Algebra and Number Theory seminar
Passcode: please drop me (s.sasaki@qmul.ac.uk) a line until I join the 21st century and set up a registration system that sends you the passcode automatically.
I personally maintain https://researchseminars.org/seminar/QMULANTS. Please look at the link for the most uptodate information.
We occsaionally recorded Zoom talks (with permission, of course). The links to the video are as follows:
(Mandi Scheffer Fry) https://media.qmplus.qmul.ac.uk/media/Mandi_Schaeffer_Fry/1_k9x5qiiw
(Alex Betts) https://media.qmplus.qmul.ac.uk/media/Alex_Betts/1_6h1poe9a
(Paul Nelson) https://qmulacuk.zoom.us/rec/share/R9lFh3l5opAXrK0K1eV0VkI8hfdyIXyQwETwoID6tH6WlSo6Ld_YmmyAXHn6xr.LT6XSZwcNbfCzieE (passcode fS+Kp*93)

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?

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.

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.

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".

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.

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.

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.

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.

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.

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.

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. 
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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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. 
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.

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.

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.

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.

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.