Past

Hirschman-Widder densities may be viewed as the probability density functions of positive linear combinations of independent and identically distributed exponential random variables. They also arise naturally in the study of Pólya frequency functions, which are integrable functions that give rise to totally positive Toeplitz kernels. This talk will introduce the class of Hirschman-Widder densities and discuss some of its properties. We will demonstrate connections to Schur polynomials and to orbital integrals. We will conclude by describing the rigidity of this class under composition with polynomial functions.

 This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).

Computers have been used to process natural language for many years. This talk considers two historical examples of computers used rather to play with human language, one well-known and the other a new archival discovery: Strachey’s 1952 love letters program, and a poetry programming competition held at Newcastle University in 1968. Strachey’s program used random number generation to pick words to fit into a template, resulting in letters of varying quality, and apparently much amusement for Strachey. The poetry competition required the entrants, mostly PhD students, to write programs whose output or source code was in some way poetic: the entries displayed remarkable ingenuity. Various analyses of Strachey’s work depict it as a parody of attitudes to love, an artistic endeavour, or as a technical exploration. In this talk I will consider how these apply to the Newcastle competition and add my own interpretations.

In 2004, Diaconis and Gamburd computed statistics of secular coefficients in the circular unitary ensemble. They expressed the moments of the secular coefficients in terms of counts of magic squares. Their proof relied on the RSK correspondence. We'll present a combinatorial proof of their result, involving the characteristic map. The combinatorial proof is quite flexible and can handle other statistics as well. We'll connect the result and its proof to old and new questions in number theory, by formulating integer and function field analogues of the result, inspired by the Random Matrix Theory model for L-functions.

Partly based on the arXiv preprint https://arxiv.org/abs/2102.11966

Global singularity theory is a classical subject which classifies singularities of maps between manifolds, and describes topological reasons for their appearance. I will start with explaining a central problem of the subject regarding multipoint and multisingularity loci, then give an introduction into some recent major developments by Kazarian, Rimanyi, Szenes and myself.

Regularization of linear least squares problems is necessary in a variety of contexts. However, the optimal regularization parameter is usually unknown a priori and is often to be determined in an ad hoc manner, which may involve solving the problem for multiple regularization parameters. In this talk, we will discuss three randomized algorithms, building on the sketch-and-precondition framework in randomized numerical linear algebra (RNLA), to efficiently solve this set of problems. In particular, we consider preconditioners for a set of Tikhonov regularization problems to be solved iteratively. The first algorithm is a Cholesky-based algorithm employing a single sketch for multiple parameters; the second algorithm is SVD-based and improves the computational complexity by requiring a single decomposition of the sketch for multiple parameters. Finally, we introduce an algorithm capable of exploiting low-rank structure (specifically, low statistical dimension), requiring a single sketch and a single decomposition to compute multiple preconditioners with low-rank structure. This algorithm avoids the Gram matrix, resulting in improved stability as compared to related work.

Combining physics with machine learning is a rapidly growing field of research. Thereby, most work focuses on leveraging machine learning methods to solve problems in physics. Here, however, we focus on the reverse direction of leveraging structure of physical systems (e.g. dynamical systems modeled by ODEs or PDEs) to construct novel machine learning algorithms, where the existence of highly desirable properties of the underlying method can be rigorously proved. In particular, we propose several physics-inspired deep learning architectures for sequence modelling as well as for graph representation learning. The proposed architectures mitigate central problems in each corresponding domain, such as the vanishing and exploding gradients problem for recurrent neural networks or the oversmoothing problem for graph neural networks. Finally, we show that this leads to state-of-the-art performance on several widely used benchmark problems.

Time-optimal control can be used to improve driving efficiency for autonomous
vehicles and it enables us explore vehicle and driver behaviour in extreme
situations. Due to the computational cost and limited scope of classical
optimal control methods we have seen new interest in applying reinforcement
learning algorithms to autonomous driving tasks.
In this talk we present methods for translating time-optimal vehicle control
problems into reinforcement learning environments. For this translation we
construct a sequence of environments, starting from the closest representation
of our optimisation problem, and gradually improve the environments reward
signal and feature quality. The trained agents we obtain are able to generalise
across different race tracks and obtain near optimal solutions, which can then
be used to speed up the solution of classical time-optimal control problems.

In the first half of the talk, we will explore the concept of a characteristic class of a subvariety in a smooth ambient space. We will focus on the so-called stable envelope class,  in cohomology, K theory, and elliptic cohomology (due to Okoukov-Maulik-Aganagic). Stable envelopes have rich algebraic combinatorics, they are at the heart of enumerative geometry calculations, they show up in the study of associated (quantum) differential equations, and they are the main building blocks of constructing quantum group actions on the cohomology of moduli spaces.

In the second half of the talk, we will study a generalization of Nakajima quiver varieties called Cherkis’ bow varieties. These smooth spaces are endowed with familiar structures: holomorphic symplectic form, tautological bundles, torus action. Their algebraic combinatorics features a new powerful operation, the Hanany-Witten transition. Bow varieties come in natural pairs called 3d mirror symmetric pairs. A conjecture motivated by superstring theory predicts that stable envelopes on 3d mirror pairs are equal (in a sophisticated sense that involves switching equivariant and Kahler parameters). I will report on a work in progress, with T. Botta, to prove this conjecture.

When you ask someone what maths is, their answer will massively depend on how they’ve been exposed to maths up until that point. From a 10-year-old who will tell you it’s adding up numbers, to a Fields medalist who may say to you about the idea of abstraction of logical ideas, there is no clear consensus as to the “right” answer to this question. Our individual journeys as mathematicians give us a clear idea about what it means to us, and this affects how we then communicate our ideas to an audience of other mathematicians and the general public. However, a pitfall that we easily fall into as a result is forgetting that others can understand maths in a different way to ourselves, and by only offering our preferred perspective, we are missing out on the chance to effectively communicate our ideas.

In this talk, I will explore how our individual understanding of what mathematics is can shape our methods of communication. I will review which methods of communication mathematicians utilise, and show examples where each method does well, and not so well.  Examples of communication methods include writing equations, plotting graphs, creating diagrams and storytelling. Given this, I will cover how by using a collection of these different methods, you can increase the impact of your research by engaging with the various different mindsets your audience may have on what mathematics is.

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

In the plane, we know that area of a set is monotone with respect to the inclusion but perimeter fails, in general. If we consider only bounded and convex sets, then also the perimeter is monotone. This property allows us to estimate the minimum number of convex components of a nonconvex set.

When studying integral functionals of the calculus of variations, convexity with respect to minors of the Jacobian matrix is a nice tool for proving existence and regularity of minimizers.

Sometimes it happens that the infimum of a functional on a set is less then the infimum taken on a dense subset: we usually refer to it as Lavrentiev phenomenon. In order to avoid it, convexity helps a lot.

Gerbes are geometric objects describing the third integer cohomology group of a manifold and the B-field in string theory. Like line bundles, they admit connections and gauge symmetries. In contrast to line bundles, however, there are now isomorphisms between gauge symmetries: the gauge group of a gerbe is a smooth 2-group. Starting from a hands-on example, I will explain gerbes and some of their properties. The main topic of this talk will then be the study of symmetries of gerbes on a manifold with G-action, and how these symmetries assemble into smooth 2-group extensions of G. In the last part, I will survey how this construction can be used to provide a new smooth model for the String group, via a theory of ∞-categorical principal bundles and group extensions.

We consider the problem of finding the best memoryless stochastic policy for an infinite-horizon partially observable Markov decision process (POMDP) with finite state and action spaces with respect to either the discounted or mean reward criterion. We show that the (discounted) state-action frequencies and the expected cumulative reward are rational functions of the policy, whereby the degree is determined by the degree of partial observability. We then describe the optimization problem as a linear optimization problem in the space of feasible state-action frequencies subject to polynomial constraints that we characterize explicitly. This allows us to address the combinatorial and geometric complexity of the optimization problem using tools from polynomial optimization. In particular, we estimate the number of critical points and use the polynomial programming description of reward maximization to solve a navigation problem in a grid world. The talk is based on recent work with Johannes Müller.

The so called Drinfeld conjecture states that the complement to very stable bundles has pure codimension one in the moduli space of vector bundles. In this talk I will explain a constructive proof in rank three, and discuss if/how it generalises to wobbly fixed points of the nilpotent cone as defined by Hausel and Hitchin. This is joint work with Pauly (Nice).

In this talk I will describe a systematic approach, introduced in our recent work 2111.08022, to construct Lagrangian descriptions for a class of strongly interacting N=2 SCFTs. I will review the main ingredients of these constructions, namely brane tilings and the connection to gauge theories. For concreteness, I will then specialize to the case of the simplest of such geometrical setups, as in the paper, even though our approach should be much more general. I will comment on some low rank examples of the theories we built, that are well understood by (many) alternative approaches and conclude with some open questions and ideas for future directions to explore.

I will report on my most recent results  with Jeremie Szeftel and Elena Giorgi which conclude the proof of the nonlinear, unconditional, stability of slowly rotating Kerr metrics. The main part of the proof, announced last year, was conditional on results concerning boundedness and decay estimates for nonlinear wave equations. I will review the old results and discuss how the conditional results can now be fully established.

We will look at the branching of irreducible representations of symmetric groups from the perspective of Okounkov-Vershik, and then look at Hecke algebras, affine Hecke algebras and cyclotomic Hecke algebras, in particular how the graded Grothendieck groups of their module categories “are” irreducible highest weight modules for affine $sl_l$, where $l$ is the “quantum characteristic”, and the branching graph is a highest weight crystal (for affine $sl_l$). The Fock space realisation of the highest weight crystal will get us back to  the Young graph for in the case of the symmetric group that we considered at the beginning.

During growth and development, tissue dynamics, such as tissue folding, cell intercalations and oriented cell divisions, are critical for shaping tissues and organs. However, less is known about how tissues regulate their dynamics during tissue homeostasis and repair, to maintain their shape after development. In this talk, we will discuss how differential growth rates can generate precise folds in tissues. We will also discuss how tissues respond to mechanical perturbations, such as stretching or wounding, by altering their actomyosin contractile structures, to change tissue dynamics, and thus preserve tissue shape and patterning. We combine genetics, biophysics and computational modelling to study these processes.

The questions we would like to answer are as follows:

1. Given three distributions pdf1, pdf2 and pdf-so, is it always possible to find a joint-distribution consistent with those 3 one-dimensional distributions?
2. Assuming that we are in a situation where (1) holds, can we find a nonparametric joint-distribution consistent with the 3 given one-dimensional distributions?
3. If (2) leads to an under-determined problem, can we find a joint-distribution that is “as close as possible” to the historical joint distribution?
4. Can we achieve (3) with a neural network?
5. If we observe the marginal and spread distributions for multiple maturities T, can we specify the evolution of pdf(T), possibly using neural differential equations?

In General Relativity, an impulsive gravitational wave is a localized and singular solution of the 

Einstein equations modeling the spacetime distortions created by a strongly gravitating source.
I will present a comprehensive theory allowing for ternary interactions of such impulsive gravitational waves in translation-symmetry, offering the first examples of such an interaction.  

The proof combines new techniques from harmonic analysis, Lorentzian geometry, and hyperbolic PDEs that are helpful to treat highly anisotropic low-regularity questions beyond the considered problem.  

This is joint work with Jonathan Luk.

Hilbert-Schmidt independence criterion (HSIC) is among the most widely-used approaches in machine learning and statistics to measure the independence of random variables. Despite its popularity and success in numerous applications, quite little is known about when HSIC characterizes independence. I am going to provide a complete answer to this question, with conditions which are often easy to verify in practice.

This talk is based on joint work with Bharath Sriperumbudur.