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?

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.

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.

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".    

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. 

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.

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.

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.

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.

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. 

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.

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.

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.