Past

Abstract. I will talk about projects in which we combine heuristics with computational data to develop a theory in problems where it was previously hard to be confident of the guesses that there are in the literature.

1/ "Speculations about the number of primes in fast growing sequences". Starting from studying the distribution of primes in sequences like $2^n-3$, Jon Grantham and I have been developing a heuristic to guess at the frequency of prime values in arbitrary linear recurrence sequences in the integers, backed by calculations.

If there is enough time I will then talk about:

2/ "The spectrum of the $k$th roots of unity for $k>2$, and beyond".  There are many questions in analytic number theory which revolve around the "spectrum", the possible mean values of multiplicative functions supported on the $k$th roots of unity. Twenty years ago Soundararajan and I determined the spectrum when $k=2$, and gave some weak partial results for $k>2$, the various complex spectra.  Kevin Church and I have been tweaking MATLAB's package on differential delay equations to help us to develop a heuristic theory of these spectra for $k>2$, allowing us to (reasonably?) guess at the answers to some of the central questions.

I will reflect on my time as Professor of Numerical Analysis.

I’m excited to share with everyone some new, unpublished work that we are just in the process of wrapping up and could use everyone’s reactions. It is a reconciliation of evolutionary game theory and ecological dynamics that I have wrestled with since I moved from an evolution program into an ecology-heavy department. It always seemed like, depending on the problem I was thinking about, I had to change my perspective and approach it as either an evolutionary game theorist, or an ecologist; and yet I had this nagging feeling that, at its core, the problem was often one and the same, and therefore one theoretical framework should suffice. So when should one write down an n-type replicator equation and when should one write down an n-species Lotka-Volterra system; and what does it mean mathematically and biologically when one has made such a choice? In the process of reconciling, I also got a deeper appreciation of what is and is not a proper game, such as a Prisoner’s Dilemma. These findings can help shed light on previously puzzling empirical findings.

Asymptotic dimension was introduced by Gromov as an invariant of finitely generated groups. It can be shown that if two metric spaces are quasi-isometric then they have the same asymptotic dimension. In 1998, the asymptotic dimension achieved particular prominence in geometric group theory after a paper of Guoliang Yu, which proved the Novikov conjecture for groups with finite asymptotic dimension. Unfortunately, not all finitely generated groups have finite asymptotic dimension. 

In this talk, we will introduce some basic tools to compute the asymptotic dimension of groups. We will also find upper bounds for the asymptotic dimension of a few well-known classes of finitely generated groups, such as hyperbolic groups, and if time permits, we will see why one-relator groups have asymptotic dimension at most two.

I will explain how to construct an Euler system for imaginary quadratic fields using Eisenstein series and their cohomology classes. This illustrates a template for a construction that should yield many new Euler systems.

A C*-algebra has the lifting property (LP) if any unital completely positive map into a quotient C*-algebra admits a completely positive lift. The local lifting property (LLP), introduced by Kirchberg in the early 1990s, is a weaker, local version of the LP.  I will present a method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C*-algebras of countable groups with (relative) property (T). This allows us to derive that the full C*-algebras of the groups $Z^2\rtimes SL_2(Z)$ and $SL_n(Z)$, for n>2, do not have the LLP. The same method allows us to prove that the full C*-algebras of a large class of groups with property (T), including those admitting a probability measure preserving action with non-vanishing second real-valued cohomology, do not have the LP.  In a different direction, we prove that the full C*-algebras of any non-finitely presented groups with property (T) do not have the LP. Time permitting, I will also discuss a connection with the notion of Hilbert-Schmidt stability for countable groups. This is based on a joint work with Pieter Spaas and Matthew Wiersma.

I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of S-duality translates to conjectural symmetries between these contributions.  One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a certain quiver variety, the instanton moduli space of torsion-free framed sheaves on $\mathbb{P}^2$. Using a recent identity of Kuhn-Leigh-Tanaka, we deduce constraints on Vafa-Witten invariants conjectured by Göttsche-Kool-Laarakker. One consequence is a formula for the contribution of the vertical component to refined Vafa-Witten invariants in rank 2.

PhD_course_Roux_2.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

We begin with a seemingly simple question: how can one distinguish a surface group from other cyclic amalgamations of two free groups? This question will prompt a (geometrically flavoured) investigation of virtual homological properties of graphs of free groups amalgamated along cyclic edge groups, where surface subgroups play a key role. 

We next turn to study limit groups and residually free groups through their finite quotients, and apply our findings to the study of profinite rigidity within these classes of groups. In particular, we will sketch out why a direct product of free and surface groups cannot have the same finite quotients as any other finitely presented residually free group.

If time permits, we will discuss other possible characterizations of surface groups among limit groups. The talk is based on joint work with Ismael Morales.

The computation of matrix function is an important task appearing in many areas of scientific computing. We propose two algorithms, one for banded matrices and the other for general sparse matrices that approximate f(A) using matrix-vector products only. Our algorithms are inspired by the decay bound for the entries of f(A) when A is banded or sparse. We show its exponential convergence when A is banded or sufficiently sparse and we demonstrate its performance using numerical experiments.

For many stationary-state PDEs, solutions can be shown to satisfy certain key identities or structures, with physical interpretations such as the dissipation of energy. By reformulating these systems in terms of new auxiliary functions, finite-element models can ensure these structures also hold exactly for the numerical solutions. This approach is known to improve the solutions' accuracy and reliability.

In this talk, we extend this auxiliary function approach to the transient case through a finite-element-in-time interpretation. This allows us to develop novel structure-preserving timesteppers for various transient problems, including the Navier–Stokes and MHD equations, up to arbitrary order in time.

A factorisation $x=u_1u_2\cdots$ of an infinite word $x$ on alphabet $X$ is called ‘super-monochromatic’, for a given colouring of the finite words $X^{\ast}$ on alphabet $X$, if each word $u_{k_1}u_{k_2}\cdots u_{k_n}$, where $k_1<\cdots<k_n$, is the same colour. A direct application of Hindman’s theorem shows that if $x$ is eventually periodic, then for every finite colouring of $X^{\ast}$, there exist a suffix of $x$ that admits a super-monochromatic factorisation. What about the converse?

In this talk we show that the converse does indeed hold: thus a word $x$ is eventually periodic if and only if for every finite colouring of $X^{\ast}$ there is a suffix of $x$ having a super-monochromatic factorisation. This has been a conjecture in the community for some time. Our main tool is a Ramsey result about alternating sums. This provides a strong link between Ramsey theory and the combinatorics of infinite words.

Joint work with Imre Leader and Luca Q. Zamboni

The irreducible smooth representations of p-adic reductive groups and the enhanced Langlands parameters of these latter can both be partitioned into series indexed by "cuspidal data". On the representation side, cuspidality refers to supercuspidal representations of Levi subgroups, while on the Galois side, it refers to "cuspidal unipotent pairs", as introduced by Lusztig, in certain subgroups of the Langlands dual groups.

In addition, on both sides, the elements in a given series are in bijection with the simple modules of a generalized affine Hecke algebra. 

The cuspidal data on one side are expected to be in bijection with the cuspidal data on the other side. We will formulate conditions on this bijection that will guarantee the existence of a bijection between the simple modules of the attached generalized affine Hecke algebras. For the exceptional group of type G_2 and for all pure inner forms of quasi-split classical groups, the Hecke algebras are actually isomorphic.

Hydrocephalus is a serious medical condition which causes an excess of cerebrospinal fluid (CSF) to build up within the brain. A common treatment for congenital hydrocephalus is to implant a permanent drainage shunt, removing excess CSF to the stomach where it can be safely cleared. However, this treatment carries the risk of vascular brain tissues such as the Choroid Plexus (CP) being dragged into the shunt during drainage, causing it to block, and also preventing the shunt from being easily replaced. In this talk I present results from our fluid-structure interaction model which simulates the deflection of the CP during the operation of the hydrocephalus shunt. We seek to improve the shunt component by optimising the geometry with respect to CP deflection.

The ε expansion was invented more than 50 years ago and has been used extensively ever since to study aspects of renormalization group flows and critical phenomena. Its most famous applications are found in theories involving scalar fields in 4−ε dimensions. In this talk, we will discuss the structure of the ε expansion and the fixed points that can be obtained within it. We will mostly focus on scalar theories, but we will also discuss theories with fermions as well as line defects. Our motivation is based on the goal of classifying conformal field theories in d=3 dimensions. We will describe recently discovered universal constraints obtained within the framework of the ε expansion and show that a “heavy handed" quest for fixed points yields a plethora of new ones. These fixed points reveal aspects of the structure of the ε expansion and suggest that a classification of conformal field theories in d=3 is likely to be highly non-trivial.

We present an inter-twining relation between operators on loop ensembles and connections.

Scattering amplitudes in string theory are written as formal integrals of correlations functions over the moduli space of punctured Riemann surfaces. It's well-known, albeit not often emphasized, that this prescription is only approximately correct because of the ambiguities in defining the integration domain. In this talk, we propose a resolution of this problem for one-loop open-string amplitudes and present their first evaluation at finite energy and scattering angle. Our method involves a deformation of the integration contour over the modular parameter to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop planar and non-planar type-I superstring four-point amplitudes, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants and directly verify long-standing conjectures about their behavior at high energies.

Let $V$ be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let $[V]$ be the vector space of complex valued functions on $V$ and let $[V]_{\mathbb Z}$ be the subgroup of $[V]$ consisting of integer valued functions. We show that there exists a Z-basis of $[V]_{\mathbb Z}$ consisting of characteristic functions of certain explicit isotropic subspaces of $V$ such that the matrix of the Fourier transform from $[V]$ to $[V]$ with respect to this basis is triangular. This continues the tradition started by Hermite who described eigenvectors for the Fourier transform over real numbers.

A central topic of study in analytic number theory is the behaviour of the Riemann zeta function. Many theorems and conjectures in this area are closely connected to concepts from probability theory. In this talk, we will discuss several results on the typical size of the zeta function on the critical line, over different scales. Along the way, we will see the role that is played by some probabilistic phenomena, such as the central limit theorem and multiplicative chaos.

Recently Linares-Otto-Tempelmayr have unveiled a very interesting algebraic structure which allows to define a new class of rough paths/regularity structures, with associated applications to stochastic PDEs or ODEs. This approach does not consider trees as combinatorial tools but their fertility, namely the function which associates to each integer k the number of vertices in the tree with exactly k children. In a joint work with J-D Jacques we have studied this algebraic structure and shown that it is related with a general and simple class of so-called post-Lie algebras. The construction has remarkable properties and I will try to present them in the simplest possible way.

Fusion 2-categories were introduced by Douglas and Reutter so as to define a state-sum invariant of 4-manifolds. Categorifying a result of Douglas, Schommer-Pries and Snyder, it was conjectured that, over an algebraically closed field of characteristic zero, every fusion 2-category is a fully dualizable object in an appropriate symmetric monoidal 4-category. I will sketch a proof of this conjecture, which will proceed by studying, and in fact classifying, the Morita equivalence classes of fusion 2-categories. In particular, by appealing to the cobordism hypothesis, we find that every fusion 2-category yields a fully extended framed 4D TQFT. I will explain how these theories are related to the ones constructed using braided fusion 1-categories by Brochier, Jordan, and Snyder.

Finite quotient singularities have a long history in mathematics, intertwining algebraic geometry, hyperkähler geometry, representation theory, and integrable systems.  I will highlight the correspondences at play here and how they culminate in Nakajima quiver varieties, which continue to attract interest in geometric representation theory and physics.  I will motivate some recent work of G. Bellamy, A. Craw, T. Schedler, H. Weiss, and myself in which we show that, remarkably, all of the resolutions of a particular finite quotient singularity are realized by a certain Nakajima quiver variety, namely that of the 5-pointed star-shaped quiver.  I will place this work in the wider context of the search for McKay-type correspondences for finite subgroups of $\mathrm{SL}(n,\mathbb{C})$ on the one hand, and of the construction of finite-dimensional-quotient approximations to meromorphic Hitchin systems and their integrable systems on the other hand.  The Hitchin system perspective draws upon my prior joint works with each of J. Fisher and L. Schaposnik, respectively. Time permitting, I will speculate upon the symplectic duality of Higgs and Coulomb branches in this setting.

Recently, a relation was introduced connecting codes of various types with the space of abelian (Narain) 2d CFTs. We extend this relation to provide holographic description of code CFTs in terms of abelian Chern-Simons theory in the bulk. For codes over the alphabet Z_p corresponding bulk theory is, schematically, U(1)_p times U(1)_{-p} where p stands for the level. Furthermore, CFT partition function averaged over all code theories for the codes of a given type is holographically given by the Chern-Simons partition function summed over all possible 3d geometries. This provides an explicit and controllable example of holographic correspondence where a finite ensemble of CFTs is dual to "topological/CS gravity" in the bulk. The parameter p controls the size of the ensemble and "how topological" the bulk theory is. Say, for p=1 any given Narain CFT is described holographically in terms of U(1)_1^n times U(1)_{-1}^n Chern-Simons, which does not distinguish between different 3d geometries (and hence can be evaluated on any of them). When p approaches infinity, the ensemble of code theories covers the whole Narain moduli space with the bulk theory becoming "U(1)-gravity" proposed by Maloney-Witten and Afkhami-Jeddi et al.