Past

A set in the Euclidean plane is called an integer distance set if the distance between any pair of its points is an integer.  All so-far-known integer distance sets have all but up to four of their points on a single line or circle; and it had long been suspected, going back to Erdős, that any integer distance set must be of this special form. In a recent work, joint with Marina Iliopoulou and Sarah Peluse, we developed a new approach to the problem, which enabled us to make the first progress towards confirming this suspicion.  In the talk, I will discuss the study of integer distance sets, its connections with other problems, and our new developments.

Nakajima quiver varieties form an important class of examples of conical symplectic singularities. For example, such varieties of dimension 2 are Kleinian singularities. Starting from this, I will describe a combinatorial approach to classifying the next case, affine quiver varieties of dimension 4. If time permits, I will try to say the implications we obtained and how can one compute the number of crepant symplectic resolutions of these varieties. This is a joint project with Samuel Lewis.

I will discuss the recent progress in the numerical bootstrap of the 3d Ising CFT using the correlation functions of stress-energy tensor and the relevant scalars. This numerical bootstrap setup gives excellent results which are two orders of magnitude more accurate than the previous world's best. However, it also presents many significant technical challenges. Therefore, in addition to describing in detail the numerical results of this work, I will also explain the state-of-the art numerical bootstrap methods that made this study possible. Based on arXiv:2411.15300 and work in progress.

When it comes to the nonlinear heat equation u_t - \Delta u = u^p, a sharp condition for the stability of positive radial steady states was derived in the classical paper by Gui, Ni and Wang.  In this talk, I will present some recent joint work with Daniel Devine that focuses on a more general system of reaction-diffusion equations (which is also also known as the parabolic Henon-Lane-Emden system).  We obtain a sharp condition that determines the stability of positive radial steady states, and we also study the separation property of these solutions along with their asymptotic behaviour at infinity.

A generalisation of Wiles’ famous modularity theorem, the Fontaine-Mazur conjecture, predicts that two dimensional representations of the absolute Galois group of the rationals, with a few specific properties, exactly correspond to those representations coming from classical modular forms. Under some mild hypotheses, this is now a theorem of Kisin. In this talk, I will explain how one can p-adically interpolate the objects on both sides of this correspondence to construct an eigensurface and “trianguline” Galois deformation space, as well as outline a new approach to proving a theorem of Emerton, that these spaces are often isomorphic.

We study the spectrum of the Laplacian on finite-area hyperbolic surfaces of large volume, focusing on small eigenvalues i.e. those below 1/4. I will discuss some recent results and open problems in this area. Based on joint works with Michael Magee and with Joe Thomas.
 

We provide sharp bounds in the supremum- and Hölder norm for parameter-dependent stochastic integrals. As an application we obtain novel long-term bounds for stochastic partial differential equations as well as novel bounds on the space-time modulus of continuity of the stochastic heat equation. This concerns joint work with Joris van Winden (TU Delft).

The fundamental group of a complex variety is finitely presented. The talk will survey algebraic variants (in fact, distant corollaries) of this fact, in the context of variants of the etale fundamental group. We will then zoom in on "tame" etale fundamental groups of p-adic analytic spaces. Our main result is that it is (topologically) finitely generated (for a quasi-compact and quasi-separated rigid space over an algebraically closed field).  The proof uses logarithmic geometry beyond its usual scope of finitely generated monoids to (eventually) reduce the problem to the more classical one of finite generation of tame fundamental groups of algebraic varieties over the residue field. This is joint work with Katharina Hübner, Marcin Lara, and Jakob Stix.

Attention-based models, such as Transformer, excel across various tasks but lack a comprehensive theoretical understanding, especially regarding token-wise sparsity and internal linear representations. To address this gap, we introduce the single-location regression task, where only one token in a sequence determines the output, and its position is a latent random variable, retrievable via a linear projection of the input. To solve this task, we propose a dedicated predictor, which turns out to be a simplified version of a non-linear self-attention layer. We study its theoretical properties, by showing its asymptotic Bayes optimality and analyzing its training dynamics. In particular, despite the non-convex nature of the problem, the predictor effectively learns the underlying structure. This work highlights the capacity of attention mechanisms to handle sparse token information and internal linear structures.

This is a joint work with Pierre Marion, Gérard Biau and Raphaël Berthier

Informally, monodromy captures the behavior of objects when one circles around a singularity. In persistent homology, non-trivial monodromy has been observed in the case of biparameter filtrations obtained by sublevel sets of a continuous function [1]. One might consider the fundamental group of an admissible open subspace of all lines defining linear one-parameter reductions of a bi-parameter filtration. Monodromy occurs when this fundamental group acts non-trivially on the persistence space, i.e. the collection of all the persistence diagrams obtained for each linear one-parameter reduction of the bi-parameter filtration. Here, under some tameness assumptions, we formalize the monodromy behavior in algebraic terms, that is in terms of the persistence module associated with a bi-parameter filtration. This allows to translate monodromy in terms of persistence module presentations as bigraded modules. We prove that non-trivial monodromy involves generators within the same summand in the direct sum decomposition of a persistence module. Hence, in particular interval-decomposable persistence modules have necessarily trivial monodromy group.

Despite the stereotype of the lone genius working by themselves, most professional mathematicians collaborate with others. But when you're learning maths as a student, is it OK to work with other people, or is that cheating? How do you build the skills and confidence to collaborate effectively? And where does AI fit into all this? In this session, we'll explore ways in which you can get the most out of collaborations with your fellow students, whilst avoiding inadvertently passing off other people's work as your own.

Hilbert’s fourteenth problem is concerned with whether invariant rings under algebraic group actions are finitely generated. A number of examples have been constructed since the mid-20th century which demonstrate that this is not always the case. However such examples by their nature are difficult to construct, and we know little about their underlying structure. This talk aims to provide an introduction to the topic of Hilbert’s fourteenth problem, as well as the finite generation ideal - a key tool used to further understand these counterexamples. We focus particularly on the example constructed by Daigle and Freudenberg at the turn of the 21st century, and describe the work undertaken to compute the finite generation ideal of this example. 

I will explain about how tubings of rooted trees can solve Dyson-Schwinger equations, and then summarize the two newer results in this direction, how to incorporate distinct insertion places and how when the Mellin transform is a reciprocal of a polynomial with rational roots, then one can use combinatorial techniques to obtain a system of differential equations that is perfectly suited to resurgent analysis.

Based on arXiv:2408.15883 (with Michael Borinsky and Gerald Dunne) and arXiv:2501.12350 (with Nick Olson-Harris).

The epithelial to mesenchymal transition (EMT) is a key cellular process underlying cancer progression, with multiple intermediate states whose molecular hallmarks remain poorly characterized. In this talk, I will describe AI-powered and ecology-inspired methods recently developed by us to provide a multi-scale view of the epithelial-mesenchymal plasticity in cancer from single cell and spatial transcriptomics data. First, we employed a large language model similar to the one underlying chatGPT but tailored for biological data (inspired by scBERT methodology), to predict individual stable states within the EMT continuum in single cell data and dissect the regulatory processes governing these states. Secondly, we leveraged spatial transcriptomics of breast cancer tissue to delineate the spatial relationships between cancer cells occupying distinct states within the EMT continuum and various hallmarks of the tumour microenvironment. We introduce a new tool, SpottedPy, that identifies tumour hotspots within spatial transcriptomics slides displaying enrichment in processes of interest, including EMT, and explores the distance between these hotspots and immune/stromal-rich regions within the broader environment at flexible scales. We use this method to delineate an immune evasive quasi-mesenchymal niche that could be targeted for therapeutic benefit. Our insights may inform strategies to counter immune evasion enabled by EMT and offer an expanded view of the coupling between EMT and microenvironmental plasticity in breast cancer.

We develop a new construction of complete non-compact 8-manifolds with holonomy equal to $\Spin(7)$. As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal $T^2$-bundles over asymptotically conical Calabi Yau manifolds. The resulting metrics have a new geometry at infinity that we call asymptotically $T^2$-fibred conical ($AT^2C$) and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperkähler geometry. We use the construction to produce infinite diffeomorphism types of $AT^2C$ $\Spin(7)$-manifolds and to produce the first known example of complete toric $\Spin(7)$-manifold.

A major open problem in the model theory of valued fields is to gain an understanding of the first-order theory of the power series field F((t)), where F denotes a finite field. For sufficiently "nice" henselian valued fields, the Ax-Kochen/Ershov philosophy allows to reduce questions of elementary equivalence and elementary embeddings to the analogous questions about the value group and residue field (or related structures). In my talk, I will present a new such principle which applies in particular to a large class of algebraic extensions of F((t)), albeit not to F((t)) itself. The talk is based on joint work with Konstantinos Kartas and Jonas van der Schaaf.

We investigate the strong data processing inequalities of contractive Markov Kernels under a specific f-divergence, namely the E-gamma-divergence. More specifically, we characterize an upper bound on the E-gamma-divergence between PK and QK, the output distributions of contractive Markov kernel K, in terms of the E-gamma-divergence between the corresponding input distributions P and Q. Interestingly, the tightest such upper bound turns out to have a non-multiplicative form. We apply our results to derive new bounds for the local differential privacy guarantees offered by the sequential application of a privacy mechanism to data and we demonstrate that our framework unifies the analysis of mixing times for contractive Markov kernels.

There is an idea, going back to work of Krasner, that p-adic fields tend to function fields as absolute ramification tends to infinity. We will present a new way of rigorizing this idea, as well as give applications to the local Langlands correspondence of Fargues–Scholze.

 Many optimal control, estimation and design problems can be formulated as so-called dynamic optimization problems, which are optimization problems with differential equations and other constraints. State-of-the-art methods based on collocation, which enforce the differential equations at only a finite set of points, can struggle to solve certain dynamic optimization problems, such as those with high-index differential algebraic equations, consistent overdetermined constraints or problems with singular arcs. We show how numerical methods based on integrating the differential equation residuals can be used to solve dynamic optimization problems where collocation methods fail. Furthermore, we show that integrated residual methods can be computationally more efficient than direct collocation.

This seminar takes place at RAL (Rutherford Appleton Lab). 

In this talk, we discuss the critical threshold phenomena in a large class of one-dimensional pressureless Euler-Poisson (EP) equations with non-vanishing background states. First, we establish local-in-time well-posedness in appropriate regularity spaces, specifically involving negative Sobolev spaces, which are adapted to ensure the neutrality condition holds. We show that this negative homogeneous Sobolev regularity is necessary by proving an ill-posedness result in classical Sobolev spaces when this condition is absent. Next, we examine the critical threshold phenomena in pressureless EP systems that satisfy the neutrality condition. We show that, in the case of attractive forcing, the neutrality condition further restricts the sub-critical region, reducing it to a single line in the phase plane. Finally, we provide an analysis of the critical thresholds for repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system in the context of cold plasma ion dynamics, where the electron density is governed by the Maxwell-Boltzmann relation. This talk is based on joint work with Dong-ha Kim, Dowan Koo, and Eitan Tadmor.

I will discuss two classes of effective, macroscopic models in elasticity: (i) 1D models applicable to thin structures, and (ii) homogenized 2D or 3D continua applicable to materials with a periodic microstructure. In both systems, the separation of scales calls for the definition of macroscopic models that slave fine-scale fluctuations to an effective, macroscopic deformation field. I will show how such models can be established in a systematic and rigorous way based on a two-scale expansion that accounts for nonlinear and higher-order (i.e. deformation gradient) effects. I will further demonstrate that the resulting models accurately predict nonlinear effects, finite size effects and localization for a set of examples. Finally, I will discuss two challenges that arise when solving these effective models: (1) missed boundary layer effects and (2) negative stiffness associated with higher-order terms.

In this talk, I will present low-regularity numerical methods for nonlinear dispersive PDEs, with unfiltered schemes analyzed in Sobolev spaces and filtered schemes in discrete Bourgain spaces, offering effective handling of low-regularity and even rough solutions. I will highlight the significance of exploring structure-preserving low-regularity schemes, as this is a crucial area for further research.

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form $\si(x)=g x$, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

A classical approach to studying the topology of a manifold is through the analysis of its submanifolds. The realm of 3-manifolds is particularly rich and diverse, and we aim to explore the complexity of surfaces within a given 3-manifold. After reviewing the fundamental definitions of the Thurston norm, we will present a constructive method for computing it on Seifert fibered manifolds and extend this approach to graph manifolds. Finally, we will outline which norms can be realized as the Thurston norm of some graph manifold and examine their key properties.