Past

Hilbert complexes are chains of spaces linked by operators, with properties that are crucial to establishing the well-posedness of certain systems of partial differential equations. Designing stable numerical schemes for such systems, without resorting to nonphysical stabilisation processes, requires reproducing the complex properties at the discrete level. Finite-element complexes have been extensively developed since the late 2000's, in particular by Arnold, Falk, Winther and collaborators. These are however limited to certain types of meshes (mostly, tetrahedral and hexahedral meshes), which limits options for, e.g., local mesh refinement.

In this talk we will introduce the Discrete De Rham complex, a discrete version of one of the most popular complexes of differential operators (involving the gradient, curl and divergence), that can be applied on meshes consisting of generic polytopes. We will use a simple magnetostatic model to motivate the need for (continuous and discrete) complexes, then give a presentation of the lowest-order version of the complex and sketch its links with the CW cochain complex on the mesh. We will then briefly explain how this lowest-order version is naturally extended to an arbitrary-order version, and briefly present the associated properties (Poincaré inequalities, primal and adjoint consistency, commutation properties, etc.) that enable the analysis of schemes based on this complex.

Come along for free pizza and to hear about the Mathematrix events this term.

We present a framework to generate and evaluate thematic recommendations based on multilayer network representations of knowledge graphs (KGs).  We represent the relative importance of different types of connections (e.g., Directing/acting) with an intuitive salience matrix that can be learnt from data, tuned to incorporate domain knowledge, address different use cases, or respect business logic. We apply an adaptation of the personalised PageRank algorithm to multilayer network models of KGs to generate item-item recommendations. These recommendations reflect the knowledge we hold about the content, and are suitable for thematic or cold-start settings.

Evaluating thematic recommendations from user data presents unique challenges. Our method only recommends items that are 'thematically' related; that is, easily reachable following connections in the KG. We develop a variant of the widely-used Normalised Discounted Cumulative Gain (NDCG) to evaluate recommendations based on user-item ratings, respecting their thematic nature.

We apply our methods to a KG of the movie industry and MovieLens ratings and in an internal AB test. We learn the salience matrix and demonstrate that our approach outperforms existing thematic recommendation methods and is competitive with collaborative filtering approaches.

Profinite rigidity is essentially the study of which groups can be distinguished from each other by their finite quotients. This talk is meant to give a gentle introduction to the area - I will explain which questions are the right ones to ask and give an overview of some of the main results in the field. I will assume knowledge of what a group presentation is.

In this talk, I will discuss the notion of quantum limits from different viewpoints: Cordes' work on the Gelfand theory for pseudo-differential operators dating from the 70’s as well as the micro-local defect measures and semi-classical measures of the 90’s. I will also explain my motivation and strategy to obtain similar notions in subRiemannian or subelliptic settings. 

I will describe the interaction between a single soliton and a gas of solitons, providing for the first time a mathematical justification for the kinetic theory as posited by Zakharov in the 1970s.  Then I will explain how to use random matrix theory to introduce randomness into a large collection of solitons.

We formulate and sketch the proof of the K-theoretic Farrell-Jones Conjecture for
for the Hecke algebras of reductive p-adic groups. This is the first time that
a version of the farrell-Jones Conjecture for topological groups is formulated. It implies that
the reductive projective class group of the Hecke algebra of a reductive p-adic group
is the colimit of these for all compact open subgroups. This has been proved rationally by
Bernstein and Dat using representation theory. The main applications of our result
will concern the theory of smooth representations
In particular we will prove a conjecture of Dat.

The proof is much more involved than the one for instance for discrete CAT(0)-groups.
We will only give a very brief sketch of it and the new problems occurring in the setting of
totally disconnected groups. Most of the talk will be devoted
an introduction to the Farrell-Jones Conjecture and the theory of
smooth representations of reductive p-adic groups, and
discussion of  applications.

This is a joint project with Arthur Bartels.

An occupational hazard of mathematicians is the investigation of objects that are "optimal" in a mathematically precise sense, yet may be far from optimal in practice. This talk will discuss an extreme example of this effect: Gauss-Hermite quadrature on the real line. For large numbers of quadrature points, Gauss-Hermite quadrature is a very poor method of integration, much less efficient than simply truncating the interval and applying Gauss-Legendre quadrature or the periodic trapezoidal rule. We will present a theorem quantifying this difference and explain where the standard notion of optimality has failed.

In this talk we explore structural results about sets with small doubling in k dimensions. We start in the continuous world with a sharp stability result for the Brunn-Minkowski inequality conjectured by Figalli and Jerison and work our way to the discrete world, where we discuss the natural extension: we show that non-degenerate sets in Z^k with doubling close to 2^k are close to convex progressions i.e. convex sets intersected with a sub-lattice. This talk is based on joint work with Peter van Hintum and Hunter Spink.

Let F be a p-adic field, and k an algebraically closed field of characteristic l different from p. In this talk, we will first give a category decomposition of Rep_k(SL_n(F)), the category of smooth k-representations of SL_n(F), with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F), which is not always a block decomposition in general. We then give a block decomposition of the supercuspidal subcategory, by introducing a partition on each GL_n(F)-equivalent supercuspidal class through type theory, and we interpret this partition by the sense of l-blocks of finite groups. We give an example where a block of Rep_k(SL_2(F)) is defined with several SL_2(F)-equivalent supercuspidal classes, which is different from the case where l is zero. We end this talk by giving a prediction on the block decomposition of Rep_k(A) for a general p-adic group A.

Starting from the principle of locality in quantum field theory, which
states that an object is influenced directly only by its immediate

surroundings, I will first briefly review some features of the notion of
locality arising in physics and mathematics. These are then encoded
in  locality relations, given by symmetric binary relations whose graph
consists of pairs of "mutually independent elements".

I will mention challenging questions that arise from  enhancing algebraic
structures to their locality counterparts, such as i) when  is the quotient
of a locality vector space by a linear subspace, a locality vector space, if
equipped with the quotient locality relation,  ii) when does  the locality
tensor product of two locality vector spaces  define a locality vector
space. These are discussed in recent joint work  with Pierre Clavier, Loïc
Foissy and Diego López.

Locality morphisms, namely maps that factorise on   products of  pairs of
"mutually independent" elements, play a key role in the context of
renormalisation in
multiple variables. They include "locality evaluators", which we use to

consistently evaluate meromorphic germs in several variables at
their poles. I will  also report on recent joint work with Li Guo and Bin
Zhang. which gives a classification of locality evaluators on certain
classes of algebras of meromorphic germs.

*** Cancelled *** We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean field limits of recent Ensemble Kalman methods for inverse problems. We show how the transport problem splits into two separate minimisation problems: one for the evolution of mean and covariance of the interpolating curve, and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities, as well as the dynamics of the associated gradient flows. This is joint work of Martin Burger, Matthias Erbar, Daniel Matthes and André Schlichting.

Modular forms are holomorphic functions on the upper half plane satisfying a transformation property under the action of Mobius transformations. While they are a priori complex-analytic objects, they have applications to number theory thanks to their connection with Galois representations. Weight one modular forms are special because their Galois representations factor through a finite quotient. In this talk, we will explain a different degeneracy: they contribute to the cohomology of a line bundle over the modular curve in degrees 0 and 1. We propose an arithmetic explanation for this: an action of a unit group associated to the Galois representation of the modular form. This extends the conjectures of Venkatesh, Prasanna, and Harris. Time permitting, we will discuss a generalization to Hilbert modular forms.

Bestvina--Feighn proved that Aut(F_n) is a rational duality group, i.e. there is a Q[Aut(F_n)]-module, called the rational dualizing module, and a form of Poincare duality relating the rational cohomology of Aut(F_n) to its homology with coefficients in this module. Bestvina--Feighn's proof does not give an explicit combinatorial description of the rational dualizing module of Aut(F_n). But, inspired by Borel--Serre's description of the rational dualizing module of arithmetic groups, Hatcher--Vogtmann constructed an analogous module for Aut(F_n) and asked if it is the rational dualizing module. In work with Miller, Nariman, and Putman, we show that Hatcher--Vogtmann's module is not the rational dualizing module.

We study elastic manifolds with self-repelling terms and estimate their effective radius. This class of manifolds is modelled by a self-repelling vector-valued Gaussian free field with Neumann boundary conditions over the domain [−N,N]^d∩Z^d, that takes values in R^D. Our main results state that for two dimensional domain and range (D=2 and d=2), the effective radius R_N​ of the manifold is approximately N. When the dimension of the domain is d=2 and the dimension of the range is D=1, the effective radius of the manifold is approximately N^{4/3}. This verifies the conjecture of Kantor, Kardar and Nelson (Phys. Rev. Lett. ’86). We also provide results for the case where d≥3 and D≤d. These results imply that self-repelling elastic manifolds with a low dimensional range undergo a significantly stronger stretching than in the case where d=D. 

This is a joint work with Carl Mueller.

For geometric variational problems one often only has weak, rather than strong, compactness results and hence has to deal with the problem that sequences of (almost) critical points $u_j$ can converge to a limiting object with different topology.

A major challenge posed by such singular behaviour is that the seminal results of Simon on Lojasiewicz inequalities, which are one of the most powerful tools in the analysis of the energy spectrum of analytic energies and the corresponding gradient flows, are not applicable.

In this talk we present a method that allows us to prove Lojasiewicz inequalities in the singular setting of almost harmonic maps that converge to a simple bubble tree and explain how these results allow us to draw new conclusions about the energy spectrum of harmonic maps and the convergence of harmonic map flow for low energy maps from surfaces of positive genus into general analytic manifolds.

I will discuss the use of partitioned schemes for neural networks. This work is in the tradition of multrate numerical ODE methods in which different components of system are evolved using different numerical methods or with different timesteps. The setting is the training tasks in deep learning in which parameters of a hierarchical model must be found to describe a given data set. By choosing appropriate partitionings of the parameters some redundant computation can be avoided and we can obtain substantial computational speed-up. I will demonstrate the use of the procedure in transfer learning applications from image analysis and natural language processing, showing a reduction of around 50% in training time, without impairing the generalization performance of the resulting models. This talk describes joint work with Tiffany Vlaar.

Liouville conformal field theory models two-dimensional gravity with a cosmological constant and conformal matter. In its timelike regime, it reproduces the characteristic negative kinetic term of the conformal factor of the metric in the Einstein-Hilbert action, the sign which infamously makes the gravity path integral ill-defined. In this talk, I will first discuss the perturbative computation of the timelike Liouville partition function around the sphere saddle and propose an all-orders result. I will then turn to the disk and present the bulk 1-point functions of this CFT, and discuss possible interpretations in terms of boundary conditions.

Waldspurger proved maximality and minimality results for certain generalised Springer representations of $\text{Sp}(2n,\mathbb{C})$. We will discuss analogous results for $G = \text{SO}(N,\mathbb{C})$ and sketch their proofs.

Let $C$ be a unipotent class of $G$ and $E$ an irreducible $G$-equivariant local system on $C$. Let $\rho$ be the generalised Springer representation corresponding to $(C,E)$. We call $C$ the support of $\rho$. It is well-known that $\rho$ appears in the top cohomology of a certain variety. Let $\bar\rho$ be the representation obtained by summing the cohomology groups of this variety.

We show that if $C$ is parametrised by an orthogonal partition consisting of only odd parts, then $\bar\rho$ has a unique irreducible subrepresentation $\rho^{\text{max}}$ whose support is maximal among the supports of the irreducible subrepresentations of $\rho^{\text{max}}$. We also show that $\text{sgn}\otimes\rho^{\text{max}}$ is the unique subrepresentation of $\text{sgn}\otimes\bar\rho$ with minimal support. We will also present an algorithm to compute $\rho^{\text{max}}$.

I will review the basic assumptions and spell out the arguments that lead to the bound on the Regge growth of gravitational scattering amplitudes. I will discuss the Regge bounds both at fixed transfer momentum and smeared over it. Our basic conclusion is that gravitational scattering amplitudes admit dispersion relations with two subtractions. For a sub-class of smeared amplitudes, black hole formation reduces the number of subtractions to one. Finally, I will discuss bounds on local growth derived using dispersion relations. This talk is based on https://arxiv.org/abs/2202.08280.

Irreversible processes are accompanied by an increase in the internal entropy of a continuum, and as such the entropy production function is fundamental in determining the overall state of the system. In this talk, it will be shown that the entropy production function can be utilized for a variational analysis of certain dissipative continua in two different ways. Firstly, a novel unified Lagrangian-Hamiltonian formalism is constructed giving phase space extra structure, and applied to the study of fluid flow and brittle fracture.  Secondly, a maximum entropy production principle is presented for simple bodies and its implications to the study of fluid flow discussed. 

Second-order regularity results are established for solutions to elliptic equations and systems with the principal part having a Uhlenbeck structure and square-integrable right-hand sides. Both local and global estimates are obtained. The latter apply to solutions to homogeneous Dirichlet problems under minimal regularity assumptions on the boundary of the domain. In particular, if the domain is convex, no regularity of its boundary is needed. A critical step in the approach is a sharp pointwise inequality for the involved elliptic operator. This talk is based on joint investigations with A.Kh.Balci, L.Diening, and V.Maz'ya.

A common observation that wider (in the number of hidden units/channels/attention heads) neural networks perform better motivates studying them in the infinite-width limit.

Remarkably, infinitely wide networks can be easily described in closed form as Gaussian processes (GPs), at initialization, during, and after training—be it gradient-based, or fully Bayesian training. This provides closed-form test set predictions and uncertainties from an infinitely wide network without ever instantiating it (!).

These infinitely wide networks have become powerful models in their own right, establishing several SOTA results, and are used in applications including hyper-parameter selection, neural architecture search, meta learning, active learning, and dataset distillation.

The talk will provide a high-level overview of our work at Google Brain on infinite-width networks. In the first part I will derive core results, providing intuition for why infinite-width networks are GPs. In the second part I will discuss challenges and solutions to implementing and scaling up these GPs. In the third part, I will conclude with example applications made possible with infinite width networks.

The talk does not assume familiarity with the topic beyond general ML background.