The colloquium of the Centre for Mathematical Sciences, Lund University, normally runs once a month, Wednesdays from 14.00 until 15.00 in the Hörmander, Riesz or Gårding lecture halls. It is aimed at the entire Centre for Mathematical Sciences with overview talks by renowned experts about exciting mathematical topics. The purpose of our colloquium is twofold: firstly, it is to provide an inspiring overview of a specific field of mathematics, secondly, it is to bring together students and staff from the entire department and to serve as the proverbial waterhole where contacts are made and maintained. For more information, see the guidelines for colloquium speakers.

The colloquium is organized by Dragi Anevski, Magnus Goffeng, Magnus Oskarsson, Tony Stillfjord and Anitha Thillaisundaram. Feel free to contact any one of us for questions or suggestions for colloquia speakers. See also the information for suggesting colloquium speakers.

September 3 at MH:Riesz

Speaker

Erik Mohlin (Lund)

Title

Game theory: models of human behaviour and social institutions

Abstract

The talk will provide a short introduction to game theory, with a special focus on models of learning and evolution and their role in motivating the predictive value of equilibrium concepts. Specifically, I aim to cover the following: 

• Representing strategic interactions as strategic-form games or extensive-form games.
• Nash equilibrium analysis in strategic-form games and backward induction in extensive-form games. 
• Justification of Nash equilibrium via trial-and-error learning. 
• Evolutionary game theory with the replicator dynamic. 
• Experimental game theory.

September 24 at MH:Riesz

Speaker

Dan Segal (Oxford)

Title

Taming infinity

Abstract

To see a world in a grain of sand

And a heaven in a wild flower,

Hold infinity in the palm of your hand

And eternity in an hour. (William Blake)

A ramble through various mathematical topics, illustrating how group theory sometimes provides finite answers to a priori ‘infinite’ questions.

The key idea is that complicated structures can be understood with the help of symmetries. The groups of symmetries then take on a life of their own: curiosity drives us to find out as much as we can about these interesting objects.

Topics include Galois Theory, where a finite group controls the solutions of a polynomial equation, and number theory, where sometimes the infinitely many solutions of an equation over the integers can be organised into finitely many orbits of a finitely generated group.

Along the way I will mention Lie groups, which arise in geometry; arithmetic groups, which are the ‘integer points’ of Lie groups; and finite simple groups, which turn out mostly to be the analogous structures over finite fields.

Also discussed will be profinite groups, a very different kind of topological group: these are a way of ‘compactifying’ some infinite families of finite groups.

October 15 at MH:Hörmander

Speaker

Markos Katsoulakis (Massachusetts)

Title

Hamilton-Jacobi Equations, Mean-Field Games, and Uncertainty Quantification for Robust Machine Learning

Abstract

Hamilton-Jacobi (HJ) equations and Mean-Field Games (MFGs) provide a natural mathematical language that unites ideas from stochastic control, optimal transport, and information theory for analyzing, designing, and improving the robustness of many modern machine learning models. We show how fundamental classes of generative models, including continuous-time normalizing flows and score-based diffusion models, emerge intrinsically from MFG formulations under different particle dynamics, cost functionals, information-theoretic divergences, and probability metrics, with analogies and connections to Wasserstein gradient-flows. The forward-backward PDE structure of MFGs offers both analytical insights and informs the development of faster, data-efficient and robust algorithms. In particular, the regularity theory of HJ equations, combined with model-uncertainty quantification, provides provable performance and robustness guarantees for generative models and complex neural architectures such as transformers. Our theoretical analysis is complemented by extensive numerical validations, with diverse examples from applied mathematics and widely used ML benchmarks.

November 5 at MH:Hörmander

Speaker

Edward Crane (Bristol)

Title

Recurrence, transience and anti-concentration of Rademacher walks

Abstract

Joint work with Satyaki Bhattacharya (Lund PhD student) and Tom Johnston (University of Bristol).

 Suppose you start with a fortune X0 = 0 and make a sequence of bets of different (deterministic) sizes a1, a2, a3, ..., on the outcome of a fair coin toss. In the nth bet, your fortune Xn increases by an if the coin shows heads, and it decreases by an if the coin shows tails.  Your fortune can turn negative, but you have an unlimited overdraft so you always can keep betting.  Does your  fortune keep returning close to 0, infinitely often?  In other words, is  lim inf |Xn| finite with probability 1?  The answer depends on the sequence a = (a1, a2, a3,...) in an interesting way.

Surprisingly, this very classical-sounding probability problem has not received much attention. As far as we know, it was first investigated by Stas Volkov and Satyaki Bhattacharya in a 2023 paper. I will describe several new results about this question, just posted at arXiv:2510.24568, and a few open problems.

November 26 at MH:Hörmander

Speaker

Carola-Bibiane Schönlieb (Cambridge)

Title

Mathematical imaging: from partial differential equations to deep learning for images

Abstract

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of nonlinear partial differential equations, inverse problems, harmonic, stochastic, and statistical analysis, and optimization, just to name a few. Applications of mathematical imaging are profound and arise in biomedicine, material sciences, astronomy, digital humanities, as well as many technological developments such as autonomous driving, facial screening and many more. 

In this talk I will discuss my perspective onto mathematical imaging, share my fascination and vision for the subject. I will then zoom into one research problem that I am currently most excited about and that we helped make first advances on: the mathematical formalisation of machine learned approaches for solving inverse imaging problems.