Past events

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.

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.

Many practical Bayesian inference problems fall into the simulation-based or "likelihood-free" setting, where evaluations of the likelihood function or prior density are unavailable or intractable; instead one can only draw samples from the joint parameter-data prior. Learning conditional distributions is essential to the solution of these problems. 
To this end, I will discuss a powerful class of methods for conditional density estimation and conditional simulation based on transportation of measure. An important application for these methods lies in data assimilation for dynamical systems, where transport enables new approaches to nonlinear filtering and smoothing. 
To illuminate some of the theoretical underpinnings of these methods, I will discuss recent work on monotone map representations, optimization guarantees for learning maps from data, and the statistical convergence of transport-based density estimators.
 

Neural network architectures play a key role in determining which functions are fit to training data and the resulting generalization properties of learned predictors. For instance, imagine training an overparameterized neural network to interpolate a set of training samples using weight decay; the network architecture will influence which interpolating function is learned. 

In this talk, I will describe new insights into the role of network depth in machine learning using the notion of representation costs – i.e., how much it “costs” for a neural network to represent some function f. Understanding representation costs helps reveal the role of network depth in machine learning. First, we will see that there is a family of functions that can be learned with depth-3 networks when the number of samples is polynomial in the input dimension d, but which cannot be learned with depth-2 networks unless the number of samples is exponential in d. Furthermore, no functions can easily be learned with depth-2 networks while being difficult to learn with depth-3 networks. 

Together, these results mean deeper networks have an unambiguous advantage over shallower networks in terms of sample complexity. Second, I will show that adding linear layers to a ReLU network yields a representation cost that favors functions with latent low-dimension structure, such as single- and multi-index models. Together, these results highlight the role of network depth from a function space perspective and yield new tools for understanding neural network generalization. 

I shall say some stuff about quasilinear reaction-diffusion equations, motivated by tissue growth in particular.

Questions in algebra, while deep and interesting, can be incredibly difficult. Thankfully, when studying the representation theory of the symmetric groups, one can often take algebraic properties and results and write them in the language of combinatorics; where one has a wide variety of tools and techniques to use. In this talk, we will look at the specific example of the submodule structure of 2-part Specht modules in characteristic 2, and answer which hook Specht modules are uniserial in characteristic 2. We will not need to assume the Riemann hypothesis for this talk.

A structure is omega-categorical if its theory has a unique countable model (up to isomorphism). We will survey some old results concerning the Apps-Wilson structure theory for omega-categorical groups and state a conjecture of Wilson from the 80s on omega-categorical characteristically simple groups. We will also discuss the analogous of Wilson’s conjecture for Lie algebras and present some connections with the restricted Burnside problem.

Optimal transport (OT) proves to be a powerful tool for non-parametric calibration: it allows us to take a favourite (non-calibrated) model and project it onto the space of all calibrated (martingale) models. The dual side of the problem leads to an HJB equation and a numerical algorithm to solve the projection. However, in general, this process is costly and leads to spiky vol surfaces. We are interested in special cases where the projection can be obtained semi-analytically. This leads us to the martingale equivalent of the seminal fluid-dynamics interpretation of the optimal transport (OT) problem developed by Benamou and Brenier. Specifically, given marginals, we look for the martingale which is the closest to a given archetypical model. If our archetype is the arithmetic Brownian motion, this gives the stretched Brownian motion (or the Bass martingale), studied previously by Backhoff-Veraguas, Beiglbock, Huesmann and Kallblad (and many others). Here we consider the financially more pertinent case of Black-Scholes (geometric BM) reference and show it can also be solved explicitly. In both cases, fast numerical algorithms are available.

Based on joint works with Julio Backhoff, Benjamin Joseph and Gregoire Leoper.  

This talk reports a work in progress. It will be done on a board.