Past

We present results (joint with Hiroshi isozaki, Matti lassas, and Teemu Tyni) on reconstruction of certain nonlinear wave operators from knowledge of their far field effect on incoming waves. The result depends on the reformulation of the problem as a non-linear Goursat problem in the Penrose conformal compactification, for suitably small incoming waves. The non-linearity is exploited to generate secondary waves, which eventually probe the geometry of the space-time. Some extensions to cosmological space-times will also be discussed.  Time permitting, we will contrast these results with near-field inverse scattering obtained for only linear waves, where no non-linearity can be exploited, and the methods depend instead on unique continuation. (The latter joint with Ali Feizmohammadi and Lauri Oksanen). 

In this talk, I will discuss Figalli and Gigli’s formulation of optimal transport between non-negative Radon measures in the setting of metric pairs. This framework allows for the comparison of measures with different total masses by introducing an auxiliary set that compensates for mass discrepancies. Within this setting, classical characterisations of optimal transport plans extend naturally, and the resulting spaces of measures are shown to be complete, separable, geodesic, and non-branching, provided the underlying space possesses these properties. Moreover, we prove that the spaces of measures 
equipped with the $L^2$-optimal partial transport metric inherit non-negative curvature in the sense of Alexandrov. Finally, generalised spaces of persistence diagrams embed naturally into these spaces of measures, leading to a unified perspective from which several known geometric properties of generalised persistence diagram spaces follow. These results build on recent work by Divol and Lacombe and generalise classical results in optimal transport.

The main challenges in constructing a holographic correspondence for asymptotically flat spacetimes lie in the null nature of the conformal boundary and the non-conservation of gravitational charges in the presence of bulk radiation. In this talk, I shall demonstrate that there exists a systematic and mathematically robust approach to understanding and deriving the associated flux-balance laws from intrinsic boundary geometric considerations — an aspect of crucial importance for flat-space holography, as I shall argue during the presentation. 

For self-containment, I shall begin by reviewing key aspects of the geometry at null infinity, which has been termed conformal Carroll geometry. Reviving Ashtekar’s old statement, I shall emphasise that boundary affine connections possess degrees of freedom that precisely serve as the sources encoding radiation from a holographic perspective. I shall conclude by deriving flux-balance laws in an effective field theory framework at the boundary, employing novel techniques that introduce “hypermomenta” as responses to fluctuations in the boundary connection. The strength of our formalism lies in its ability to perform all computations in a manifestly coordinate- and Weyl-invariant manner within the framework of Sir Penrose’s conformal compactification.

Many justifications can be offered for the study of the history of mathematics. Here we focus on three, each of them illustrated by a specific historical example: it can aid in the learning of mathematics; it can prompt the development of new mathematics; and last but certainly not least – it's fun and interesting!

The category PAb of profinite abelian groups is an abelian category with many nice properties, which allows us to do most of standard homological algebra. The category PAb naturally embeds into the category TAb of topological abelian groups, but TAb is not abelian, nor does it have a satisfactory theory of tensor products. On the other hand, PAb also naturally embeds into the category CondAb of "condensed abelian groups", which is an abelian category with nice properties. We will show that the embedding of profinite modules into condensed modules (actually, into "solid modules") preserves usual homological notions such Ext and Tor, so that the condensed world might be a better place to study profinite modules than the topological world.

Recent progress in the development of equivariant neural network architectures predominantly used for machine learning interatomic potentials (MLIPs) has opened new possibilities in the development of data-driven coarse-graining strategies. In this talk, I will first present our work on the development of learning potential energy surfaces and other physical quantities, namely the Hyperactive Learning framework[1], a Bayesian active learning strategy for automatic efficient assembly of training data in MLIP and ACEfriction [2], a framework for equivariant model construction based on the Atomic Cluster Expansion (ACE) for learning of configuration-dependent friction tensors in the dynamic equations of molecule surface interactions and Dissipative Particle Dynamics (DPD). In the second part of my talk, I will provide an overview of our work on the simulation and analysis of Generalized Langevin Equations [3,4] as obtained from systematic coarse-graining of Hamiltonian Systems via a Mori-Zwanzig projection and present an outlook on our ongoing work on developing data-driven approaches for the construction of dynamics-preserving coarse-grained representations.

References:

[1] van der Oord, C., Sachs, M., Kovács, D.P., Ortner, C. and Csányi, G., 2023. Hyperactive learning for data-driven interatomic potentials. npj Computational Materials

[2] Sachs, M., Stark, W.G., Maurer, R.J. and Ortner, C., 2024. Equivariant Representation of Configuration-Dependent Friction Tensors in Langevin Heatbaths. to appear in Machine Learning: Science & Technology

[3] Leimkuhler, B. and Sachs, M., 2022. Efficient numerical algorithms for the generalized Langevin equation. SIAM Journal on Scientific Computing

[4] Leimkuhler, B. and Sachs, M., 2019. Ergodic properties of quasi-Markovian generalized Langevin equations with configuration-dependent noise and non-conservative force. In Stochastic Dynamics Out of Equilibrium: Institut Henri Poincaré, 2017 

Due to connections to flat space holography, Carrollian geometry, physics and fluid dynamics have received an explosion of interest over the last two decades. In the Carrollian limit of vanishing speed of light c, relativistic fluids reduce to a set of PDEs called the Carrollian fluid equations. Although in general these equations are not well understood, and their PDE theory does not appear to have been studied, in dimensions 1+1 it turns out that there is a duality with the Galilean compressible Euler equations in 1+1 dimensions inherited from the isomorphism of the Carrollian (c to 0) and Galilean (c to infinity) contractions of the Poincar\'e algebra. Under this duality time and space are interchanged, leading to different dynamics in evolution. I will discuss recent work with N. Athanasiou (Thessaloniki), M. Petropoulos (Paris) and S. Schulz (Pisa) in which we establish the first rigorous PDE results for these equations by introducing a notion of Carrollian isentropy and studying the equations using Lax’s method and compensated compactness. In particular, I will explain that there is global existence in rough norms but finite-time blow-up in smoother norms.

Galilean and Carrollian algebras are dual contractions of the Poincaré algebra. They act on two-dimensional Newton--Cartan and Carrollian manifolds and are isomorphic. A consequence of this property is a duality correspondence between one-dimensional Galilean and Carrollian fluids. I will describe the algebras and the dynamics of these systems as they emerge from the relevant  limits of Lorentzian hydrodynamics, and explore the advertised duality relationship. This interchanges longitudinal and transverse directions with respect to the flow velocity, and permutes equilibrium and out-of-equilibrium observables, unveiling specific features of Carrollian physics. I will also discuss the hydrodynamic-frame invariance in Lorentzian systems and its fate in the Galilean and Carrollian avatars.

Rank-based models for equity markets are reduced-form models where the asset dynamics depend on the rank that asset occupies in the investment universe. Such models are able to capture certain stylized macroscopic properties of equity markets, such as stability of the capital distribution curve and collision rates of stock rank switches. However, when calibrated to real equity data the models possess undesirable features such as an "Atlas stock" effect; namely the smallest security has an unrealistically large drift. Recently, Campbell and Wong (2024) identified that listings and delistings (i.e. entrances and exists) of securities in the market are important drivers for the stability of the capital distribution curve. In this work we develop a framework for ranked-based models with listings and delistings and calibrate them to data. By incorporating listings and delistings the calibration procedure no longer leads to an "Atlas stock" behaviour. Moreover, by studying an appropriate "local model", focusing on a specific target rank, we are able to connect collisions rates with a notion of particle density, which is more stable and easier to estimate from data than the collision rates. The calibration results are supported by novel theoretical developments such as a new master formula for functional generation of portfolios in this setting. This talk is based on joint work in progress with Martin Larsson and Licheng Zhang.  

A way to study rational points on a variety is by looking at their image in the p-adic points. Some natural questions that arise are the following: is there any obstruction to weak approximation on the variety? Which primes might be involved in it? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces.

The Landau equation stands as one of the fundamental equations in kinetic theory and plays a key role in plasma physics. However, computing it presents significant challenges due to the complexity of the Landau operator,  the dimensionality, and the need to preserve the physical properties of the solution. In this presentation, I will introduce deep learning assisted particle methods aimed at addressing some of these challenges. These methods combine the benefits of traditional structure-preserving techniques with the approximation power of neural networks, aiming to handle high dimensional problems with minimal training. 

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.

On this talk we will focus on the family of aggregation-diffusion equations

$$\frac{\partial \rho}{\partial t} = \mathrm{div}\left(\mathrm{m}(\rho)\nabla (U'(\rho) + V) \right).$$

Here, $\mathrm{m}(s)$ represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. $U$ corresponds to the diffusive potential and includes all the porous medium cases, i.e. $U(s) = \frac{1}{m-1} s^m$ for $m > 0$ or $U(s) = s \log (s)$ if $m = 1$. $V$ corresponds to the attractive potential and it is such that $V \geq 0$, $V \in W^{2, \infty}$.

Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. Furthermore, since this problem has a formal gradient-flow structure, we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu.

The talk presents joint work with Prof. J.A. Carrillo and Prof. D.  Gómez-Castro.

Inversion problems, such as full waveform inversion (FWI), based on wave propagation, are computationally costly optimization processes used in many applications, ranging from seismic imaging to brain tomography. In most of these uses, high-order methods are required for both accuracy and computational efficiency. Within finite element methods (FEM), using high(er)-order can provide accuracy and the usage of flexible meshes. However, FEM are rarely employed in connection with unstructured simplicial meshes because of the computational cost and complexity of code implementation. They are used frequently with quadrilateral or hexahedral spectral finite elements, but the mesh adaptivity on those elements has not yet been fully explored. In this work, we address these challenges by developing software that leverages accurate higher-order mass-lumped simplicial elements with a mesh-adaption parameter, allowing us to take advantage of the computational efficiency of newer mass-lumped simplicial elements together with waveform-adapted meshes and the accuracy of higher-order function spaces. We also calculate these mesh-related parameters and develop software for high-order spectral element methods, allowing mesh flexibility. We will also discuss future developments. The open-source code was implemented using the Firedrake framework and the Unified Form Language (UFL), a mathematical-based domain specific language, allowing flexibility in a wide range of wave-based problems. 

The beautiful displays exhibited by fish schools and bird flocks have long fascinated scientists, but the role of their complex behavior remains largely unknown. In particular, the influence of hydrodynamic interactions on schooling and flocking has been the subject of debate in the scientific literature. I will present a model for flapping wings that interact hydrodynamically in an inviscid fluid, wherein each wing is represented as a plate that executes a prescribed time-periodic kinematics. The model generalizes and extends thin-airfoil theory by assuming that the flapping amplitude is small, and permits consideration of multiple wings through the use of conformal maps and multiply-connected function theory. We find that the model predictions agree well with experimental data on freely-translating, flapping wings in a water tank. The results are then used to motivate a reduced-order model for the temporally nonlocal interactions between schooling wings, which consists of a system of nonlinear delay-differential equations. We obtain a PDE as the mean-field limit of these equations, which we find supports traveling wave solutions. Generally, our results indicate how hydrodynamics may mediate schooling and flocking behavior in biological contexts.

We introduce ultrasolid modules, a variant of complete topological vector spaces. In this setting, we will prove some results in commutative algebra and apply them to the deformation of algebraic varieties in the language of derived algebraic geometry.

We present an existence and uniqueness result for nonlinear Fokker--Planck equations driven by rough paths. These equations describe the evolution of the probability distributions associated with McKean--Vlasov stochastic dynamics under (rough) common noise.  A key motivation comes from the study of interacting particle systems with common noise, where the empirical measure converges to a solution of such a nonlinear equation. 
Our approach combines rough path theory and the stochastic sewing techniques with Lions' differential calculus on Wasserstein spaces.

This is joint work with Peter K. Friz and Wilhelm Stannat.

The Critical 2d Stochastic Heat Flow arises as a non-trivial solution
of the Stochastic Heat Equation (SHE) at the critical dimension 2 and at a phase transition point.
It is a log-correlated field which is neither Gaussian nor a Gaussian Multiplicative Chaos.
We will review the phase transition of the 2d SHE, describe the main points of the construction of the Critical 2d SHF
and outline some of its features and related questions. Based on joint works with Francesco Caravenna and Rongfeng Sun.

A classical result of Dixmier and Douady enables us to classify locally trivial bundles of C*-algebras with compact operators as fibres via methods in homotopy theory. Dadarlat and Pennig have shown that this generalises to the much larger family of bundles of stabilised strongly self-absorbing C*-algebras, which are classified by the first group of the cohomology theory associated to the units of complex topological K-theory. Building on work of Evans and Pennig I consider Z/pZ-equivariant C*-algebra bundles over Z/pZ-spaces. The fibres of these bundles are infinite tensor products of the endomorphism algebra of a Z/pZ-representation. In joint work with Pennig, we show that the theory refines completely to this equivariant setting. In particular, we prove a full classification of the C*-algebra bundles via equivariant stable homotopy theory.

I will discuss various results and conjectures about off-diagonal hypergraph Ramsey numbers, focusing on recent developments.

I am interested in studying moduli spaces and associated enumerative invariants via degeneration techniques. Logarithmic geometry is a natural language for constructing and studying relevant moduli spaces. In this talk I  will explain the logarithmic Hilbert (or more generally Quot) scheme and outline how the construction helps study enumerative invariants associated to Hilbert/Quot schemes- a story we now understand well. Time permitting, I will discuss some challenges and key insights for studying moduli of stable vector bundles/ sheaves via similar techniques - a theory whose details are still being worked out. 

In this talk I will give an introduction to analogues to the classical finiteness conditions FP_n for totally disconnected locally compact groups. I will present a construction of non-discrete tdlc groups of arbitrary finiteness length. As a bi-product we also obtain a new collection of (discrete) Thompson-like groups which contains, for all positive integers n, groups of type FP_n but not of type FP_{n+1}. This is joint work with I. Castellano, B. Marchionna, and Y. Santos-Rego.

Networks provide a powerful language to model interacting systems by representing their units as nodes and the interactions between them as links. Interactions can be connotated in several ways, such as binary/weighted, undirected/directed, etc. In the present talk, we focus on the positive/negative connotation - modelling trust/distrust, alliance/enmity, friendship/conflict, etc. - by considering the so-called signed networks. Rooted in the psychological framework of the balance theory, the study of signed networks has found application in fields as different as biology, ecology, economics. Here, we approach it from the perspective of statistical physics by extending the framework of Exponential Random Graph Models to the class of binary un/directed signed networks and employing it to assess the significance of frustrated patterns in real-world networks. As our results reveal, it critically depends on i) the considered system and ii) the employed benchmark. For what concerns binary directed networks, instead, we explore the relationship between frustration and reciprocity and suggest an alternative interpretation of balance in the light of directionality. Finally,  leveraging the ERGMs framework, we propose an unsupervised algorithm to obtain statistically validated projections of bipartite signed networks, according to which any, two nodes sharing a statistically significant number of concordant (discordant) motifs are connected by a positive (negative) edge, and we investigate signed structures at the mesoscopic scale by evaluating the tendency of a configuration to be either `traditionally' or `relaxedly' balanced.

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.