Past

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.

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves. I will also mention some applications to rational points on certain families of curves.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

There is a wide range of problems in continuum mechanics that involve transformation fronts, which are non-stationary interfaces between two different phases in a phase-transforming or a chemically-transforming material. From the mathematical point of view, the considered problems are represented by systems of non-linear PDEs with discontinuities across non-stationary interfaces, kinetics of which depend on the solution of the PDEs. Such problems have a significant industrial relevance – an example of a transformation front is the localised stress-affected chemical reaction in Li-ion batteries with Si-based anodes. Since the kinetics of the transformation fronts depends on the continuum fields, the transformation front propagation can be decelerated and even blocked by the mechanical stresses. This talk will focus on three topics: (1) the stability of the transformation fronts in the vicinity of the equilibrium position for the chemo-mechanical problem, (2) a fictitious-domain finite-element method (CutFEM) for solving non-linear PDEs with transformation fronts and (3) an applied problem of Si lithiation.

From a model theoretic point of view, local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are much less well understood than their characteristic zero counterparts - the fields of real, complex and p-adic numbers. I will discuss different approaches to axiomatize and decide at least their existential theory in various languages and under various forms of resolution of singularities. This includes new joint work with Sylvy Anscombe and Philip Dittmann.

This will be a gentle introduction to the theory of pseudo-Anosov  flows on 3-manifolds, as seen from the perspective of a topologist and not a dynamicist.

I will start by considering geodesic flows on the unit tangent bundles of hyperbolic surfaces. This will lead to a definition of an Anosov and then a pseudo-Anosov flow on a 3-manifold. After discussing a couple of examples, I will outline some connections between pseudo-Anosov flows and other aspects of 3-manifold topology/ geometry/ group theory.

The notion of topological order was introduced by Xiao-Gang Wen in order to capture the features of the exotic phases of matter given by fractional quantum Hall phases. I will motivate why the corresponding mathematical structures are higher categories with additional properties. In 2+1-dimensions, I will explain in details how the definition of fusion category arises from physical and geometrical intuitions about topological orders. Finally, I will sketch how the notion of higher fusion category emerges in higher dimensions.

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo p. This is joint work with Justin Desrochers available at https://arxiv.org/abs/2201.04606.

In this talk we will discuss the correlations of the Riemann Zeta in various ranges, and prove a new result for correlations of squares. This problem is closely related to correlations of the characteristic polynomial of CUE with a very subtle difference. We will explain where this difference comes from, and what it means for the moments of moments of the Riemann Zeta, and its maximum in short intervals.

We develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut, resulting in a compact cut space which is needed for localisation. We introduce the notion of a moment polyptych associated to a hypertoric variety and prove that the necessary isotropy data can be read off from it. Finally, the equivariant Hirzebruch-Riemann-Roch formula is applied to the cut spaces and expresses the dimension of the equivariant quantisation space as a finite sum over the fixed-points. This is joint work with Johan Martens.

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue coloring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Trujić showed that if $H$ is a graph on $n$ vertices and maximum degree three, then $\hat{r}(H) = O(n^{8/5})$, improving upon the bound of $n^{5/3 + o(1)}$ by Kohayakawa, Rödl, Schacht and Szemerédi. In our paper, we show that $\hat{r}(H)\leq n^{3/2+o(1)}$. While the previously used host graphs were vanilla binomial random graphs, we prove our result by using a novel host graph construction.
We also discuss why our bound is a natural barrier for the existing methods.
This is joint work with Kalina Petrova.

Chimera states have attracted significant attention as symmetry-broken states exhibiting the coexistence of coherence and incoherence. Despite the valuable insights gained by analyzing specific systems, the understanding of the physical mechanism underlying the emergence of chimeras has been incomplete. In this presentation, I will argue that an important class of stable chimeras arise because coherence in part of the system is sustained by incoherence in the rest of the system. This mechanism may be regarded as a deterministic analog of noise-induced synchronization and is shown to underlie the emergence of so-called strong chimeras. These are chimera states whose coherent domain is formed by identically synchronized oscillators. The link between coherence and incoherence revealed by this mechanism also offers insights into the dynamics of a broader class of states, including switching chimera states and incoherence-mediated remote synchronization.