Algebra and Number Theory seminar

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 p-groups. Some applications to invariant theory will also be given.

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 2A-axes of the 196884-dimensional 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 two-generated Majorana algebras.

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.

The so-called 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 B₀ of the wreath product S_pwr 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 B₀ 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 Khovanov-Lauda-Rouquier algebra and the resulting grading on the group algebra of the symmetric group. This result generalizes a theorem of Chuang-Kessar, which applies to the case w

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.

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 n-bases of G, that is, generating sets of G of size n. The question can be modified and n-bases 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)=²G₂(q), in particular their maximal subgroups, in terms of their 2-transitive permutation representations of degree q³+1.

Introduced in 2008 by Khovanov and Lauda, and independently by Rouquier, the KLR algebras are a family of infinite-dimensional 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 quasi-hereditary. In this talk I'll explain what it means to be affine quasi-hereditary 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.

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.

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.

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.

*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.

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 (Duistermaat-Heckman 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 Khovanskii-Pukhlikov formula that relates the
volume and the number of integer points of a smooth lattice polytope.

In this talk we shall be concerned with the induced simple modules of the 0-Hecke algebras of types A and B.

The irreducible representations of 0-Hecke algebras were classified and shown to be one-dimensional by Norton in 1979.

To understand the structure of a finite-dimensional 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.

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.

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 K-theoretic 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.

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 "non-trivial"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 point-primitive projective plane is either classical (Desarguesian) or has a
prime number of points and a severely restricted automorphism group. I will review some on-going work with John Bamberg, Stephen Glasby, Luke
Morgan, Cheryl Praeger and Csaba Schneider that aims to classify the point-primitive generalised quadrangles, hexagons and octagons.

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!

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.

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.

I will give an introduction to 2-representation theory and will give an overview of the state of the art for finitary 2-categories, which should be seen as 2-analogues of finite-dimensional algebras.

There is a well-known 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.

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 local-global method fails within families of polynomial equations arising from the norm map between fields, as seen in Galois theory. This is joint work with Tim Browning.

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.

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.

Let M be a map on a connected, closed and orientable surface X. If B is a subset of the edge-set 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 delta-matroid, 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 Galois-theoretic aspect of maps on surfaces (in the sense of Grothendieck's programme on dessins d'enfants).

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 Malvenuto-Reutenauer's algebra, via Loday-Ronco's algebra, to Solomon's descent algebra.
In this talk, we present an extension of this framework to acyclic k-triangulations 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.

Abstract: We study symplectic invariants of the open symplectic manifolds X_Γ obtained by plumbing
cotangent bundles of 2-spheres 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 non-Dynkin, 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.

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 1-dimensional subspaces spanned by the fundamental (or rank-1) tensors in V form the so-called 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 G-orbits 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 G-line 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 so-called Veronese variety. Interestingly, several of the G-line 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.

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 long-standing records.

This is joint work with Dolica Akello-Egwel, Klas Markstrom, and Søren Riis.

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.

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.

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.

In 1923, Artin posed a conjecture about the finite-dimensional 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 2-dimensional 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.

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 high-profile 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).

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 non-permutations $s$, 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?

In 1987, J.-P. Serre made some remarkably precise conjectures (known commonly as `Serre's conjecture') about weights and levels of two-dimensional (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.

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.

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.

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 well-known 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. 

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 right-ordered 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. 

Suppose that G is a no trivial finite group, p is a prime and P is a Sylow p-subgroup of G. Let Q be the largest normal p-subgroup of G and suppose that C(Q) \leq Q. Clearly, P contains a non-trivial normal subgroup that is normal in G, for example Q, but does P contain a non-trivial 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 ZJ-Theorem (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_4-free 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.

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.

I will discuss some recent work where we obtain an explicit pullback formula that gives an integral representation for the twisted standard L-function for a holomorphic vector-valued 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 L-values 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 L-values. All of this is joint work with Ameya Pitale and Ralf Schmidt.

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.

The famous Birch & Swinnerton-Dyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the so-claeed analytic rank, which is the order of vanishing of the asociated L-function at the central point. In this talk, we shall discuss the analytic rank of automorphic L-functions in an "alternate universe".    

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.

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.  

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 K-theory. 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.

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, square-integrable 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\'e-type 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 sup-norm 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.

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.

We offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of L-functions for GL_1 with the third moment of L-functions for GL_2 over number fields, studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of pre-trace formula with test functions on Mat_2(\A) instead of GL_2(\A), on whose spectral side the matrix coefficients are the standard Godement-Jacquet zeta integrals.

The Langlands program is a far-reaching 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 p-adic groups. I will provide an overview of our understanding of the representations of p-adic 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.

(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.

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 re-proof of the Mordell Conjecture by Brian Lawrence and Akshay Venkatesh.

The strong form of Serre's conjecture states that every two-dimensional 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 Breuil-Mézard conjecture we give a third interpretation of this minimal weight as the smallest k>1 such that the representation has a crystalline lift of Hodge-Tate type (0, k-1). 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.

The McKay conjecture is one of the main open conjectures in the realm of the local-global 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 p-subgroup. 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 character-theoretic problems.

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.

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 non-archimedean analysis and geometry.

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 Iwaniec--Sarnak, 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^2-norm of a theta function defined using the correspondence of Eichler, Shimizu and Jacquet--Langlands.  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 Harcos--Templier and Blomer--Michel.  I will describe the proof in the simplest case.

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 n-dimensional Galois representations which are essentially conjugate self-dual. 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 Bellaiche-Chenevier in the regular case. 

I will explain our result and discuss its proof, which uses V. Lafforgue's notion of pseudocharacters and invariant theory.

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.

There is a well-known 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 so-called vertical Sato-Tate 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.

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).

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 Gan-Gross-Prasad (GGP) periods for GPin and GSpin. In this talk,  I will talk about the multiplicity-at-most-one theorem for the local GGP periods for GPin and GSpin.

The Breuil-Mezard 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 Hodge-Tate 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 Breuil-Kisin modules.

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. 

(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 semi-simple. 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.

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, time-permitting, discuss some open questions.

(AB) In the 1990s, Cogdell and Piatetski-Shapiro proved various theorems characterising the automorphic representations of GL(n) over a number field using analytic properties of the associated Rankin-Selberg L-functions. The most well known of these assumes properties of the twists by representations of GL(n-2), 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 Piatetski-Shapiro that uses twists by representations of GL(n-1) with greatly restricted ramification.

(EN) The cyclotomic Hecke algebra is a "higher level" version of the Iwahori-Hecke 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 closed-form character formulas of the irreducible representations. However, a certain type of representation called "calibrated" is more tractable: those representations on which the Jucys-Murphy elements act semisimply. We classify the calibrated representations in terms of their Young diagrams, give a multiplicity-free formula for their characters, and homologically construct them via BGG resolutions. This is joint work with Chris Bowman and José Simental.

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!

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 measure-classification 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. El-Baz, B. Huang, J. Marklof and A. Strömbergsson. 

In a joint work with D. Le, B. V. Le Hung, S. Morra and C. Park, we prove under standard Taylor--Wiles 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 5n-generic Fontaine--Laffaille weight at $v$.

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 p-torsion coefficients. These Galois representations are expected to satisfy local-global 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 local-global compatibility in the crystalline case under mild technical assumptions. This relies on a new idea of
using P-ordinary 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 Fontaine-Laffaille cases.

 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 G-extensions 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.  

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 1-dimensional connected algebraic group. This is joint work with Dony Varghese.

A subset of R^d is formally called self-similar if it is equal to the union of finitely many re-scaled, 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 self-affine. In general, self-affine sets remain far less well-understood than self-similar sets. This talk will describe some algebraic conditions which make the dimension of a self-affine set "defective", and finishes with some open questions of an algebraic nature which are relevant to the theory of self-affine sets.

This is joint work with Marni Mishna, Sheila Sundaram, and Stefan Trandafir.

Passcode: 244487

Since the seminal works of Wiles and Taylor-Wiles, 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 Taylor-Wiles 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.

(Reference: arXiv:2107.09478)

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 FC-type. 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://qmul-ac-uk.zoom.us/j/81420780676?pwd=d2xJb2xncUxDWkRkVXVwRk1ZbTVpZz09

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.

About a dozen years ago Ozaki proved the following theorem: Given any finite p-group G, there exists a number field K such that the Galois group over K of the p-Hilbert 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.

Since the seminal works of Wiles and Taylor-Wiles, 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 Taylor-Wiles 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.

The classification of the irreducible representations of GL_n(F), F non-archimedean local field
is one of the highlights of the Bernstein-Zelevinsky 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 Bernstein-Zelevinsky theory is standard modules and their irreducible socles.
More recently, representations with irreducible socles show up prominently in the work of
Kang-Kashiwara-Kim-Oh 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

The Diophantine exponent on algebraic groups, d'après Ghosh--Gorodnik--Nevo, measures the complexity of rational points needed to approximate generic real points. For the group SL(n) the best-known exponent so far was n-1, 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.

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.

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 degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.

Likely to have no seminar. 

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.

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.

The theory of complex representations of p-adic 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.

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.

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. 

No seminar

Fundamental work by Duke-Rudnick-Sarnak and Eskin-McMullen 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 infinite-volume homogeneous spaces, and one of their most striking applications: counting in Apollonian circle packings.

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In the past century, moments of L-functions have been important in number theory and are well-motivated 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 L-functions and uncover the interesting underlying structures (some of them can be modeled by random matrix theory). 

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Let q be a natural number. The strong approximation theorem for
$SL_n(Z)$ says that the modulo $q$ map $SL_n(Z) \to SL_n(Z/qZ)$ is
onto. This leads to the following research problem: Given a parameter T,
look at the (finite) set of matrices $B_T:={A\in SL_n(Z): ||A|| \le
T}$, where $||.||$ is some matrix norm. We are interested in
understanding the image of $B_T$ in $SL_n(Z/qZ)$, for T a function of
q. Such studies were initiated (in a more general context) by
Duke-Rudnick-Sarnak, and further developed by many others, notably
Gorodnik-Nevo. We will focus on the problem of covering $SL_n(Z/qZ)$ with the image
of $B_T$, and explain the connection of the problem to the Generalized
Ramanujan Conjecture in automorphic forms.
Based on joint work with Subhajit Jana.

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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 S-arithmetic set-up, where S is a finite set of valuations of rationals.

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No seminar due to clash with school colloquium.

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.

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 off-diagonal 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 non-degenerate rank 8 case.

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 Koukoulopoulos-Maynard 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.

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.

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.

We will discuss a problem concerning the distribution of rational points near 'nice' fractals similar to the middle-third Cantor set. Among the others, we shall see some Khinchine as well as Besicovitch-Jarnik type results about Diophantine approximation on fractals.

Elliptic curves have played a central role in the development
of algebraic number theory and there is an elegant theory of the
endomorphisms of elliptic curves.  Generalising to the
higher-dimensional analogues of elliptic curves, called abelian
varieties, more complex phenomena occur. When we consider abelian varieties varying in families, there are often
only finitely many members of the family whose endomorphism ring is
larger than the endomorphism ring of a generic member.  The Zilber-Pink
conjecture, generalising the André-Oort conjecture, predicts precisely
when this finiteness occurs. In this talk, I will discuss some of the progress which has been made on
the Zilber-Pink conjecture, including results of Daw and myself about
families with multiplicative degeneration.

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.

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 p-adic 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 Fontaine-Mazur conjecture.

The Kodaira-Spencer 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 Kodaira-Spencer isomorphism generalizes (at least) to the setting of Hilbert modular varieties, with applications to the construction and integrality properties of Hecke operators.

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.

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 1-dimensional p-adic family of eigenforms as we p-adically deform the weight (captured by the eigencurve). This object studies the p-adic 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 2-dimensional families, 8 of them in 1-dimensional families, and 8 of them in no family at all.

I will discuss the relation between algebraic and geometric modularity. Let F be a totally real field and \rho a two-dimensional 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.

In the talk, I will discuss an arithmetic property of L-functions namely relations of rationality for special values of L-functions 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 Rankin-Selberg L-functions 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 L-functions by giving a cohomological interpretation to an integral representing a critical L-value in terms of Poincare pairing.

Recent progress in birational geometry has allowed for the development of a moduli theory for higher dimensional varieties patterned on the classical moduli theory of stable curves.

I will spend some time explaining these new developments, before changing gears and considering the related problem of constructing moduli spaces of fibrations.  I will explain through some concrete examples some interesting (and problematic) phenomena in the moduli of fibrations, and then explain a perspective on these phenomena through a broader framework on moduli of foliations.

I will discuss joint work with Ivan Cheltsov, Maksym Fedorchuk and Kento Fujita. 

Family 4.1 in the Mori-Mukai classification of Fano 3-folds consists of hypersurfaces of multi degree (1,1,1,1) in (P^1)^4 a product of four copies of P^1. Smooth members of the family are K-polystable and belong to a 3-dimensional component of the K-moduli space of smoothable Fano 3-folds of anticanonical degree 24, which I will describe in this talk. 

Our description of the K-moduli is informed by the appearance of such Fano 3-folds in surprising contexts: as geometric avatars of entangled states of 4 qubits in quantum computing on the one hand, and as moduli spaces of parabolic vector bundles on elliptic curves on the other.

Differential Galois groups encode symmetries of differential equations the same way classical Galois groups encode symmetries of polynomial equations. The latter has been generalised to the setting of schemes by the celebrated work of A. Grothendieck. However, Galois theory of differential equations has resisted such a generalisation to differential schemes. In this talk I will talk about the Galois theory of differential schemes we have developed in the recent work with Ivan Tomasic.

Gotzmann's persistence theorem is a useful tool for finding equations of the Hilbert scheme parameterising subschemes of projective space. From the commutative algebra perspective, there is a natural way to generalise such Hilbert schemes to any smooth projective toric variety. A key example we will discuss is the Hilbert scheme parameterising subschemes of the product of projective spaces. We will see how Gotzmann's persistence theorem generalises to this setting.

A conjecture of Malle predicts an asymptotic formula for the number of number fields with given Galois group and bounded discriminant. Malle conjectured the shape of the formula but not the leading constant. We present a new conjecture on the leading constant motivated by a version for algebraic stacks of Peyre's constant from Manin's conjecture. This is joint work with Tim Santens.

The moduli space of polarized abelian surfaces is an example of a Shimura variety, and we can consider its mod p reduction as a variety in positive characteristic. Using their moduli interpretation I construct a new family of differential operators on some vector bundles over them. I will emphasize the new phenomena that occur in characteristic p vs the complex numbers. These have some arithmetic applications to the weight part of Serre's conjecture.