihomogeneous isolated
singularity W with a group G of its diagonal symmetries. The state space of this theory
is the equivariant Milnor ring of W and the corresponding invariants can be viewed as
analogs of the GromovWitten invariants for the noncommutative space associated with the pair (W,G).
In the case of simple singularities of type A they control the intersection theory on the
moduli space of higher spin curves.
The construction is based on derived categories of (equivariant) matrix factorizations of W. 
23/03/2015 4:30 PM103Karel Casteels (Kent)Combinatorial Models of Quantum Matrix AlgebrasSeminar series:
We will describe combinatorial "models" that can be used to study various quantum algebras (for example quantum matrices, quantum symmetric and skewsymmetric matrices, the quantum grassmannian and more). For all of these algebras, there is an action of an algebraic torus by automorphisms and a description of the torusinvariant prime ideals is a key step towards understanding the full prime spectrum due to work of Goodearl and Letzter. We will discuss how the above combinatorial models can be used to calculate Grobner bases of all torusinvariant prime ideals, as well as provide other useful information. Portions of this talk are joint work with Stephane Launois.

26/05/2015 4:00 PM103Dr Michael Wibmer (RWTH Aachen)Étale difference algebraic groupsSeminar series:
The talk will begin with an introduction to difference algebraic groups, i.e., groups defined by algebraic difference equations. Like étale algebraic groups can be described as finite groups with a continuous action of the absolute Galois group of the base field, étale difference algebraic groups can be described as certain profinite groups with some extra structure. Étale difference algebraic groups satisfy a decomposition theorem that shows that they can all be build from étale algebraic groups and finite groups equipped with an endomorphism.

16/06/2015 11:00 AM103Jack EdmondsExistential Polytime and Polyhedral CombinatoricsSeminar series:
Combinatorial structure of paths, marriage, routes for Chinese postmen, traveling salesmen, and itinerant preachers, optimum systems of trees and branchings.
Repeats Tuesday through Friday, same room, same time.
For more, see http://www.maths.qmul.ac.uk/~fink/Edmonds2015.html .

28/09/2015 5:00 PMMaths 103Hossein Abbaspour (Nantes)Frobenius algebras and their Hochschild homologySeminar series:

18/01/2016 4:30 PM103Pierre Dechant (York)A new construction of E8 and the other exceptional root systemsSeminar series:
Lie groups&algebras, Coxeter/reflection groups and root systems are closely related, and feature prominently throughout mathematics and physics, in particular the exceptional ones. We argue that the root system concept is the most useful for our purposes, and that since an inner product is implicit when considering reflections, one can always construct the Clifford algebra over the underlying vector space. Clifford algebra has a very simple reflection formula and via the CartanDieudonne theorem provides a double cover of the orthogonal transformations. In particular, in 3D the Clifford algebra is 8dimensional and its even subalgebra is 4dimensional. Starting from a 3D root system one can therefore construct groups of 4D or 8D objects under Clifford multiplication. The 4D ones can in general be shown to be root systems with interesting automorphism groups  in particular D4, F4, H4 are induced from A3, B3, H3  and for the 8D case one can show (via R. Wilson's reduced inner product) that the Clifford double cover of the 120 reflections in H3 yields the 240 roots of E8.

25/01/2016 4:30 PM(No seminar – School Colloquium)

15/02/2016 4:30 PM(No seminar – School Colloquium)

29/02/2016 4:30 PM103Rob Wilson (QMUL)[Cancelled]
*Clifford algebra or Lie algebra: what are the Dirac matrices?*
The Dirac equation reconciles quantum mechanics with special relativity, by describing the wave functions of particles like the electron travelling at relativistic speeds. It is a PDE with coefficients which are 4 x 4 complex matrices called the Dirac gamma matrices. Conventional wisdom states that these matrices generate the Clifford algebra Cl(3,1) for the quadratic form with signature (3,1). However, their use in physics requires multiplying some of the Clifford algebra by i, thereby destroying the Clifford algebra structure. I argue that it makes more sense to say the gamma matrices generate the Lie algebra so(5,1). This viewpoint potentially throws light on the nature of the weak force, and thereby on the nature of mass and charge.

07/03/2016 4:30 PM103Matthias Lenz (Fribourg)On splines and vector partition functions
Vector partition functions and their continuous analogues (multivariate splines) appear
in many different fields, including approximation theory (box splines and their discrete analogues),
symplectic geometry and representation theory (DuistermaatHeckman measure and
weight multiplicity function/Kostant's partition function),
and discrete geometry (volumes and number of integer points of convex polytopes).I will start by presenting the theory of the spaces spanned by the local pieces of these
piecewise (quasi)polynomial functions and point out connections with matroid theory.
This theory has been developed in the 1980s by Dahmen and Micchelli. Later it has been
put in a broader context by De Concini, Procesi, Vergne and others.
Then I will present a refined version of the KhovanskiiPukhlikov formula that relates the
volume and the number of integer points of a smooth lattice polytope. 
14/03/2016 4:30 PM(No seminar – School Colloquium)

21/03/2016 4:30 PM103Imen Belmokhtar (QMUL)The structure of induced simple modules of 0Hecke algebras
In this talk we shall be concerned with the induced simple modules of the 0Hecke algebras of types A and B.
The irreducible representations of 0Hecke algebras were classified and shown to be onedimensional by Norton in 1979.
To understand the structure of a finitedimensional module, one would ideally like to know its full submodule lattice; this is easily computable for small dimensions but much harder for larger ones. Given certain conditions, a smaller poset encoding the submodule lattice can be rather easily obtained.
We shall discuss the theory allowing us to get this smaller poset and build on results by Fayers in the type A case to state new results in type B.

03/10/2016 5:00 PMMaths LTNo seminar owing to School colloquium

10/10/2016 4:30 PMFB 3.11Leonard H. Soicher (QMUL)Block intersection polynomials and strongly regular graphs
I will give a brief introduction to block intersection polynomials, and
then discuss their application to the study of strongly regular graphs,
in particular describing recent joint work with Gary Greaves on
new upper bounds for the clique numbers of strongly regular graphs
in terms of their parameters. No previous knowledge of strongly
regular graphs will be assumed. 
17/10/2016 4:30 PMFB 3.11Robert A. Wilson (QMUL)Principal ideal domains and Euclidean domains

24/10/2016 4:30 PMFB 3.11Wajid Mannan (QMUL)Group presentations, representations over the integers and homotopy

31/10/2016 5:00 PMMaths LTNo seminar owing to School colloquium

14/11/2016 4:30 PMFB 3.11Cecilia Busuioc (QMUL)Ktheory and Arithmetic
In this talk, I will give a brief account of the deep connection between the geometry of modular curves and the arithmetic of cyclotomic fields, originally conjectured by R. Sharifi.
The main idea relies on a Ktheoretic construction of modular symbols that enjoys further
generalisations to a GL_n setting. This is the subject of a work in progress with G. Stevens and O. Patashnick. 
21/11/2016 4:30 PMFB 3.11John N. Bray (QMUL)Representations of some finite groups

28/11/2016 4:30 PMFB 3.11Tomasz Popiel (QMUL)Symmetries of generalised polygons
Generalised polygons are point?line incidence geometries introduced by Jacques Tits in an attempt to find geometric models for finite simple
groups of Lie type. A famous theorem of Feit and G. Higman asserts that the only "nontrivial"examples are generalised triangles (projective
planes), quadrangles, hexagons and octagons. In each case, there are "classical" examples associated with certain Lie type groups, and in the
latter two cases these are the only known examples. The classical examples are highly symmetric; in particular, their automorphism groups act
transitively on flags and primitively on both points and lines. There have been various attempts to classify generalised polygons subject to
symmetry assumptions whether weaker, stronger, or just different to those mentioned above and perhaps one of the strongest results in this
direction is a theorem of Kantor from 1987, asserting that a pointprimitive projective plane is either classical (Desarguesian) or has a
prime number of points and a severely restricted automorphism group. I will review some ongoing work with John Bamberg, Stephen Glasby, Luke
Morgan, Cheryl Praeger and Csaba Schneider that aims to classify the pointprimitive generalised quadrangles, hexagons and octagons. 
05/12/2016 5:00 PMMaths LTNo seminar owing to School colloquium

08/05/2017 3:00 PMW316Roozbeh Hazrat (Western Sydney University)Leavitt path algebras
From a directed graph one can generate various algebras that capture the movements along the graph. One such algebra is the Leavitt path algebra.
Despite being introduced only 10 years ago, Leavitt path algebras have arisen in a variety of different contexts as diverse as analysis, symbolic dynamics, noncommutative geometry and representation theory. In fact, Leavitt path algebras are algebraic counterpart to graph C*algebras, a theory which has become an area of intensive research globally. There are strikingly parallel similarities between these two theories. Even more surprisingly, one cannot (yet) obtain the results in one theory as a consequence of the other; the statements look the same, however the techniques to prove them are quite different (as the names suggest, one uses Algebra and other Analysis). These all suggest that there might be a bridge between Algebra and Analysis yet to be uncovered.
In this talk, we introduce Leavitt path algebras and try to classify them by means of (graded) Grothendieck groups. We will ask nice questions!

25/09/2017 4:30 PMQueens W316Steve Lester (QMUL)Superscars for wave functions of a point scatterer on the torus
A fundamental problem in Quantum Chaos is to understand the distribution of mass of Laplace eigenfunctions on a given smooth Riemannian manifold in the limit as the eigenvalue tends to infinity. In this talk I will consider a Laplace operator perturbed by a delta potential (point scatterer) on the torus and describe the distribution of mass of the eigenfunctions of this operator. It turns out that in this setting, the distribution of mass of the eigenfunctions is related to properties of integers which are representable as sums of two squares. I will describe this relationship and indicate how tools from analytic number theory such as sieve methods and the theory of multiplicative functions can be used to study the relevant properties of such integers.

09/10/2017 4:30 PMQueens W316Tobias Berger (Sheffield)Paramodularity of abelian surfaces
The key ingredient in Wiles' proof of Fermat's last theorem was to establish the modularity of elliptic curves. Despite many impressive advances in the Langlands programme the analogous question of modularity for abelian varieties of dimension 2 is far from settled. I will report on work in progress with Kris Klosin on the modularity of Galois representations $G_{\mathbf{Q}} \to {\rm GSp}_4(\mathbf{Q}_p)$ that are residually reducible. I will explain, in particular, how this can be used in certain cases to verify Brumer and Kramer's paramodular conjecture for abelian surfaces over Q with a rational torsion point of order p.

16/10/2017 4:30 PMQueens W316Vanessa Miemietz (UEA)2representations of finitary 2categories
I will give an introduction to 2representation theory and will give an overview of the state of the art for finitary 2categories, which should be seen as 2analogues of finitedimensional algebras.

23/10/2017 4:30 PMQueens 316Peter Humphries (UCL)The Conductor and the Newform for Representations of GL_n(R) and GL_n(C)
There is a wellknown theory of decomposing spaces of automorphic forms into subspaces spanned by newforms and oldforms, and associated to a newform is its conductor. This theory can be reinterpreted as a local statement, and generalised to GL_n, as distinguishing certain vectors in a generic irreducible admissible representation of GL_n(F), where F is a nonarchimedean local field, and associating to this representation a conductor (or rather, a conductor exponent). Such a local theory was previously not well understood for archimedean fields. In this talk, I will introduce this theory in this hitherto unexplored setting.

13/11/2017 4:30 PMQueens W316Rachel Newton (Reading)Counting failures of a localglobal principle
The search for rational solutions to polynomial equations is ongoing for more than 4000 years. Modern approaches try to piece together 'local' information to decide whether a polynomial equation has a 'global' (i.e. rational) solution. I will describe this approach and its limitations, with the aim of quantifying how often the localglobal 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.

20/11/2017 4:30 PMQueens W316Matthew Fayers (QMUL)Irreducible projective representations of symmetric groups which remain irreducible in characteristic 2
For any finite group G and any prime p, it is interesting to ask which ordinary irreducible representations remain irreducible modulo p. For the symmetric and alternating groups this problem was solved several years ago. Here we look at projective representations of symmetric groups, or equivalently representations of double covers of symmetric groups, focussing on characteristic 2 (which behaves very differently from odd characteristic). I'll give the classification of which irreducibles remain irreducible in characteristic 2, and describe some of the methods used in the proof. I'll assume some basic knowledge of representation theory, but I'll introduce projective representations and double covers from scratch.

04/12/2017 4:30 PMQueens W316Tomasz Popiel (QMUL)TBA

26/01/2015 4:30 PM103Sarah Brodsky (TU Berlin)Moduli of Tropical Plane Curves
Tropical curves have been studied under two perspectives; the first perspective defines a tropical curve in terms of the tropical semifield T=(R∪{∞}, max, +), and the second perspective defines a tropical curve as a metric graph with a particular weight function on its vertices. Joint work with Michael Joswig, Ralph Morrison, and Bernd Sturmfels, we study which metric graphs of genus g can be realized as smooth, plane tropical curves of genus g with the motivation of understanding where these two perspectives meet.
Using Polymake, TOPCOM, and other computational tools, we conduct our study by constructing a map taking smooth, plane tropical curves of genus g into the moduli space of metric graphs of genus g and studying the image of this map. In particular, we focus on the cases when g=2,3,4,5. In this talk, we will introduce tropical geometry, discuss the motivation for this study, our methodology, and our results.

30/10/2017 4:30 PMQueens W316Leonard Soicher (QMUL)Synchronization, primitive permutation groups, and graph colouring exploiting symmetry
The notion of a synchronizing permutation group arose from attempts to prove the longstanding Černý conjecture in automata theory. The class of synchronizing permutation
groups is of interest in its own right, and lies strictly between the classes of finite primitive permutation groups and finite 2transitive groups. I will discuss my recent determination of the synchronizing permutation groups of degree at most 255, using my newly developed algorithms and programs for proper vertexkcolouring a graph making use of that graph's automorphism group.This seminar may be of interest to combinatorialists as well as algebraists.

26/01/2015 4:30 PM103Sarah Brodsky (TU Berlin)Moduli of Tropical Plane Curves
Tropical curves have been studied under two perspectives; the first perspective defines a tropical curve in terms of the tropical semifield T=(R∪{∞}, max, +), and the second perspective defines a tropical curve as a metric graph with a particular weight function on its vertices. Joint work with Michael Joswig, Ralph Morrison, and Bernd Sturmfels, we study which metric graphs of genus g can be realized as smooth, plane tropical curves of genus g with the motivation of understanding where these two perspectives meet.
Using Polymake, TOPCOM, and other computational tools, we conduct our study by constructing a map taking smooth, plane tropical curves of genus g into the moduli space of metric graphs of genus g and studying the image of this map. In particular, we focus on the cases when g=2,3,4,5. In this talk, we will introduce tropical geometry, discuss the motivation for this study, our methodology, and our results.

02/03/2015 4:30 PMNo seminar ― School colloquium

01/02/2016 4:30 PM103Goran Malić (Manchester)Maps on surfaces, matroids and Galois theory
Let M be a map on a connected, closed and orientable surface X. If B is a subset of the edgeset of M such that X\B is connected, then we say that B is a base of M. The collection of all bases of M form a deltamatroid, also known as a Lagrangian matroid. Analogously to matroids, there are two rich families of Lagrangian matroids: those that arise from embedded graphs, and those that arise from maximal isotropic subspaces of symplectic vector spaces.
Aside from the usual contraction and deletion operations, Lagrangian matroids admit twists; in the case of embedded graphs, twists of Lagrangian matroids correspond to the operation of partial duality, introduced by Chmutov in 2009. A partial dual of a map M is a map with only some of the edges dualised, and it can be interpreted as an intermediate step between M and its dual map M*.
In this talk I shall explain the relationship between maps, Lagrangian matroids, their twists, and partial duals. I shall also talk about a family of abstract tropical curves that arises from a map and its partial duals, and how it fits with the Galoistheoretic aspect of maps on surfaces (in the sense of Grothendieck's programme on dessins d'enfants).

08/02/2016 4:30 PM103Vincent Pilaud (CNRS & LIX, École Polytechnique)Brick polytopes, lattice quotients, and Hopf algebras
This talk is motivated by the deep connections between the combinatorial properties of permutations, binary trees, and binary sequences. Namely, classical surjections from permutations to binary trees (BST insertion) and from binary trees to binary sequences (canopy) yield:
∙ lattice morphisms from the weak order, via the Tamari lattice, to the boolean lattice;
∙ normal fan coarsenings from the permutahedron, via Loday's associahedron, to the parallelepiped generated by the simple roots;
∙ Hopf algebra inclusions from MalvenutoReutenauer's algebra, via LodayRonco's algebra, to Solomon's descent algebra.
In this talk, we present an extension of this framework to acyclic ktriangulations of a convex (n+2k)gon, or equivalently to acyclic pipe dreams for the permutation (1, …, k, n+k, …, k+1, n+k+1, …, n+2k). These objects are in bijection with the classes of the congruence of the weak order on S_n defined as the transitive closure of the rewriting rule U a c V_1 b_1 ⋯ V_k b_k W = U c a V_1 b_1 ⋯ V_k b_k W, for letters a < b_1, …, b_k < c and words U, V_1, …, V_k, W on [n]. It enables us to transport the known lattice and Hopf algebra structures from the congruence classes to these acyclic pipe dreams. We will describe the cover relations in this lattice and the product and coproduct of this algebra in terms of pipe dreams. We will also recall the connection to the geometry of the brick polytope. 
22/02/2016 4:30 PM103Yankı Lekili (King's)Koszul duality patterns in Floer theory
Abstract: We study symplectic invariants of the open symplectic manifolds X_Γ obtained by plumbing
cotangent bundles of 2spheres according to a plumbing tree Γ. For any tree Γ, we calculate
(DG)algebra models of the Fukaya category F(X_Γ) of closed exact Lagrangians in X_Γ and the
wrapped Fukaya category W(X_Γ). When Γ is a Dynkin tree of type An or Dn (and conjecturally
also for E6 , E7, E8 ), we prove that these models for the Fukaya category F(X_Γ) and W(X_Γ) are
related by (derived) Koszul duality. As an application, we give explicit computations of symplectic
cohomology of X_Γ for Γ = An, Dn , based on the Legendrian surgery formula. In the
case that Γ is nonDynkin, we merely obtain a spectral sequence that converges to symplectic
cohomology whose E2 page is given by the Hochschild cohomology of the preprojective algebra
associated to the corresponding Γ. This is joint work with Tolga Etgü. 
30/03/2015 5:30 PMNo seminar ― School colloquium

05/10/2015 5:30 PM(No seminar – School Colloquium)

19/10/2015 5:30 PM(No seminar – School Colloquium)

07/12/2015 4:30 PM(No seminar – School Colloquium)

04/12/2017 4:30 PMQueens W316Tomasz Popiel (QMUL)The symmetric representation of lines in PG(F^3 ⊗ F^3)
Tensors have numerous applications in areas such as complexity theory and data analysis, where it is often necessary to understand ‘decompositions’ and/or ‘canonical forms’ of tensors in certain tensor product spaces. Such problems are often studied over the complex numbers, but there are also reasons to to study them over finite fields, including connections with classifications of semifields. In this talk, I will discuss the following problem. Consider the vector space V of 3x3 matrices over a finite field F, i.e. the tensor product of F^3 with itself. The 1dimensional subspaces spanned by the fundamental (or rank1) tensors in V form the socalled Segre variety in the projective space PG(V), and the setwise stabiliser G in PGL(V) of this variety may be identified with PGL(3,F) acting via g in G taking a matrix representative A to g^TAg. The Gorbits of points and lines in the ambient projective space PG(V) were determined by Michel Lavrauw and John Sheekey (Linear Algebra Appl. 2015). I will discuss joint work with Michel Lavrauw in which we determine which of the Gline orbits can be represented by symmetric 3x3 matrices, i.e. we classify the orbits of lines in PG(V) under the setwise stabiliser K of the socalled Veronese variety. Interestingly, several of the Gline orbits that have such ‘symmetric representatives’ split under the action of K, and in many cases this splitting depends on the characteristic of F. Connections are also drawn with old work of Jordan, Dickson and Campbell on the classification of ternary quadratic forms.

08/10/2018 4:30 PMQueens' Building, Room: W316Charles R. LeedhamGreen (QMUL)Condorcet domains
A Condorcet domain of degree $d$ is a subset of the symmetric group of degree $d$ satisfying a condition that relates to the mathematics of choice. I have no interest in the mathematics of choice, but these objects turn out to have interesting properties.
The main challenge has been to find large Condorcet domains of given degree, and we have been using various techniques, from supercomputers to cardboard, with some theoretical ideas thrown in, to break some longstanding records.
This is joint work with Dolica AkelloEgwel, Klas Markstrom, and Søren Riis.

15/10/2018 4:30 PMQueens' Building, Room: W316Yegor Stepanov (QMUL)Octonions, Albert vectors and the groups of type E_6(F).
We discuss a uniform construction of the groups $\mathrm{E}_6(F)$, where $F$ is any field. In particular, we illuminate some of the subgroup structure of these groups.

22/10/2018 4:30 PMQueens' Building, Room: W316Adam Harper (Warwick).Prime number races with very many competitors.
The prime number race is the competition between different coprime residue classes mod $q$ to contain the most primes, up to a point $x$. Rubinstein and Sarnak showed, assuming two $L$function conjectures, that as $x$ varies the problem is equivalent to a problem about orderings of certain random variables, having weak correlations coming from number theory. In particular, as $q \rightarrow \infty$ the number of primes in any fixed set of $r$ coprime classes will achieve any given ordering for $\sim1/r!$ values of $x$. In this talk I will try to explain what happens when $r$ is allowed to grow as a function of $q$, concentrating on the lack of uniformity that can arise. This is joint work with Kevin Ford and Youness Lamzouri.

29/10/2018 5:00 PMQueens' Building, Room: W316Christopher D. Bowman (Kent).Unitary simples of symmetric groups, Hecke algebras, and Cherednik algebras. NONSTANDARD start time of 5 pm.
In this talk we review some new results concerning the structure of simple modules (and in particular unitary simple modules) for symmetric groups and their deformations over fields of arbitrary characteristic. If time permits, we will discuss applications in calculating resolutions, (graded) Betti numbers, and CM regularity of certain highly symmetric algebraic varieties.

05/11/2018 4:30 PMQueens' Building, Room: W316Andrew Booker (Bristol)Two results on Artin representations
In 1923, Artin posed a conjecture about the finitedimensional complex representations of Galois groups of number fields (now called Artin representations). This conjecture, most cases of which are still open, is one of the main motivating problems behind the Langlands programme. After a brief introduction to these topics, I will discuss two recent related results. The first, joint with Min Lee and Andreas Strömbergsson, is a classification of the 2dimensional Artin representations of small conductor, based on some new explicit versions of the Selberg trace formula. The second extends theorems of Sarnak and Brumley to the effect that certain modular forms with algebraic Fourier coefficients must be associated to Artin representations.

19/11/2018 4:30 PMQueens' Building, Room: W316Simon R. Blackburn (Royal Holloway)The Walnut Digital Signature Algorithm
Walnut is a digital signature algorithm that was first proposed in 2017 by Anshel, Atkins, Goldfeld and Gunnells. The algorithm is based on techniques from braid group theory, and is one of the submissions for the highprofile NIST Post Quantum Cryptography standardisation process. The talk will describe Walnut, and some of the attacks that have been mounted on it. No knowledge of cryptography or the braid group will be assumed. Based on joint work with Ward Beullens (KU Leuven).

03/12/2018 4:30 PMQueens' Building, Room: W316Peter J. Cameron (St Andrews)Permutation groups and regular semigroups
How does the group of units shape the structure of a semigroup? This is a question on which progress was very slow, but the increased knowledge of finite groups resulting from the Classification of Finite Simple Groups has opened new lines of progress. I will talk mainly about the following question. What properties of a permutation group $G$ guarantee that, for all nonpermutations $s$, or all in some specified class (say, rank $k$, or given image), the semigroup $\langle G,s\rangle$ is regular, or has some other property of interest?

27/09/2019 3:00 PMG.O. Jones Building, Room 410 A&BShu Sasaki (QMUL)Serre's conjecture about weights of mod $p$ modular forms
In 1987, J.P. Serre made some remarkably precise conjectures (known commonly as `Serre's conjecture') about weights and levels of twodimensional (modular) mod $p$ Galois representations of the absolute Galois group of $\mathbb{Q}$. They have been completely proved by C. Khare and J.P. Wintenberger (2009) building on the work of many mathematicians (A. Wiles, R. Taylor, and M. Kisin to name a few), but they have also inspired a good deal of new mathematics.
I will explain what Serre's conjecture actually says and what it means in the context of the Langlands philosophy. I will then discuss my recent joint work with F. Diamond about a (geometric) generalisation of Serre's conjecture to the Hilbert case, while focusing more on its combinatorial/algebraic aspects. 
04/10/2019 3:00 PMMathematics Building, Room: MB503Sinéad Lyle (UEA)Representations of the full transformation monoid
The transformation monoid $T_n$ consists of all maps from the set $\{1, 2, \ldots, n\}$ to itself. Consider the algebra $\mathbb{C} T_n$. This algebra has dimension $n^n$ and it is not semisimple for $n \geq 2$. However it is standardly based (in the sense of Du and Rui) and its representations are controlled by those of its maximal subgroups, the symmetric groups $S_d$ where $1 \leq d \leq n$. In this talk, we shall discuss some of the facts which are known about the representations of the transformation monoid and how they are related to those of the symmetric groups.

11/10/2019 3:00 PMMathematics Building, Room: MB503Konstanze Rietsch (KCL)The tropical critical point and toric mirror symmetry (joint with Jamie Judd)
Call a (generalised) Puiseaux series positive if the leading term is a positive real number. Suppose we are given a Laurent polynomial f(x_1,..., x_n) over the field of generalised Puiseaux series, and that f has positive coefficients. We show that under a mild hypothesis on the Newton polytope such a Laurent polynomial has a unique positive critical point. We apply this result to toric varieties. Suppose X is a projective toric variety with moment polytope P. Then one can associate to X a Laurent polynomial f by mirror symmetry. The unique positive critical point of f gives rise by tropicalisation to a canonically associated point in the interior of P. We interpret this point in two ways.

18/10/2019 3:00 PMMathematics Building, Room: MB503Stacey Law (Oxford)Linear characters of Sylow subgroups of the symmetric group
Let $p$ be an odd prime and let $n$ be a natural number. We determine the irreducible constituents of the permutation module induced by the action of the symmetric group $S_n$ on the cosets of a Sylow $p$subgroup $P = P_n$. In the course of this work, we also prove a symmetric group analogue of a wellknown result of Navarro for $p$solvable groups on a conjugacy action of $N_G(P)$. Before describing some consequences of these results, we will give an overview of the background and recent related results in the area.

25/10/2019 3:00 PMMathematics Building, Room: MB503Ian Chiswell (QMUL)Ordered groups and related classes
Although the idea of an ordered group goes back to the 19th century, they have been of interest in recent decades because of connections with topology (eg existence of certain foliations in $3$manifolds, knot theory, braid groups). More general classes have since been introduced (such as rightordered groups and unique product groups). We consider the relations between these classes and the more recently introduced class of diffuse groups, which has several characterisations.

01/11/2019 3:00 PMMathematics Building, Room: MB503Paul Flavell (Birmingham)A characteristic subgroup of a $Qd(p)$free group
Suppose that G is a no trivial finite group, p is a prime and P is a Sylow psubgroup of G. Let Q be the largest normal psubgroup of G and suppose that C(Q) \leq Q. Clearly, P contains a nontrivial normal subgroup that is normal in G, for example Q, but does P contain a nontrivial characteristic subgroup that is normal in G? This is an important question whose answer has several applications, for example in the revised proof of the Odd Order Theorem by Bender, Glauberman, and Peterfalvi.
Let Qd(p) denote the semidirect product of SL_2(p) with its natural module. Then Qd(p) demonstrates that the answer is no in general – but it turns out that this is the only obstruction. Glauberman’s celebrated ZJTheorem (1966) gives an affirmative answer for groups that do not involve Qd(p) in the case that p is odd. Glauberman’s proof is quite complex. It was suspected that the answer is again yes in the case p=2 provided G does not involve Qd(2) (which is isomorphic to S_4). This case turned out to be even more complex than for odd p. Indeed a proof had to wait until 1996 with Stellmacher’s celebrated S_4free Theorem. More recently Glauberman and Solomon gave a much simplified proof for odd p. We will report on joint work with Stellmacher that gives a new proof for p=2.

08/11/2019 3:00 PMMathematics Building, Room: MB503Rowena Paget (Kent)Some questions about plethysm
The symmetric group S_{mn} acts naturally on the collection of set partitions of a set of size mn into n sets each of size m. The irreducible constituents of the associated ordinary character are largely unknown; in particular they are the subject of the longstanding Foulkes Conjecture. There are equivalent reformulations using polynomial representations of infinite general linear groups or using plethysms of symmetric functions. I will review plethysm from these three perspectives before presenting recent work with Chris Bowman and another project with Mark Wildon.

15/11/2019 3:00 PMMathematics Building, Room: MB503Abhishek Saha (QMUL)Critical Lvalues and congruence primes for Siegel modular forms of degree 2
I will discuss some recent work where we obtain an explicit pullback formula that gives an integral representation for the twisted standard Lfunction for a holomorphic vectorvalued Siegel cusp form of degree n and arbitrary level. By specializing our integral representation to the case n=2, we prove an algebraicity result for the critical Lvalues in that case. I will also talk of some ongoing work that extends this idea to prove congruences between Hecke eigenvalues of two Siegel cusp forms modulo primes dividing a certain quotient of Lvalues. All of this is joint work with Ameya Pitale and Ralf Schmidt.

29/11/2019 3:00 PMMathematics Building, Room: MB503James Maynard (Oxford)On the DuffinSchaeffer conjecture
How well can you approximate reals with fractions coming from some chosen set? In general this problem is impossibly hard, but almost 80 years ago Duffin and Schaeffer conjectured that if you allow for a small exceptional set, there is actually a beautiful simplicity: regardless of the setup, either almost all reals can be approximated or almost none, and there is a simple way of telling which case holds. I'll talk about recent work with D. Koukoulopoulos which establishes this conjecture.

06/12/2019 3:00 PMMathematical Building, Room: MB503Hung Bui (Manchester)Analytic rank of automorphic Lfunctions
The famous Birch & SwinnertonDyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the soclaeed analytic rank, which is the order of vanishing of the asociated Lfunction at the central point. In this talk, we shall discuss the analytic rank of automorphic Lfunctions in an "alternate universe".

13/12/2019 3:00 PMMathematics Building, Room: MB503Anna Seigal (Oxford)Tensors under congruence action
Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry, and numerical optimization to the group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. Based on joint work with Max Pfeffer and Bernd Sturmfels.

24/01/2020 3:00 PMMathematics Building, Room: MB503Joni Teravainen (Oxford)Higher order uniformity of the Möbius function
In a recent work, Matomaki, Radziwill and Tao showed that the Mobius function is discorrelated with linear exponential phases on almost all short intervals. I will discuss joint work where we generalize this result to ``higher order phase functions", so as a special case the Mobius function is shown not to correlate with polynomial phases on almost all short intervals. As an application, we show that the number of sign patterns that the Liouville function takes grows superpolynomially.

31/01/2020 3:00 PMMathematics Building, Room: MB503Sarah Zerbes (UCL)Euler systems and the Bloch—Kato conjecture for GSp(4)Euler systems are compatible families of Galois cohomology classes attached to a global Galois representation, and they play an important role in proving cases of the Bloch—Kato conjecture.In my talk, I will review the construction of an Euler system attached to the spin representation of a genus 2 Siegel modular form. I will then sketch a proof of new cases of the Bloch—Kato conjecture in analytic rank 0. This is the consequence of an explicit reciprocity law, relating the Euler system to values of a padic Lfunction. This is joint work with David Loeffler and Chris Skinner.

14/02/2020 3:00 PMMathematics Building, Room: MB503Behrang Noohi (QMUL)Categorical calculus and representation theory
Using category theory, one can rephrase basic concepts of representation theory of groups in a geometric language, allowing one to import ideas from geometry to prove results in representation theory. For instance, an analogue of Stokes' theorem in calculus gives rise to interesting formulas in representation theory, some of which happen to be related to topological quantum field theory and twisted Ktheory. I will not speak about the latter two (to keep the talk elementary), but instead will mention some simple applications to (twisted) representation theory of finite groups.

20/03/2020 3:00 PMMathematics Building, Room: MB503Jolanta Marzec (Darmstadt) CANCELLEDConstruction of Poincarétype series by generating kernels
Let $\Gamma\subset\text{\rm PSL}_2(\mathbb{R})$ be a Fuchsian group of the first kind whose fundamental domain $\Gamma\backslash\mathbb{H}$ is of finite volume, and let $\widetilde\Gamma$ be its cover in $\SL_2(\mathbb{R})$. Consider the space of twice continuously differentiable, squareintegrable functions on $\mathbb{H}$, which transform in a suitable way with respect to a multiplier system of weight $k\in\mathbb{R}$ under the action of $\widetilde\Gamma$. The space of such functions admits action of the hyperbolic Laplacian $\Delta_k$ of weight $k$. Following an approach of Jorgenson, von Pippich and Smajlovi\'c (where $k=0$), we use spectral expansion associated to $\Delta_k$ to construct wave distribution and then identify conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincar\'etype series. As we will show, one of advantages of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive supnorm bounds for the eigenfunctions in the discrete spectrum of $\Delta_k$. This is joint work with Y. Kara, M. Kumari, K. Maurischat, A. Mocanu and L. Smajlovi\'c.

07/02/2020 3:00 PMMathematics Building, Room: MB503Matthew Young (MPIM, Bonn)Characters in higher Real representation theory
In the first part of the talk I will introduce the Real (in the sense of Atiyah) representation theory of a higher finite group on a higher category. I will then describe a geometric character theory for higher Real representations and explain its relevance to problems in the topology of unoriented manifolds. Partially based on joint works with Behrang Noohi and Dmitriy Rumynin.

27/03/2020 3:00 PMMathematics Building, Room: MB503Clément Dupont (Montpellier) CANCELLED

28/02/2020 3:00 PMMathematics Building, Room: MB503Han Wu (QMUL)On Motohashi's formula
We offer a new perspective of the proof of a Motohashitype formula relating the fourth moment of Lfunctions for GL_1 with the third moment of Lfunctions for GL_2 over number fields, studied earlier by MichelVenkatesh and Nelson. Our main tool is a new type of pretrace formula with test functions on Mat_2(\A) instead of GL_2(\A), on whose spectral side the matrix coefficients are the standard GodementJacquet zeta integrals.

06/03/2020 3:00 PMMathematics Building, Room: MB503CANCELLED

21/02/2020 3:00 PMMathematics Building, Room: MB503Jessica Fintzen (Cambridge)Representations of padic groups
The Langlands program is a farreaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of padic groups. I will provide an overview of our understanding of the representations of padic groups, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.

13/03/2020 3:00 PMMathematics Building, Room: MB503Alice Pozzi (UCL)The values of the DedekindRademacher cockle at real multiplication points
A rigid meromorphic cocycle is a class in the first cohomology of the group SL_2(Z[1/p]) acting on the nonzero rigid meromorphic functions on the Drinfeld padic upper half plane by Mobius transformation. Rigid meromorphic cocycles can be evaluated at points of real multiplication, and their values conjecturally lie in the ring class field of real quadratic fields, suggesting striking analogies with the classical theory of complex multiplication.
In this talk, we study derivatives of a padic family of Hilbert Eisenstein series, in analogy to the work of Gross and Zagier. We relate its diagonal restriction to certain values of rigid meromorphic cocycles at real multiplication points. We explain how a refinement of this strategy, relying on the study of certain Galois deformation rings, can be used to show the algebraicity of the real multiplication values of the DedekindRademacher rigid meromorphic cocyle. This is joint work with Henri Darmon and Jan Vonk.

07/10/2020 1:00 PMZoomYukako Kezuka (Regensburg)The arithmetic of twists of the Fermat elliptic curveThe BirchSwinnertonDyer conjecture is one of the most celebrated open problems in number theory. In this talk, I will explain some recent progress on the study of this conjecture for the classical family of elliptic curves E of the form x^3+y^3=N for a positive integer N prime to 3. They are cubic twists of the Fermat elliptic curve x^3+y^3=1, and admit complex multiplication by the ring of integers of Q(sqrt{3}). First, I will explain the Tamagawa number divisibility satisfied by the central Lvalues, and exhibit a curious relation between the 3part of the TateShafarevich group of E and the number of prime divisors of N which are inert in Q(sqrt{3}). I will then explain my joint work with Yongxiong Li where we study in more detail the case when N=2p or 2p^2 for an odd prime number p congruent to 2 or 5 modulo 9. For these curves, we establish the 3part of the BirchSwinnertonDyer conjecture and a relation between the ideal class group of Q(sqrt[3]{p}) and the 2Selmer group of E, which can be used to study nontriviality of the 2part of the TateShafarevich group.

14/10/2020 1:00 PMZoomJay Taylor (Southern California)Unitriangularity of Decomposition Matrices of Unipotent Blocks
(joint work with O. Brunat and O. Dudas). A distinguishing feature of the representation theory of finite groups is the ability to take an (ordinary) irreducible representation over a field of characteristic zero and reduce modulo a prime to get a (modular) representation over a field of characteristic p>0. Whilst the original ordinary representation was irreducible the resulting modular representation may be far from irreducible. The (p)decomposition matrix is a rectangular matrix with rows labelled by ordinary irreducible representations and columns labelled by modular irreducible representations. A row of the matrix gives the multiplicities of the modular irreducible representations in a composition series for the reduced ordinary representation.
Understanding the decomposition matrix is of central importance in the modular representation theory of finite groups. The focus of this talk will be the case of finite reductive groups G(q), such as GL_n(q), with the representations taken over a field whose characteristic does not divide q. We will present a recent result showing that, under mild restrictions on p and q, the decomposition matrix has a particular unitriangular shape. 
21/10/2020 4:00 PMZoom (819 9044 6856)Rob Silversmith (Northeastern)Studying subschemes of affine/projective space via matroidsGiven a homogeneous ideal I in a polynomial ring, one may apply the following combinatorial operation: for each degree d, make a list of all subsets S of the set of degreed monomials such that S is the set of nonzero coefficients of an element of I. For each d, this set of subsets is a combinatorial object called a matroid. As d varies, the resulting sequence of matroids is called the tropicalization of I.I will discuss some of the many questions one can ask about tropicalizations of ideals, and how they are related to some classical questions in combinatorial algebraic geometry, such as the classification of torus orbits on Hilbert schemes of points in C^2. Some unexpected combinatorial objects appear: e.g. when studying tropicalizations of subschemes of P^1, one is led to Schur polynomials and binary necklaces.

28/10/2020 1:00 PMZoomAlex Betts (Bonn)Galois and the LawrenceVenkatesh method
In a letter to Faltings, Grothendieck defined the set of ``Galois sections'' associated to a curve of genus at least 2 over a number field, which is conjectured to be equal to the set of rational points. However, this set remains very mysterious, and we do not even know  except in a few specific cases  whether it is finite. In this talk, I will discuss ongoing work with Jakob Stix in which we obtain partial results in this direction. The method we employ is based on the recent reproof of the Mordell Conjecture by Brian Lawrence and Akshay Venkatesh.

04/11/2020 1:00 PMZoomHanneke Wiersema (KCL)Minimal weights of mod p Galois representations
The strong form of Serre's conjecture states that every twodimensional continuous, odd, irreducible mod p representation of the absolute Galois group of Q arises from a modular form of a specific minimal weight, level and character. In this talk we use modular representation theory to prove the minimal weight is equal to a notion of minimal weight inspired by work of Buzzard, Diamond and Jarvis. Moreover, using the BreuilMézard conjecture we give a third interpretation of this minimal weight as the smallest k>1 such that the representation has a crystalline lift of HodgeTate type (0, k1). Finally, we will report on work in progress where we study similar questions in the more general setting of mod p Galois representations over a totally real field.

11/11/2020 4:00 PMZoom (819 9044 6856)Mandi Schaeffer Fry (Denver)The McKay—Navarro Conjecture: The Conjecture That Keeps on Giving!
The McKay conjecture is one of the main open conjectures in the realm of the localglobal philosophy in character theory. It posits a bijection between the set of irreducible characters of a group with p’degree and the corresponding set in the normalizer of a Sylow psubgroup. In this talk, I’ll give an overview of a refinement of the McKay conjecture due to Gabriel Navarro, which brings the action of Galois automorphisms into the picture. A lot of recent work has been done on this conjecture, but possibly even more interesting is the amount of information it yields about the character table of a finite group. I’ll discuss some recent results on the McKay—Navarro conjecture, as well as some of the implications the conjecture has had for other interesting charactertheoretic problems.

18/11/2020 1:00 PMZoomDustin Clausen (Copenhagen)Condensed sets
I'll give an introduction to the category of condensed sets, whose objects are similar to topological spaces but whose formal properties are similar to those of the category of sets. I'll give the definition, explain the relation to topological spaces, and sketch how one can make some computations. This is joint work with Peter Scholze.

25/11/2020 1:00 PMZoomDustin Clausen (Copenhagen)Nonarchimedean analysis and geometry
Buliding on the previous talk, I'll define a full subcategory of condensed abelian groups called "solid" abelian groups, and explain how it yields a very convenient base category for nonarchimedean analysis and geometry.

02/12/2020 1:00 PMZoomPaul Nelson (Zurich)Theta functions, fourth moments of eigenforms and the supnorm problem
I will discuss joint work with Raphael Steiner and Ilya Khayutin in which we study the sup norm problem for GL(2) eigenforms in the squarefree level aspect. Unlike the standard approach to the problem via arithmetic amplification following IwaniecSarnak, we apply a method, introduced earlier in other aspects by my collaborators, which consists of identifying a fourth moment over a family of eigenforms evaluated at the point of interest with the L^2norm of a theta function defined using the correspondence of Eichler, Shimizu and JacquetLanglands. After solving some counting problems (involving both "linear" sums as in traditional approaches and new "bilinear" sums), we obtain a bound comparable to the fourth root of the volume, improving upon the trivial square root bound and the nontrivial cube root bound established by HarcosTemplier and BlomerMichel. I will describe the proof in the simplest case.

09/12/2020 1:00 PMZoomTobias Berger (Sheffield)Oddness of limits of automorphic Galois representations
For classical modular forms f one knows that the associated Galois representation $\rho_f:G_{\mathbf{Q}} \to {\rm GL}_2(\overline{\mathbf{Q}}_p)$ is odd, in the sense that ${\rm det}(\rho(c))=1$ for any complex conjugation $c$.
There is a similar parity notion for ndimensional Galois representations which are essentially conjugate selfdual. In joint work with Ariel Weiss (Hebrew University) we prove that the Galois representations associated to certain irregular automorphic representations of U(a,b) are odd, generalizing a result of BellaicheChenevier in the regular case.
I will explain our result and discuss its proof, which uses V. Lafforgue's notion of pseudocharacters and invariant theory.

16/12/2020 1:00 PMZoomJack Shotton (Durham)Shimura curves and Ihara's lemma
Ihara's lemma is a statement about the structure of the mod l cohomology of modular curves that was the key ingredient in Ribet's results on level raising. I will motivate and explain its statement, and then describe joint work with Jeffrey Manning on its extension to Shimura curves.

12/03/2021 4:00 PMZoomManami Roy (Fordham)Counting cuspidal automorphic representations of GSp(4)
There is a wellknown connection between the Siegel modular forms of degree 2 and the automorphic representations of GSp(4). Using this relationship and the available dimension formulas for the spaces of Siegel cusp forms of degree 2, we count a specific set of cuspidal automorphic representations of GSp(4). Consequently, we obtain an equidistribution result for a family of cuspidal automorphic representations of GSp(4). This kind of equidistribution result is analogous to the socalled vertical SatoTate conjecture for GL(2). The method of counting automorphic representations is also helpful for computing dimensions of some spaces of Siegel cusp forms, which are not yet known. The talk is based on a joint work with Ralf Schmidt and Shaoyun Yi.

19/03/2021 4:00 PMZoomAriel Pacetti (Aveiro)Modularity of abelian surfacesThe paramodular conjecture states a relation between rational abelian surfaces (without extra endomorphisms) and some siegel modular forms. It is a generalization of the 1dimensional case, namely the ShimuraTaniyama conjecture. In this talk I will explain the conjecture, its relation to modularity of elliptic curves over quadratic fields, the state of the art of the conjecture and some mention some proven cases. If time allows, I will present a Bianchi newform over Q(\sqrt{7}) with rational eigenvalues which is attached to an abelian surface over Q( √ −7) (and explain its relation with the conjecture).

26/02/2021 4:00 PMZoomJun Su (Cambridge)Arithmetic group cohomology with generalised coefficientsCohomology of arithmetic subgroups, with algebraic representations as coefficients, has played an important role in the construction of Langlands correspondence. Traditionally the first step to access these objects is to view them as cohomology of sheaves on locally symmetric spaces and hence connect them with spaces of functions. However, sometimes infinite dimensional coeffients also naturallhy arise, e.g. when you try to attach elliptic curves to weight 2 eigenforms on GL_2/an imaginary cubic field, and the sheaf theoretic viewpoint might no longer be fruitful. In this talk we'll explain a very simple alternative understanding of the connection between arithmetic group cohomology (with finite dimensional coefficients) and function spaces, and discuss its application to infinite dimensional coefficients.

16/04/2021 4:00 PMZoomAshay Burungale (Caltech)An even parity instance of the Goldfeld conjecture
In 1979 D. Goldfeld conjectured: 50% of the quadratic twists of an elliptic curve over the rational numbers have analytic rank zero. We present the first instance  the congruent number elliptic curves (joint with Y. Tian).

02/04/2021 4:00 AMZoomShuichiro Takeda (Missouri)Multiplicityatmostone theorem for GSpin and GPin
Let V be a quadratic space over a nonarchimedean local field of characteristic 0. The orthogonal group O(V) and the special orthogonal group SO(V) have a unique nontrivial GL_1 extension called GPin(V) and GSpin(V), respectively. Let W\subseteq V be a subspace of codimension 1. Then there are natural inclusions GPin(W)\subseteq GPin(V) and GSpin(W)\subseteq GSpin(V). One can then consider the GanGrossPrasad (GGP) periods for GPin and GSpin. In this talk, I will talk about the multiplicityatmostone theorem for the local GGP periods for GPin and GSpin.

26/03/2021 4:00 PMZoomRobin Bartlett (Munster)BreuilMezard identities in moduli spaces of BreuilKisin modules
The BreuilMezard conjectures predicts relations between certain cycles in the moduli space of mod p Galois representations, in terms of the representation theory of GLn(Fq). In this talk I will consider the special case where the cycles in question come from two dimensional crystalline representations with small HodgeTate weights. Under these assumptions I will explain how the topological aspects of these identities can be obtained from analagous identities appearing, first inside the affine Grassmannian, and then in moduli spaces of BreuilKisin modules.

01/10/2021 2:30 PMMB503Tara Fife (QMUL)
To each circuit of a matroid, we can define a tropical hyperplane. The intersection of these hyperplane yields a tropical linear space, namely the Bergman fan of the matroid. If the tropical hyperplanes associated with a subset, $\mathscr{B}$ of the circuit set of $M$ is the same tropical linear space, then $\mathscr{B}$ is a tropical basis of $M$. Tropical basis need not be minimal. Josephine Yu and Debbie Yuster described minimal tropical basis for several classes of matroids and asked for explicit minimal tropical basis for the class of transversal matroids. The talk will begin with an introduction to matroids, including a careful definition of tropical basis. We give explicit minimal tropical basis for two special subclasses of transversal matroids.

08/10/2021 2:30 PMOnlineKoji Shimizu (Berkeley)Robba cohomology for dagger spaces in positive characteristicWe will discuss a padic cohomology theory for rigid analytic varieties with overconvergent structure (dagger spaces) over a local field of characteristic p. After explaining the motivation, we will define a site (Robba site) and discuss its basic properties.

15/10/2021 2:30 PMMB503Diego Millan Berdasco (QMUL) 14301500, and Tim Davis (QMUL) 15001530(DMB) Problems on decomposition numbers of the symmetric group, and (TD) The Fourier coefficients of Hilbert modular forms at cusps
(DMB) The most important open problem in the representation theory of the symmetric group in positive characteristic is finding the decomposition numbers; i.e., the multiplicity of the simple modules as composition factors of the Specht modules. In characteristic 0 the Specht modules are just the simple modules of the symmetric group algebra, but in positive characteristic they may no longer be simple, nor the algebra semisimple. We will survey briefly the rich interplay between representation theory and combinatorics of integer partitions, present recent and ongoing work on decomposition numbers and discuss new conjectures arising from these results.
(TD) In this talk we give an answer to the following question: given a Hilbert newform and a matrix in the Hilbert modular group what is the explicit number field which contains all the Fourier coefficients of the Hilbert newform at that cusp? This generalises a result by Brunault and Neururer who answered this question in the setting of classical newforms. We will give an overview of the method used to prove our result which differs from the method of Brunault and Neuruer and relies on the properties of local Whittaker newforms.

22/10/2021 3:00 PMOnlineMax Kutler (Ohio)Motivic and topological zeta functions of matroids
We associate to any matroid a motivic zeta function. If the matroid is representable by a complex hyperplane arrangement, then this coincides with the motivic Igusa zeta function of the arrangement. Although the motivic zeta function is a valuative invariant which is finer than the characteristic polynomial, it is not obvious how one should extract meaningful combinatorial data from the motivic zeta function. One strategy is to specialize to the topological zeta function. I will survey what is known about these functions and, timepermitting, discuss some open questions.

29/10/2021 1:00 PMMB503Andrew Booker (Bristol) 1314, and Emily Norton (Kent) 14301530(AB) A converse theorem for GL(n), and (EN) Calibrated representations of cyclotomic Hecke algebras at roots of unity
(AB) In the 1990s, Cogdell and PiatetskiShapiro proved various theorems characterising the automorphic representations of GL(n) over a number field using analytic properties of the associated RankinSelberg Lfunctions. The most well known of these assumes properties of the twists by representations of GL(n2), and was used in important applications such as the third and fourth symmetric power lifts from GL(2) by Kim and Shahidi. I will describe joint work with Krishnamurthy improving on another theorem of Cogdell and PiatetskiShapiro that uses twists by representations of GL(n1) with greatly restricted ramification.
(EN) The cyclotomic Hecke algebra is a "higher level" version of the IwahoriHecke algebra of the symmetric group. It depends on a collection of parameters, and its combinatorics involves multipartitions instead of partitions. We are interested in the case when the parameters are roots of unity. In general, we cannot hope for closedform character formulas of the irreducible representations. However, a certain type of representation called "calibrated" is more tractable: those representations on which the JucysMurphy elements act semisimply. We classify the calibrated representations in terms of their Young diagrams, give a multiplicityfree formula for their characters, and homologically construct them via BGG resolutions. This is joint work with Chris Bowman and José Simental.

05/11/2021 3:00 PMOnlineRoozbeh Hazrat (Western Sydney)Leavitt path algebras
We give a down to earth overview of these algebras which have been introduced 15 years ago and have found connections to all kind of mathematics!

12/11/2021 2:30 PMMB503Min Lee (Bristol)Effective equidistribution of rational points on expanding horospheres
In this talk, we study the behaviour of rational points on the expanding horospheres in the space of unimodular lattices. The equidistribution of these rational points is proved by Einsiedler, Mozes, Shah and Shapira (2016) and their proof uses techniques from homogeneous dynamics and relies in particular on measureclassification theorems due to Ratner. We pursue an alternative strategy based on Fourier analysis, Weil's bound for Kloosterman sums, recently proved bounds (by M. Erdélyi and Á. Tóth) for matrix Kloosterman sums, Roger’s formula and the spectral theory of automorphic functions. Our methods yield an effective estimate on the rate of convergence for a specific horospherical subgroup in any dimension.
This is a joint work with D. ElBaz, B. Huang, J. Marklof and A. Strömbergsson.

19/11/2021 2:30 PMOnlineZicheng Qian (Toronto)Moduli of FontaineLaffaille modules and a mod p localglobal compatibility result
In a joint work with D. Le, B. V. Le Hung, S. Morra and C. Park, we prove under standard TaylorWiles condition that the Hecke eigenspace attached to a mod p global Galois representation $\overline{r}$ determines the restriction of $\overline{r}$ at a place $v$ about p, assuming that $v$ is unramified over $p$ and $\overline{r}$ has a 5ngeneric FontaineLaffaille weight at $v$.

26/11/2021 2:30 PMMB503Ana Caraiani (Imperial College London)Localglobal compatibility in the crystalline case
Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GL_n/F with ptorsion coefficients. These Galois representations are expected to satisfy localglobal compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems. I will discuss joint work with J. Newton, where we establish localglobal compatibility in the crystalline case under mild technical assumptions. This relies on a new idea of using Pordinary parts, and improves on earlier results obtained in joint work with P. Allen, F. Calegari, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne in certain FontaineLaffaille cases.

03/12/2021 3:00 PMOnlineAdam Morgan (Glasgow)Integral Galois module structure of MordellWeil groups
Let E/Q be an elliptic curve, G a finite group and V a fixed finite dimensional rational representation of G. As we run over Gextensions F/Q with E(F)⊗Q isomorphic to V , how does the Z[G]module structure of E(F) vary from a statistical point of view? I will report on joint work with Alex Bartel in which we propose a heuristic giving a conjectural answer to an instance of this question, and make progress towards its proof. In the process I will relate the question to quantifying the failure of the Hasse principle in certain families of genus 1 curves, and explain a close analogy between these heuristics and Stevenhagen's conjecture on the solubility of the negative Pell equation.

10/12/2021 2:30 PMMB503Robert Kurinczuk (Sheffield)Local Langlands in families for classical groups in the banal caseThe conjectural local Langlands correspondence connects representations of padic groups to certain representations of Galois groups of local fields called Langlands parameters. In recent joint work with Dat, Helm, and Moss, we have constructed moduli spaces of Langlands parameters over Z[1/p] and studied their geometry. We expect this geometry is reflected in the representation theory of the padic group. Our main conjecture “local Langlands in families” describes the GIT quotient of the moduli space of Langlands parameters in terms of the centre of the category of representations of the padic group generalising a theorem of HelmMoss for GL(n). I will explain how after inverting the "nonbanal primes" we can prove this conjecture for the local Langlands correspondence for classical groups of Arthur, Mok, and others.

17/12/2021 3:00 PMonlineDustin Cartwright (Tennessee)Characteristic sets of matroids
A matroid is a combinatorial abstraction of the types of dependence relations that appear both as linear dependence in vector spaces and algebraic dependence in field extensions. As not all matroids can be realized in either of these ways, we can define the linear and algebraic characteristic sets of a matroid as the set characteristics of fields over which the matroid is realizable in a vector space or field extension, respectively. The focus of my talk will be the possible characteristic sets of matroids. An important tool will be the construction of algebraic matroids from the ring of endomorphisms of a 1dimensional connected algebraic group. This is joint work with Dony Varghese.

28/01/2022 3:00 PMMB503Ian Morris (QMUL)Some algebraic questions in fractal geometry
A subset of R^d is formally called selfsimilar if it is equal to the union of finitely many rescaled, translated, isometric copies of itself. If this condition is relaxed to allow the set to be equal to the union of finitely many affine images of itself then the set is instead called selfaffine. In general, selfaffine sets remain far less wellunderstood than selfsimilar sets. This talk will describe some algebraic conditions which make the dimension of a selfaffine set "defective", and finishes with some open questions of an algebraic nature which are relevant to the theory of selfaffine sets.

04/02/2022 3:00 PMMB503Rosa Winter (King's College London)Density of rational points on del Pezzo surfaces of degree 1Let X be an algebraic variety over an infinite field k. In arithmeticgeometry we are interested in the set X(k) of krational points on X. Forexample, is X(k) empty or not? And if it is not empty, is X(k) dense in Xwith respect to the Zariski topology?Del Pezzo surfaces are surfaces classified by their degree d, which is an integerbetween 1 and 9 (for d ≥ 3, these are the smooth surfaces of degree d in P^d).For del Pezzo surfaces of degree at least 2 over a field k, we know that the setof krational points is Zariski dense provided that the surface has one krationalpoint to start with (that lies outside a specific subset of the surface for degree 2).However, for del Pezzo surfaces of degree 1 over a field k, even though we knowthat they always contain at least one krational point, we do not know if the setof krational points is Zariski dense in general.I will talk about density of rational points on del Pezzo surfaces, state whatis known so far, and show a result that is joint work with Julie Desjardins,in which we give sufficient and necessary conditions for the set of krationalpoints on a specific family of del Pezzo surfaces of degree 1 to be Zariskidense, where k is finitely generated over Q.

11/02/2022 3:00 PMZoom (the link can be found in the abstract)Mercedes Rosas Celis (Universidad de Sevilla)On the quasipolynomiality of the Kronecker coefficients.The Kronecker coefficients are the structure constants for the restrictionof irreducible representations of the general linear group GL(nm,C) into irreduciblesfor the subgroup GL(n, C)xGL(m, C).I will focus on the quasipolynomial nature of the Kronecker function (the functionthat assigns to a triple of partitions, its corresponding Kronecker coefficient) usingelementary tools from polyhedral geometry. Then, I will show how to write the Kronecker function in terms of coefficients of a vector partition function, in anexplicit and selfcontained way. This approach will produce exact formulas, andan upper bound for the Kronecker coefficients in some nontrivial cases.
This is joint work with Marni Mishna, Sheila Sundaram, and Stefan Trandafir.
https://qmulacuk.zoom.us/j/81420780676?pwd=d2xJb2xncUxDWkRkVXVwRk1ZbTVpZz09Meeting ID: 814 2078 0676Passcode: 244487

18/02/2022 3:00 PMMB503James Newton (University of Oxford)Modularity over CM fields
Since the seminal works of Wiles and TaylorWiles, robust methods were developed to prove the modularity of 'polarised' Galois representations. These include, for example, those coming from elliptic curves defined over totally real number fields. Over the last 10 years, new developments in the TaylorWiles method (Calegari, Geraghty) and the geometry of Shimura varieties (Caraiani, Scholze) have broadened the scope of these methods. One application is the recent work of Allen, Khare and Thorne, who prove modularity of a positive proportion of elliptic curves defined over a fixed imaginary quadratic field. I'll review some of these developments and work in progress with Caraiani which has further applications to modularity of elliptic curves over imaginary quadratic fields.

25/02/2022 3:00 PMMB503Lennart Meier (Utrecht University)An introduction to topological modular formsTopological modular forms are a numbertheoryinspired cohomology theory, which can in particular be used to study homotopy groups of spheres and the topology of manifolds. The talk will first give an introduction to the topic and at the end point to more recent results

04/03/2022 3:00 PMZoom (the link can be found in the abstract) & MB502Farbod Shokrieh (University of Washington)Heights and moments of abelian varietiesWe give a formula which, for a principally polarized abelian
variety (A, \lambda) over the field of algebraic numbers, relates
the stable Faltings height of A with the N\'eronTate height of a
symmetric theta divisor on A. Our formula involves invariants
arising from tropical geometry. We also discuss the case of Jacobians
in some detail, where graphs and electrical networks will play a key
role. (Based on joint works with Robin de Jong.)https://qmulacuk.zoom.us/j/81420780676?pwd=d2xJb2xncUxDWkRkVXVwRk1ZbTVpZz09Meeting ID: 814 2078 0676Passcode: 244487

11/03/2022 3:00 PMMB503Ivo Dell'Ambrogio (Université de Lille)The origins of classical Green functorsWhen studying invariants of finite groups, one often replaces abelian groups and rings by Mackey functors and Green functors, respectively, which also encode the ubiquitous restriction, induction and conjugation maps. Since their introduction in (linear) representation theory by J. A. Green in 1971, Green functors have been used throughout equivariant mathematics, with examples including Burnside rings, character and representation rings, group cohomology and Tate cohomology algebras, homotopy groups of Gring spectra, algebraic Ktheory of Grings, topological Ktheory of Gspaces and GC*algebras, etc.In this talk, I will introduce the analogous higher structure of a "Green 2functor": roughly, this is a family of linear tensor categories indexed by finite groups and equipped with restriction, induction and conjugation functors satisfying some basic properties. I will then explain how all of the abovementioned classical examples of Green functors arise by some decategorification procedure (of two essentially different kinds) out of some Green 2functor occurring in Nature.
(Reference: arXiv:2107.09478)

18/03/2022 3:00 PMZoom (the link can be found in the abstract) & MB502Conchita MartínezPérez (Universidad de Zaragoza)On the Sigmainvariants for even Artin groups of FCtype
Sigma invariants are geometric invariants that one can associate to a finitely generated group that can be used to determine the homotopical and homological finiteness properties of coabelian subgroups. We will describe a sufficient condition for a character to be in the $n$th Sigma invariant for even Artin groups of FCtype. We will also explain how in some particular cases this condicion is neccessary. This is a joint work with Rubén Blasco and José Ignacio Cogolludo.
https://qmulacuk.zoom.us/j/81420780676?pwd=d2xJb2xncUxDWkRkVXVwRk1ZbTVpZz09
Meeting ID: 814 2078 0676Passcode: 244487

25/03/2022 3:00 PMMB503Ming Ng (University of Birmingham)Adelic Geometry via Topos TheoryIn this talk, I will give a leisurely introduction to the theory of classifying toposes, before introducing a new research programme (joint with Steven Vickers) of developing a version of adelic geometry via topos theory.
To elaborate, let us highlight two important aspects of the story.
First, much of the theorybuilding in number theory has been guided by the following tension: completions of a number field ought to be treated in a symmetric way (cf. Hasse LocalGlobal Principle, product formula etc.) yet there also exists important differences between the Archimedean vs. nonArchimedean completions. This raises an important question: what is the right framework for us to understand this tension? In topos theory, our main point of leverage is that every topos classifies some (logical) theory T, and contains a “generic model” of T — which is “generic” in the sense that it generates all other models of T. In our programme, we ask: is there a topos of completions of the rationals Q? How might we go about constructing this topos? What can the generic completion tell us about the relationship between Archimedean vs. nonArchimedean completions?
Second, in order to work with classifying toposes we shall need to work “geometrically” — which effectively means pulling our mathematics away the set theory. This seemingly innocuous move turns out to reveal a deep nerve connecting topology and algebra, invisible from the perspective of classical mathematics. For instance, one important step of our project involves constructing the topos of places of Q, which incidentally provides a topostheoretic account of the Arakelov compactification of Spec(Z). However, whereas the classical picture views the Archimedean place as a single point “at infinity”, our picture reveals that the Archimedean place resembles a blurred interval living below Spec(Z), raising challenging questions to our current understanding of the number theory.
This talk will discuss both aspects, along with some of their interesting implications. 
01/04/2022 3:00 PMMB503CANCELLED

08/04/2022 3:00 PMMB503Vaidehee Thatte (King's College London)Understanding the Defect via Ramification TheoryClassical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow nondiscrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a nontrivial defect is one of the main obstacles to longstanding problems, such as obtaining resolution of singularities in positive characteristic.
Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.

29/04/2022 3:00 PMMB503Ravi Ramakrishna (Cornell)On Ozaki’s Theorem
About a dozen years ago Ozaki proved the following theorem: Given any finite pgroup G, there exists a number field K such that the Galois group over K of the pHilbert class field tower is G. Ozaki’s K is totally complex. In joint work with Hajir and Maire we give a more general version of the theorem (e.g. K may be totally real) with a simpler proof.

06/05/2022 3:00 PMMB503James Newton (University of Oxford)Modularity over CM fields
Since the seminal works of Wiles and TaylorWiles, robust methods were developed to prove the modularity of 'polarised' Galois representations. These include, for example, those coming from elliptic curves defined over totally real number fields. Over the last 10 years, new developments in the TaylorWiles method (Calegari, Geraghty) and the geometry of Shimura varieties (Caraiani, Scholze) have broadened the scope of these methods. One application is the recent work of Allen, Khare and Thorne, who prove modularity of a positive proportion of elliptic curves defined over a fixed imaginary quadratic field. I'll review some of these developments and work in progress with Caraiani which has further applications to modularity of elliptic curves over imaginary quadratic fields.

27/05/2022 3:00 PMMB503Lucia Morotti (University of Hannover)Homogeneous reductions of spin representations in characteristic 3Let V be a representation of a group G in characteristic 0. Even if V is irreducible its reduction modulo p is in general not irreducible and often not even homogeneous, that is it has nonisomorphic composition factors.Given a group G a natural question is to characterise irreducible representation which remain irreducible or homogeneous when reduced to characteristic p.In this talk I will present reduction results on the classification of (almost) homogeneous reductions of spin representations of symmetric groups in characteristic 3. From this result it follows that, in characteristic 3, homogeneous reductions of spin representations of symmetric or alternating groups are actually irreducible.
This is joint work with Matthew Fayers. 
16/09/2022 3:00 PMMB503Erez Lapid (Weizmann Institute of Sciences)A binary operation on B(∞) and applications to representation theory of GL_n(F), F nonarchimedean local field.
The classification of the irreducible representations of GL_n(F), F nonarchimedean local field
is one of the highlights of the BernsteinZelevinsky theory from the 1970's.
They are also closely related to Lusztig's (dual) canonical bases of type A, indexed by irreducible
components of nilpotent varieties, or vertices of Kashiwara's crystal B(∞).
A key ingredient in BernsteinZelevinsky theory is standard modules and their irreducible socles.
More recently, representations with irreducible socles show up prominently in the work of
KangKashiwaraKimOh on monoidal categorification of cluster algebras.
I will discuss some constructions, conjectures and results aiming at understanding such socles
and irreducibility of parabolic induction.Based on joint works with Avraham Aizenbud and Alberto Minguez

23/09/2022 3:00 PMMB503Paul Johnson (Sheffield)Cores and Quotients for Stanley's Upper and Lower Hook lengthsCores and quotients of partitions were first introduced in the context of representation theory of the symmetric group, but have connections to many other areas: of importance to us is GarvanKimStanton's construction observation that they're theta functions, leading to a uniform proof of the Ramanujan congruences.Stanley introduced weighted versions of hook lengths in his study of Jack polynomials, but the analogs of the cores and quotients for these seem little studied. We explain ongoing work in this direction, some joint with Jørgen Rennemo, in connecting them with orbifold Hilbert schemes. In particular, we describe a two variable generalization of the core partition generating function that has specializations to a theta function and to a rational function.

30/09/2022 3:00 PMMB503Subhajit Jana (QMUL)Diophantine exponents and growth of automorphic forms
The Diophantine exponent on algebraic groups, d'après GhoshGorodnikNevo, measures the complexity of rational points needed to approximate generic real points. For the group SL(n) the bestknown exponent so far was n1, obtained by the same authors using homogeneous dynamics in a series of famous works, which is, however, quite far from the optimal exponent 1. We will show how the spectral theory of automorphic forms can improve the exponent to 1+O(1/n). We will also try to discuss how the growth of automorphic forms, in particular, the Eisenstein series plays a crucial role in the argument. This is joint work with Amitay Kamber.

07/10/2022 3:00 PMMB503Stacey Law (Cambridge)Sylow branching coefficients for symmetric groups
One of the key questions in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and its local subgroups. Sylow branching coefficients describe the restriction of irreducible characters of G to a Sylow subgroup P of G, and have been recently shown to characterise structural properties such as the normality of P in G. In this talk, we will discuss and present some new results on Sylow branching coefficients for symmetric groups.

14/10/2022 3:00 PMMB503Aleksander Horawa (Oxford)Motivic action on coherent cohomology of Hilbert modular varieties
A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degreeshifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.

21/10/2022 3:00 PM
Likely to have no seminar.

28/10/2022 3:00 PMMB503Tony Scholl (Cambridge)Plectic structures on Shimura varieties
Some years ago, Jan Nekovar and I formulated a set of conjectures on the cohomology of Shimura varieties which would have interesting arithmetic consequences. I will describe some of this theory and the current state of knowledge.

04/11/2022 3:00 PMMB503Navid Nabijou (QMUL)Introduction to logarithmic mapping spaces
I will give an introduction to moduli spaces in algebraic geometry, with a particular focus on spaces of stable logarithmic maps. I will mention some of my recent results on the geometry and topology of these spaces, but most of the talk will be spent explaining how to work with moduli spaces in practice, using the tools of logarithmic and tropical geometry.

11/11/2022 3:00 PMMB503Shaun Stevens (UEA)Modular representations of padic groups
The theory of complex representations of padic groups has been extensively studied, as part of the local Langlands programme, for the last 50 years or more. More recently, following pioneering work of Vignéras and motivated by the study of congruences between automorphic forms, representations over other coefficient fields, or even rings, have also been studied. I will try to describe what is currently known, in particular in terms of explicit constructions of representations and decomposition of the category of representations into blocks. Recent results that I report on are/will be joint work with Kurinczuk, Skodlerack, Helm.

18/11/2022 3:00 PMMB503Yoav Len (St Andrews)The geometry of Prym varieties
I will discuss combinatorial aspects of Prym varieties, a class of Abelian varieties that shows up in the presence of double covers of curves. Pryms have deep connections with torsion points of Jacobians, hyperkähler manifolds, lines on cubic surfaces, and spin structures. As I will explain, problems concerning Pryms may be reduced, via tropical geometry, to combinatorial games on graphs. Consequently we obtain new results in the geometry of special algebraic curves and a generalization of Krichhoff’s matrixtree theorem.

25/11/2022 3:00 PMZoom (meeting ID 82155593664)Mahdi Asgari (Oklahoma/Cornell)Convex Polytopes and the Combinatorics of the Arthur Trace FormulaThe Arthur Trace Formula (ATF), in its various incarnations, has played a very important role in Number Theory and Automorphic Forms in the last 50 years. The noninvariant ATF, the first incarnation, is an equality of two distributions, the socalled geometric and spectral sides, on suitably chosen test functions on an adelic reductive group. However, theusual trace diverges. Arthur proved, using complicated geometric/combinatorial and analytical techniques, that a truncated version of the trace, depending on a truncation parameter, is convergent when the parameter is sufficiently regular, and indeed gives a polynomial in the parameter. These facts form the basis of the developmentof his theory of the ATF.
The combinatorial aspects of Arthur’s proof, when mixed with the analytic techniques, appear somewhat mysterious. My goal in this talk is to explicate the geometric/combinatorial aspects of Arthur’s proof by recasting them in the language of convex polytopes and fans, making the geometric and combinatorial aspects more transparent and natural. Apart from the motivation and background, the talk can be considered as being purely about combinatorics of polytopes. There are connections to other areas, such as toric varieties and compactifications of locally symmetric spaces, which we are currently exploring as well. This is joint work with Kiumars Kaveh (University of Pittsburgh).

02/12/2022 3:00 PMMB503Yue Ren (Durham)Tropical Geometry of Generic Root Counts
Many systems are modellable using polynomials, and solving systems of
polynomial equations is a fundamental task in their study. A staple
method for polynomial system solving is homotopy continuation, which
constructs an easy start system and deforms it to the difficult target
system whilst keeping track of the solutions along the way. To do this
optimally requires an accurate estimate of the number of solutions,
which is generally a very difficult task. Fortunately, polynomial
systems in many applications can be assumed to be generic instances
inside a bigger family of polynomial systems. We refer to their number
of solutions as the generic root count of the family.
In this talk, we explain how the variation of polynomial systems
within a family can be exploited tropically in order to encode their
generic root count in a tropical intersection number. We further
discuss how this tropical intersection number can be computed, and
highlight the role of matroids in their computation. The main
theoretic result is a tropical generalisation of Bernstein's Theorem
to families of properly intersecting schemes. Main applications are
the steady states of chemical reaction networks, as well as the
Duffing and Kuramoto model for dampened and coupled oscillators,
respectively. 
09/12/2022 3:00 PMMB503Jack Sempliner (Imperial College London)On Shimura Varieties and Conjugation
This is joint work with Richard Taylor. In the 1970's Deligne proposed a definition of certain varieties generalizing Hilbert modular varieties, modular curves, and the Siegel varieties which associates to certain reductive groups $G/\mathbb{Q}$ equipped with some extra data a variety $Sh(G)/\mathbb{C}$ admitting an action of an appropriate algebra of Hecke operators for $G$. Deligne showed that these varieties with Hecke action actually descend to a number field (considered as a subfield of $\mathbb{C}$) parameterized by the data defining $Sh(G)$. We propose an alternate version of Deligne's Shimura data which canonically parameterizes a Shimura variety over any field of characteristic 0 (without fixing a choice of embedding into $\mathbb{C}$). To do this it is necessary to revisit the theory, which was developed by Langlands, Milne, Borovoi, and others, of conjugation of Shimura varieties, which our description casts in a somewhat new light.

16/12/2022 3:00 PMMB503/ZoomMatt, Alex, Felipe, Behrang, ShuWhat have you been up to?

27/01/2023 3:00 PM
No seminar

03/02/2023 3:00 PMMB503Samuel Edwards (Durham University)Orbital counting and dynamics on infinitevolume homogeneous spaces
Fundamental work by DukeRudnickSarnak and EskinMcMullen relates orbital counting problems for a discrete subgroup \Gamma of a Lie group G with equidistribution problems on the homogeneous space \Gamma\G. In the classical setting, \Gamma is a lattice in G, the most famous example of which being G=SL(2,R) and \Gamma=SL(2,Z). Over the last fifteen years, these ideas have been developed for discrete subgroups Γ which are not lattices. I will give an overview of some of the key concepts in homogeneous dynamics on infinitevolume homogeneous spaces, and one of their most striking applications: counting in Apollonian circle packings.

10/02/2023 3:00 PMMB503Kevin Kwan (UCL)Some Aspects of Moments
In the past century, moments of Lfunctions have been important in number theory and are wellmotivated by a variety of arithmetic applications. In this talk, we will begin with two elementary counting problems of Diophantine nature as motivation, followed by a survey of techniques in the past and the present. The main goal is to demonstrate how period integrals can be used to study moments of automorphic Lfunctions and uncover the interesting underlying structures (some of them can be modeled by random matrix theory).

17/02/2023 3:00 PMMB503Akshat Mudgal (Oxford)Energy variants of the sumproduct problemA famous conjecture of Erdős and Szemerédi suggests that for any set A of n integers and for any fixed positive integer s, either there are n^{s  o(1)} distinct sfold sums of the form a_1 + … + a_s or there are n^{so(1)} distinct sfold products of the form a_1…a_s, with a_1, ..., a_s being elements of A. While this conjecture remains open, a celebrated result of BourgainChang from 2004 implies that one can obtain at least n^{h} sfold sums or products, where h = h(s) grows with s. In 2020, their result was quantitatively improved by Pálvölgyi–Zhelezov, who showed that h >> (log s)^{1  o(1)}.In this talk, we will survey some of the recent work in this direction, including our own results which deliver an “energy version” of this problem. The latter confirms a speculation of Balog and Wooley in the integer setting.

24/02/2023 3:00 PMMB503Amitay Kamber (Cambridge)Optimal Lifting for SL_n(Z)
Let q be a natural number. The strong approximation theorem for$SL_n(Z)$ says that the modulo $q$ map $SL_n(Z) \to SL_n(Z/qZ)$ isonto. This leads to the following research problem: Given a parameter T,look at the (finite) set of matrices $B_T:={A\in SL_n(Z): A \leT}$, where $.$ is some matrix norm. We are interested inunderstanding the image of $B_T$ in $SL_n(Z/qZ)$, for T a function ofq. Such studies were initiated (in a more general context) byDukeRudnickSarnak, and further developed by many others, notablyGorodnikNevo. We will focus on the problem of covering $SL_n(Z/qZ)$ with the imageof $B_T$, and explain the connection of the problem to the GeneralizedRamanujan Conjecture in automorphic forms.Based on joint work with Subhajit Jana.

03/03/2023 3:00 PMMB503Shreyasi Datta (University of Michigan)Sarithmetic Diophantine approximation
Diophantine approximation deals with quantitative and qualitative aspects of approximating numbers by rationals. A major breakthrough by Kleinbock and Margulis in 1998
was to study Diophantine approximations for manifolds using homogeneous dynamics. After giving an overview of recent developments in this subject, I will talk about Diophantine
approximation in the Sarithmetic setup, where S is a finite set of valuations of rationals. 
10/03/2023 3:00 PM
No seminar due to clash with school colloquium.

17/03/2023 3:00 PMMB503Nicola Gambino (Manchester)The differential calculus of analytic functorsTitle: The differential calculus of analytic functorsAbstract: Joyal introduced the notion of an analytic functor on the category of sets as part of his functorial approach to enumerative combinatorics. In this context, analytic functors are a categorification of exponential power series, a point of view that is supported by a calculus which includes operations of sum, product and differentiation.In earlier joint work with Fiore, Hyland and Winskel, the notion of an analytic functor between presheaf categories was introduced as a generalisation of Joyal’s notion, and it was shown that the calculus of analytic functors not only extends to this setting, but also becomes richer, thanks to the presence of Cartesian closed structure.In this talk, I will show how Joyal’s differentiation operation can also be extended, similarly to how, in ordinary analysis, we pass from one variable to many variables. In particular, the rules for differentiation give rise to a model of the differential lambdacalculus, a logical system introduced by Ehrhard and Regnier, in which one can approximate programs by their Taylor expansion.This is based on joint work with Marcelo Fiore and Martin Hyland.

24/03/2023 3:00 PMMB503Haldun Ozgur Bayindir (City)Adjoining roots to ring spectra, algebraic Ktheory and chromatic redshiftTitle: Adjoining roots to ring spectra, algebraic Ktheory and chromatic redshiftAbstract: I will start my talk by providing background on ring spectra and algebraic Ktheory through applications to manifold topology. After this, I will discuss our method for adjoining roots to ring spectra which we exploit to obtain new algebraic Ktheory computations and to provide a conceptual approach to some of the known algebraic Ktheory computations. For this, we introduce a new definition of logarithmic THH and show that root adjunction is logarithmic THHetale.

31/03/2023 3:00 PMMB503Franco Rota (Glasgow)Towards homological mirror symmetry for log del Pezzo surfaces
Title: Towards homological mirror symmetry for log del Pezzo surfaces.
Abstract: The homological mirror symmetry conjecture predicts a duality, expressed in terms of categorical equivalences, between the complex geometry of a variety X (the B side) and the symplectic geometry of its mirror object Y (the A side).Motivated by this, we study a series of singular surfaces (called log del Pezzo). I will describe the category arising in the B side, using the McKay correspondence and explicit birational geometry. If time permits, I will discuss some preliminary results obtained on the A side, which relate to results on string junctions from the physics literature.
The description of the B side is joint with Giulia Gugiatti, while the work on the A side is in collaboration with Giulia Gugiatti and Matt Habermann.

07/04/2023 3:00 PM

14/04/2023 3:00 PMMB503Hanneke Wiersema (Cambridge)Modularity in the partial weight one caseSpeaker: Hanneke Wiersema (Cambridge)Title: Modularity in the partial weight one caseAbstract: The strong form of Serre's conjecture states that a twodimensional mod p representation of the absolute Galois group of Q arises from a modular form of a specific weight, level and character. Serre restricted to modular forms of weight at least 2, but Edixhoven later refined this conjecture to include weight one modular forms. In this talk we explore analogues of Edixhoven's refinement for Galois representations of totally real fields, extending recent work of DiamondSasaki. In particular, we show how modularity of partial weight one Hilbert modular forms can be related to modularity of Hilbert modular forms with regular weights, and vice versa. We will also discuss the applications of this to padic Hodge theory.

29/09/2023 3:00 PMNo seminarQuIPS

06/10/2023 3:00 PMMB503Jakub Dobrowolski (QMUL)Problem of solving quadratic forms in prime variables
In this talk, we will report on the work on establishing the limits of L. Zhao’s techniques for counting solutions to quadratic forms in prime variables. Zhao considered forms with rank at least 9 and showed that these equations have solutions in primes provided there are no local obstructions. We will look in detail at the degenerate cases of offdiagonal rank 1 and 2, and sketch the proof that reduces the rank lower bounds to 6 and 8 respectively. These results complement a recent breakthrough of Green on the nondegenerate rank 8 case.

13/10/2023 3:00 PMMB503Andrew Scoones (Oxford)Nonasymptotic Results in Quantitative Metric Theory of Diophantine Approximation
Many results regarding quantitative problems in the metric theory of Diophantine approximation are asymptotic, showing, for example, that the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic error term. Two common tools relied upon for giving such results are Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory", in which it is shown that a sum of certain functions is almost everywhere equal to a main term plus an asymptotic error term. The error term incorporates an implicit constant that varies from one point to another. This means that applications of these results do not give concrete bounds when applied to, say a finite sum, or when applied to counting the number of solutions up to a finite point for a given inequality. In this talk, we discuss a method to address this and make the tools and their results effective, by making the implicit constant explicit outside of an exceptional subset of Lebesgue measure at most $\delta>0$, an arbitrarily small constant set in advance. We deduce from this the effective results for Schmidt's Theorem, quantitative KoukoulopoulosMaynard Theorem and quantitative results on $M_{0}$sets; we also provide effective results regarding statistics of normal numbers and strong law of large numbers, some of which I shall discuss as an application of our main effective result. This is joint with Ying Wai Lee.

20/10/2023 3:00 PMMB503Lasse Grimmelt (Oxford)Primes in large arithmetic progressions
One important topic in analytic prime number theory is the study of the distribution of primes in arithmetic progressions. In the case of large arithmetic progressions, sums of Kloosterman sums and the spectral theory of automorphic forms are the main tools. In this talk, I will give the background behind this connection and explain how my coauthors V. Blomer, J. Li, S. Rydin Myerson and I applied it when considering additive problems with almost prime squares. The aim will be to not dive into the technical details, but instead to get an intuitive understanding of why the seemingly unrelated area of automorphic forms is so effective in this application.

27/10/2023 3:00 PMMB503Alex Walker (UCL)Shifted Convolutions of Mock Modular Forms and Hurwitz Class Numbers
The Hurwitz class numbers H(n) relate in a simple way to the class numbers of imaginary quadratic fields. The Hurwitz class numbers also appear as the coefficients of a certain mock modular form, i.e. the holomorphic part of a weak harmonic Maass form. In this talk, I will show how the Rankin—Selberg method can be adapted to understand convolutions and shifted convolutions of mock modular forms. As an application, we produce estimates for second moments and shifted convolution sums of Hurwitz class numbers.

03/11/2023 3:00 PMMB503Han Yu (Warwick)Diophantine approximation on fractals
We will discuss a problem concerning the distribution of rational points near 'nice' fractals similar to the middlethird Cantor set. Among the others, we shall see some Khinchine as well as BesicovitchJarnik type results about Diophantine approximation on fractals.

10/11/2023 3:00 PMMB503No seminar

17/11/2023 3:00 PMMB503Andrew PearceCrump (York)Characteristic polynomials, the Hybrid model, and the Ratios ConjectureIn the 1960s Shanks conjectured that the $\zeta'(\rho)$, where $\rho$ is a nontrivial zero of zeta, is both real and positive in the mean. Conjecturing and proving this result has a rich history, but efforts to generalise it to higher moments have so far failed. Building on the work of Keating and Snaith using characteristic polynomials from Random Matrix Theory, the Hybrid model of Gonek, Hughes and Keating, and the Ratios Conjecture of Conrey, Farmer, and Zirnbauer, we have been able to produce new conjectures for the full asymptotics of higher moments of the derivatives of zeta. This is joint work with Chris Hughes.

24/11/2023 3:00 PMMB503Martin Orr (Manchester)Endomorphisms of abelian varieties in families
Elliptic curves have played a central role in the developmentof algebraic number theory and there is an elegant theory of theendomorphisms of elliptic curves. Generalising to thehigherdimensional analogues of elliptic curves, called abelianvarieties, more complex phenomena occur. When we consider abelian varieties varying in families, there are oftenonly finitely many members of the family whose endomorphism ring islarger than the endomorphism ring of a generic member. The ZilberPinkconjecture, generalising the AndréOort conjecture, predicts preciselywhen this finiteness occurs. In this talk, I will discuss some of the progress which has been made onthe ZilberPink conjecture, including results of Daw and myself aboutfamilies with multiplicative degeneration.

01/12/2023 3:00 PMMB503Mark Wildon (RHUL)Plethysms of symmetric functions
The plethysm product on symmetric functions corresponds to composing polynomial representations of general linear groups. Decomposing a plethysm product of Schur functions into a linear combination of Schur functions is one of the main open problems in algebraic combinatorics. I will give an introduction to these mathematical objects emphasising the beautiful interplay between the representation theory and the combinatorics. I will end with new results obtained in joint work with Christopher Bowman (University of York) and Rowena Paget (University of Kent) on the stability of plethysm coefficients. No specialist background knowledge will be assumed.

08/12/2023 3:00 PMMB503Beth Romano (KCL)Epipelagic representations of padic groupsI will discuss some new results about epipelagic representations of padic groups and where these representations fit into the local Langlands correspondence. I will not assume background knowledge about padic groups. Instead, I will give an introduction to the topic via examples.

15/12/2023 3:00 PMMB503

26/01/2024 3:30 PMMB503Miriam Norris (Manchester)On composition factors for tensor products of simple GL(n)representationsUnderstanding the composition factors of tensor products is an important question in representation theory. The classical LittlewoodRichardson coefficients describe the composition factors of both the tensor products of simple CGLn(C)modules and the restriction of simple CGLn(C)modules to some Levi subgroup.
Now let F denote an algebraically closed field of characteristic p > 0 and let G:= GLn(F). In this case very little is known about composition factors of tensor products of simple FGmodules. However, it is thought that they may still be related to the composition factors of the restriction of simple FGmodules to some Levi subgroup. In this talk we will discuss a relationship of this kind for tensor products of simple FGLn(F)modules with the wedge square of the dual natural module and see how this might be used to find composition factors.

02/02/2024 3:30 PMMB503Alan Lauder (Oxford)Explicit points on elliptic curvesI will discuss an explicit analytic formula for points on elliptic curves. The elliptic curves are ones defined over Q, and the points are defined over different types of number fields. The formula involves modular forms of weight two (attached to the elliptic curve) and of weight one (associated to the number field), and the choice of a prime number p. The formula is proven in some cases, and conjectural in others (supported by experimental evidence, and indeed in a large part discovered in this manner). It in some sense subsumes the few known existing formulas for explicit points. The talk is joint work with Henri Darmon and Victor Rotger. I will mainly discuss some rather old work of ours, but hope to touch upon some newer work too.

09/02/2024 3:30 PMMB503Xenia Dimitrakopoulou (Warwick)Anticyclotomic padic Lfunctions for families of $U_n \times U_{n+1}$I will report on recent work on the construction of anticyclotomic $p$adic $L$functions for RankinSelberg products. I will explain how by $p$adically interpolating the branching law for the spherical pair $\left(U_n, U_n \times U_{n+1}\right),$ we can construct a $p$adic $L$function attached to cohomological automorphic representations of $U_n \times U_{n+1}$. Due to the recent proof of the unitary GanGrossPrasad conjecture, this $p$adic $L$function interpolates the square root of all critical $L$values, including anticyclotomic variation. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation.

16/02/2024 3:30 PMMB503James Kiln (QMUL)Modularity for certain trianguline Galois representations
I will explain the unpublished result of Emerton that every trianguline representation of the absolute Galois group of Q, satisfying certain conditions, arises as a twist of the Galois representation attached to an overconvergent padic cuspidal eigenform of finite slope. I will outline a new approach to prove this result by patching trianguline eigenvarieties and eigenvarieties for modular forms on GL2 to establish an “R=T” theorem in the setting of rigid analytic spaces. There are several nice consequences to such a theorem, including a new approach to deduce the classicality of overconvergent eigenforms of small slope, as well as applications to the FontaineMazur conjecture.

23/02/2024 3:30 PMMB503Fred Diamond (KCL)KodairaSpencer isomorphisms and Hecke correspondences
The KodairaSpencer isomorphism for a modular curve relates its cotangent bundle to that of the universal elliptic curve over it. I'll explain a perspective that yields a description of dualizing sheaves for integral models in certain situations of bad reduction. The resulting KodairaSpencer isomorphism generalizes (at least) to the setting of Hilbert modular varieties, with applications to the construction and integrality properties of Hecke operators.

01/03/2024 3:30 PMMB503Yu Min (ICL)Recent advances in integral padic Hodge theoryPrismatic cohomology has now become a fundamental tool in integral padic Hodge theory. In this talk, I will begin by providing a concise overview of this cohomology theory. Then, I will talk about a stacky perspective on prismatic cohomology, due to Drinfeld and BhattLurie. Various applications will be discussed along the way.

08/03/2024 3:30 PMMB503

15/03/2024 3:30 PMMB503Farhad Babaee (Bristol)Tropical and complex geometryIn this presentation, I revisit fundamental concepts and questions in tropical geometry, including tropical varieties, tropicalisation, extremality, intersection theory, realisability, etc. Then, I mention some of their analogous counterparts in complex analytic geometry.

22/03/2024 3:30 PMMB503Andrea Dotto (Cambridge/KCL)Some consequences of mod p multiplicity one for Shimura curves
The multiplicity of Hecke eigenspaces in the mod p cohomology of Shimura curves is a classical invariant, which has been computed in significant generality when the group is split at p. This talk will focus on the complementary case of nonsplit quaternion algebras, and will describe a new multiplicity one result, as well as some of its consequences regarding the structure of completed cohomology. I will also discuss applications towards the categorical mod p Langlands correspondence for the nonsplit inner form of GL_2(Q_p). Part of the talk will comprise a joint work in progress with Bao Le Hung.

29/03/2024 3:30 PMMB503GoodFriday

05/04/2024 3:30 PMMB503Chris Williams (Nottingham)CANCELLED DUE TO TRAIN STRIKE!
Let p be a prime and N an integer prime to p. Let f be an eigenform of level Gamma_0(p). For a given integer m, does there exist another eigenform g of level such that f = g mod p^m? For classical modular forms, which are automorphic forms for GL(2), the answer is yes. Even better, every such eigenform f can be deformed in a 1dimensional padic family of eigenforms as we padically deform the weight (captured by the eigencurve). This object studies the padic variation of systems of Hecke eigenvalues, rather than eigenforms, and has had profound consequences in Iwasawa theory and the Langlands program.
It is natural to ask if this holds more generally. I will describe recent joint work with Daniel Barrera and Andy Graham, where we consider the setting of symplectic automorphic forms on GL(N). In this case, it turns out the question is more subtle. For example, if 𝛑 is an automorphic representation of GL(4) of level N, there are 24 attached eigensystems at level Np. We conjecture that 8 of them deform in 2dimensional families, 8 of them in 1dimensional families, and 8 of them in no family at all.

12/04/2024 3:30 PMMB503Siqi Yang (LSGNT)Geometric modularity for paritious weights in quadratic case
I will discuss the relation between algebraic and geometric modularity. Let F be a totally real field and \rho a twodimensional mod p representation of the absolute Galois group of F that is irreducible, continuous, and totally odd. It is conjectured by Diamond and Sasaki that \rho being geometrically modular of a weight (k, l) implies algebraic modularity of the same weight, if k lies in a certain minimal cone. In this talk, I will focus on the real quadratic case, under the assumption that p is unramified in F and the weight is paritious. I will discuss the main methods in the proof, using cohomology vanishing, weight shiftings, and the properties of the stratification on mod p Hilbert modular varieties.

24/06/2024 2:00 PMMaths Seminar RoomDr Gunja SachdevaAn Automorphic translation of Deligne’s conjecture: Special values of RankinSelberg L function for GL(n) × GL(m) over a number field
In the talk, I will discuss an arithmetic property of Lfunctions namely relations of rationality for special values of Lfunctions attached to a representation. A classical example is “the value of Riemann Zeta function at all positive even integers is equal to an integral power of π up to a rational number”. As it’s generalisation, I will discuss algebraicity results for all the critical values of certain RankinSelberg Lfunctions attached
to a pair of automorphic representations for GL(n) × GL(m) over a number field. I will end the discussion by giving a methodology to briefly explain that how these algebraicity results are obtained from the theory of Lfunctions by giving a cohomological interpretation to an integral representing a critical Lvalue in terms of Poincare pairing.