Past

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.

We will give a relaxed introduction to some of the most classical dynamical systems - Anosov flows. These flows were highly influential in the development of ideas which the audience might be more familiar with. For example, Anosov flows give rise to exponential group growth and taut foliations, both of which we will discuss. Finally, we will talk about some recent work obstructing Anosov flows and their combinatorial analogs - veering triangulations

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.

For any smooth complex variety Y with an action of a finite group W, Etingof defines the global Cherednik algebra H_c and its spherical subalgebra B_c as certain sheaves of algebras over Y/W. When Y is an n-dimensional abelian variety, the algebra of global sections of B_c is a polynomial algebra on n generators, as shown by Etingof, Felder, Ma, and Veselov. This defines an integrable system on Y. In the case of Y being a product of n copies of an elliptic curve E and W=S_n, this reproduces the usual elliptic Calogero­­--Moser system. Recently, together with P. Argyres and Y. Lu, we proposed that many of these integrable systems at the classical level can be interpreted as Seiberg­­--Witten integrable systems of certain super­symmetric quantum field theories. I will describe our progress in understanding this connection for groups W=G(m, 1, n), corresponding to the case Y=E^n where E is an elliptic curves with Z_m symmetry, m=2,3,4,6. 

The Nyström method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the randomized, single-pass, streamable, cost-effective, and accurate alternative to the randomized SVD, and it facilitates the computation of several matrix low-rank factorizations. In this presentation, we take these advancements a step further by introducing a higher-order variant of Nyström's methodology tailored to approximating low-rank tensors in the Tucker format: the multilinear Nyström technique. We show that, by introducing appropriate small modifications in the formulation of the higher-order method, strong stability properties can be obtained. This algorithm retains the key attributes of the generalized Nyström method, positioning it as a viable substitute for the randomized higher-order SVD algorithm.

In this talk, I describe an exact duality between the double scaling limit of the SYK model and quantum geometry of de Sitter spacetime in three dimensions. The duality maps the so-called chord rules that specify the exact SYK correlations functions to the skein relations that govern the topological interactions between world-line operators in 3D de Sitter gravity.

This talk is part of the series of Willis Lamb Lectures in Theoretical Physics. Herman Verlinde is the Lamb Lecturer of 2024.

The massive planar Gaussian free field (GFF) is a random distribution defined on a subset of the complex plane. As a random distribution, this field a priori does not have well-defined level lines. In this talk, we give a meaning to this concept by constructing a coupling between a massive GFF and a random collection of loops, called massive CLE_4, in which the loops can naturally be interpreted as the level lines of the field. This coupling is constructed by appropriately reweighting the law of the standard GFF-CLE_4 coupling and this construction can be seen as a conditional version of the path-integral formulation of the massive GFF. We then relate massive CLE_4 to a massive version of the Brownian loop soup. This provides a more direct construction of massive CLE_4 and proves a conjecture of Camia.

A subset in the ideal boundary of a CAT(0) space is called symmetric if every complete geodesic with one ideal boundary point
in the set has both ideal boundary points in the set. In the late 80s Eberlein proved that if a Hadamard manifold contains a non-trivial closed symmetric  subset in its ideal boundary, then its holonomy group cannot act transitively. This leads to rigidty via
the Berger-Simons Theorem. I will discuss rigidity of ideal symmetric sets in the general context of locally compact geodesically complete
CAT(0) spaces.
 

Inspired by questions concerning the evolution of phase fields, we study the Allen-Cahn equation in dimension 2 with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint works with Simon Gabriel, Martin Hairer, Khoa Lê and Nikos Zygouras.