Past events

 In this short course, we will discuss the Euler equations and applications in gas dynamics and geometry. First, the basic theory of Euler equations and mixed-type problems will be reviewed. Then we will present the results on the transonic flows past obstacles, transonic flows in the fluid dynamic formulation of isometric embeddings, and the transonic flows in nozzles. We will discuss global solutions and stability obtained through various techniques and approaches. The short course consists of three parts and is accessible to PhD students and young researchers.

Dynamical interface motions are important flow patterns and fundamental free boundary problems in fluid mechanics, and have attracted huge attention in the mathematical community. Such waves for purely inviscid fluids are subject to various instabilities such as Kelvin-Helmholtz and Rayleigh-Taylor instabilities unless other stabilizing effects such as surface tension, Taylor-sign conditions or dissipations are imposed. However, in the presence of magnetic fields, it has been known that tangential magnetic fields may have stabilizing effects for free surface waves such as plasma-vacuum or plasma-plasma interfaces (at least locally in time), yet whether transversal magnetic fields (which occurs often for interfacial waves for astrophysical plasmas) can stabilize typical free interfacial waves remain to be some open problems. In this talk, I will show the stabilizing effects of the transversal magnetic fields for some interfacial waves for both compressible and incompressible multi-dimensional magnetohydrodynamics (MHD).

First, I will present the local (in time) well-posedness in Sobolev space of multi- dimensional compressible MHD contact discontinuities, which are the most typical interfacial waves for astrophysical plasma and prototypical fundamental waves for systems of hyperbolic conservations. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversally, which leads to a two-phase free boundary problem that may have nonlinear Rayleigh- Taylor instability and whose front symbols have no ellipticity. We overcome such difficulties by exploiting full the transversality of the magnetic fields and designing a nonlinear approximate problem, which yields the local well-posed without loss of derivatives and without any other conditions such as Rayleigh-Taylor sign conditions or surface tension. Second, I will discuss some results on the global well-posedness of free interface problems for the incompressible inviscid resistive MHD with transversal magnetic fields. Both plasma-vacuum and plasma-plasma interfaces are studied. The global in time well-posedness of both interface problems in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field near an equilibrium are established, which reveals the strong stabilizing effect of the transversal field as the global well- posedness of the free boundary incompressible Euler equations (without the irrotational assumptions) around an equilibrium is unknown. This talk is based on joint work with Professor Yanjin Wang. 

Apart from the binomial coefficients which are ubiquitous in many counting problems, the Catalan and Fibonacci sequences seem to appear almost as frequently. There are also well-known interpretations of the Catalan numbers as lattice paths, or as the number of ways of connecting 2n points on a circle via non-intersecting lines. We start by obtaining some identities for sums involving the Catalan sequence. In addition, we use the beautiful binomial transform which allows us to obtain several pretty identities involving Fibonacci numbers, Catalan numbers, and trigonometric sums.

Speaker: James Taylor
Title: D-Modules and p-adic Representations

Abstract: The representation theory of finite groups is a beautiful and well-understood subject. However, when one considers more complicated groups things become more interesting, and to classify their representations is often a much harder problem. In this talk, I will introduce the classical theory, the particular groups I am interested in, and explain how one might hope to understand their representations through the use of D-modules - the algebraic incarnation of differential equations.

Speaker: Anthony Webster
Title: An Introduction to Epidemiology and Causal Inference

Abstract: This talk will introduce epidemiology and causal inference from the perspective of a statistician and former theoretical physicist. Despite their studies being underpinned by deep and often complex mathematics, epidemiologists are generally more concerned by seemingly mundane information about the relationships between potential risk factors and disease. Because of this, I will argue that a good epidemiologist with minimal statistical knowledge, will often do better than a highly trained statistician. I will also argue that causal assumptions are a necessary part of epidemiology, should be made more explicitly, and allow a much wider range of causal inferences to be explored. In the process, I will introduce ideas from epidemiology and causal inference such as Mendelian Randomisation and the "do calculus", methodological approaches that will increasingly underpin data-driven population research.  

Training deep learning models involves searching for a good model over the space of possible architectures and their parameters. Discovering models that exhibit robust generalization to unseen data and tasks is of paramount for accurate and reliable machine learning. Generalization, a hallmark of model efficacy, is conventionally gauged by a model's performance on data beyond its training set. Yet, the reliance on vast training datasets raises a pivotal question: how can deep learning models transcend the notorious hurdle of 'memorization' to generalize effectively? Is it feasible to assess and guarantee the generalization prowess of deep neural networks in advance of empirical testing, and notably, without any recourse to test data? This inquiry is not merely theoretical; it underpins the practical utility of deep learning across myriad applications. In this talk, I will show that scrutinizing the training dynamics of neural networks through the lens of topology, specifically using 'persistent-homology dimension', leads to novel bounds on the generalization gap and can help demystifying the inner workings of neural networks. Our work bridges deep learning with the abstract realms of topology and learning theory, while relating to information theory through compression.

Bone regeneration processes are complex multiscale intrinsic mechanisms in bone tissue whose primary outcome is restoring function and form to a bone insufficiency. The effect of mechanics on the newly formed bone (the woven bone), is fundamental, at the tissue, cellular or even molecular scale. However, at these multiple scales, the identification of the mechanical parameters and their mechanisms of action are still unknown and continue to be investigated. This concept of mechanical regulation of biological processes is the main premise of mechanobiology and is used in this seminar to understand the multiscale response of the woven bone to mechanical factors in different bone regeneration processes: bone transport, bone lengthening and tissue engineering. The importance of a multidisciplinary approach that includes both in vivo and in silico modeling will be remarked during the seminar.

String topology is an umbrella under which lives a family of algebraic structures on the homology of the (compact-open) loop space of a closed smooth manifold, M. Of great interest are the string product and coproduct, in view of the failure of the latter to be a homotopy invariant. We will discuss some existing algebraic and geometric perspectives on these operations, and give some examples that probe the extent to which the string coproduct fails to be a homotopy invariant. We will sketch an alternative point of view on string topology as the study of the derived bornological smooth loop stack and explain why this is a promising model for the observed phenomena of string topology.

Predicting real-world phenomena often requires an understanding of their causal relations, not just their statistical associations. I will begin this talk with a brief introduction to the field of causal inference in the classical case of structural causal models over directed acyclic graphs, and causal discovery for static variables. Introducing the temporal dimension results in several interesting complications which are not well handled by the classical framework. The main component of a constraint-based causal discovery procedure is a statistical hypothesis test of conditional independence (CI). We develop such a test for stochastic processes, by leveraging recent advances in signature kernels. Then, we develop constraint-based causal discovery algorithms for acyclic stochastic dynamical systems (allowing for loops) that leverage temporal information to recover the entire directed graph. Assuming faithfulness and a CI oracle, our algorithm is sound and complete. We demonstrate strictly superior performance of our proposed CI test compared to existing approaches on path-space when tested on synthetic data generated from SDEs, and discuss preliminary applications to finance. This talk is based on joint work with Georg Manten, Cecilia Casolo, Søren Wengel Mogensen, Cristopher Salvi and Niki Kilbertus: https://arxiv.org/abs/2402.18477 .

We introduce the notion of a tensorially absorbing inclusion of C*-algebras, i.e., when a unital inclusion absorbs a strongly self-absorbing C*-algebra. This is a strong condition that ensures certain properties of both algebras (and their intermediate subalgebras) in a very strong sense. We discuss such inclusions, their non-triviality, and how often these inclusions appear.

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.

Many researchers are developing stable and accurate numerical methods for flow problems, which roughly belong to upwind methods or characteristics(-based) methods. 
The Lagrange-Galerkin method proposed and analyzed in, e.g., [O. Pironneau. NM, 1982] and [E. S\"uli. NM, 1988] is the finite element method combined with the idea of the method of characteristics; hence, it belongs to the characteristics(-based) methods. The advantages are the CFL-free robustness for convection-dominated problems and the symmetry of the resulting coefficient matrix. In this talk, we introduce stabilized Lagrange-Galerkin schemes of second order in time for viscous and viscoelastic flow problems, which employ the cheapest conforming P1-element with the help of pressure-stabilization [F. Brezzi and J. Pitk\"aranta. Vieweg+Teubner, 1984] for all the unknown functions, i.e., velocity, pressure, and conformation tensor, reducing the number of DOFs. 
Focusing on the recent developments of discretizations of the (non-conservative and conservative) material derivatives and the upper-convected time derivative, we present theoretical and numerical results.

The Fokker-Planck equation is one of the major tools of statistical physics in the description of stochastic processes, with numerous applications in physics, chemistry and biology. In the case of heterogeneous diffusions, the formulation of the equation depends on the choice of the discretization of the stochastic integral in the underlying Langevin-equation due to the multiplicative noise. In the Fokker-Planck equation, the choice of discretization then enters as a parameter in the definition of drift and diffusion terms. I show how both short- and long-time limits are affected by this choice. In the long-time limit, the existence of normalizable probability distribution functions is not always guaranteed which can be remedied by invoking elements of infinite ergodic theory. 

[1] S. Giordano, F. Cleri, R. Blossey, Phys Rev E 107, 044111 (2023)

[2] T. Dupont, S. Giordano, F. Cleri, R. Blossey, arXiv:2401.01765 (2024)

Aggregation-diffusion equations and systems have garnered much attention in the last few decades. More recently, models featuring nonlocal interactions through spatial convolution have been applied to several areas, including the physical, chemical, and biological sciences. Typically, one can establish the well-posedness of such models via regularity assumptions on the kernels themselves; however, more effort is required for many scenarios of interest as the nonlocal kernel is often discontinuous.

In this talk, I will present recent progress in establishing a robust well-posedness theory for a class of nonlocal aggregation-diffusion models with minimal regularity requirements on the interaction kernel in any spatial dimension on either the whole space or the torus. Starting with the scalar equation, we first establish the existence of a global weak solution in a small mass regime for merely bounded kernels. Under some additional hypotheses, we show the existence of a global weak solution for any initial mass. In typical cases of interest, these solutions are unique and classical. I will then discuss the generalisation to the $n$-species system for the regimes of small mass and arbitrary mass. We will conclude with some consequences of these theorems for several models typically found in ecological applications.

This is joint work with Dr. Jakub Skrzeczkowski and Prof. Jose Carrillo.

I will talk about some model-theoretic properties of Booleanizations of theories, subdirect products of structures, and sheaves of structures. I will discuss a result of Macintyre from 1973 on model-completeness, and more recent results jointly with Ehud Hrushovski and with Angus Macintyre.

Ancient Egypt is credited (along with Mesopotamia) for providing the oldest extant mathematical texts. Since the 19th century, when the first edition of the Rhind mathematical papyrus was published, it has held an important role in the historiography of mathematics. One of the earliest researchers in the field of ancient Egyptian sciences was Otto Neugebauer who has been a major influence on the early development of the field. While research in Egyptian mathematics initially focused on those aspects that could be linked to its possible successors in modern mathematics, research has also revealed various characteristics that could not easily be transferred into a modern equivalent. In addition, research on other sciences, like medicine and astronomy, has yielded further evidence that a limitation on those aspects that have successors in modern sciences will at best give an incomplete picture of ancient scholarship. This will be explored in a new long-term project, which is briefly sketched. In the context of this project, Egyptian mathematics is also studied. The talk will present an example from the terminology used in Egyptian mathematical texts to describe this area of knowledge and explore its epistemological consequences for our studies of ancient Egyptian mathematics and aim to situate it in its ancient context.

In the world of von Neumann algebras, the factors that do not have a trace, the so-called type III factors, are the most difficult to study. Some of their key structural properties are still not well-understood. In this talk, I will give a gentle introduction to Connes's Bicentralizer Problem, which is the most important open problem in the theory of type III factors. I will then present some recent progress on this problem and its applications.

We will present a matching upper and lower bound for the right tail probability of the maximum of a random model of the Riemann zeta function over short intervals.  In particular, we show that the right tail interpolates between that of log-correlated and IID random variables as the interval varies in length. We will also discuss a new normalization for the moments over short intervals. This result follows the recent work of Arguin-Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over short intervals.

Stable commutator length is a measure of homological complexity of group elements, which is known to take large values in the presence of various notions of negative curvature. We will present a new geometric proof of a theorem of Heuer on sharp lower bounds for scl in right-angled Artin groups. Our proof relates letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups) to negatively curved angle structures for surfaces estimating scl.

We consider the problem of approximating singular values of a matrix when provided with approximations to the leading singular vectors. In particular, we focus on the Generalized Nystrom (GN) method, a commonly used low-rank approximation, and its error in extracting singular values. Like other approaches, the GN approximation can be interpreted as a perturbation of the original matrix. Up to orthogonal transformations, this perturbation has a peculiar structure that we wish to exploit. Thus, we use the Jordan-Wieldant Theorem and similarity transformations to generalize a matrix perturbation theory result on eigenvalues of a perturbed Hermitian matrix. Finally, combining the above,  we can derive a bound on the GN singular values approximation error. We conclude by performing preliminary numerical examples. The aim is to heuristically study the sharpness of the bound, to give intuitions on how the analysis can be used to compare different approaches, and to provide ideas on how to make the bound computable in practice.

The talk introduces an analytical framework for examining socio-spatial segregation across various spatial scales. This framework considers regional connectivity and population distribution, using an information theoretic approach to measure changes in socio-spatial segregation patterns across scales. It identifies scales where both high segregation and low connectivity occur, offering a topological and spatial perspective on segregation. Illustrated through a case study in Ecuador, the method is demonstrated to identify disconnected and segregated regions at different scales, providing valuable insights for planning and policy interventions.

A colouring rule is a way to colour the points $x$ of a probability space according to the colours of finitely many measure preserving tranformations of $x$. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We show that proper graph colouring as a rule can be paradoxical. And we demonstrate rules defined via optimisation that are paradoxical. A connection to measure theoretic paradoxes is established.