10 December – Stavroula Makri (VU Amsterdam)
Title: Braids and fixed points
Abstract: The use of braid group theory in surface dynamics and Nielsen fixed point theory was initiated in the early 1980s and has since played a key role in studying fixed points and periodic orbits of surface homeomorphisms. In this talk, I will begin with a basic introduction to Nielsen fixed point theory and braid group theory, and then explain how a braid can be associated with a homeomorphism of a compact surface isotopic to the identity that leaves a finite set invariant. We will see how braid theory can be applied to obtain important results and information about surface homeomorphisms. In particular, I will discuss an elegant result showing that the matrix representation of a braid provides valuable information about the existence and linking behavior of its fixed points, and how we can extend it to a 3-dimensional setting.
4th of February - Havva Yoldaş (TU Delft)
Title: Connecting multiple scales through PDEs: scaling limits and hypocoercivity
Abstract: I will give a brief introduction to Hilbert’s Sixth Problem — how natural phenomena can be modelled. mathematically across different scales (microscopic, mesoscopic, macroscopic). This will be followed by an exposition of mathematical problems in kinetic theory and a summary of some key techniques such as hypocoercivity, and structure-preserving numerical schemes. I will show how these techniques allow us to address multiscale problems. Finally, I will present some applications in mathematical biology.
25th of February - Alejandro Fernández Jiménez (VU Amsterdam)
Title: An overview on gradient flows.
Abstract: We will first briefly recall the theory of gradient flows in the Euclidean setting. Afterwards, we will present its generalisation to metric spaces by Ambrosio, Gigli and Savaré and finally we will explain how a family of PDEs can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). Then, it will come a short introduction to optimal transport in which, for instance, we will define the notion of geodesic convexity and its applications towards the analysis of PDEs. Finally, we will apply this theory to a particular case and we will explain how to use gradient flows in order to obtain existence and a deterministic particle approximation for a particular fourth-order PDE.
11th of March - Wioletta Ruszel (Utrecht)
Title: Nonlocal perimeter for the long-range Ising model
Abstract: In this talk we will first talk about about isoperimetric inequalities for the classical and nonlocal case. Then we will introduce the Ising model from statistical mechanics and review some results on in the nearest-neighbour and long-range interaction case. In particular we will present some recent results on the bi-axial long-range Ising model. The bi-axial long-range Ising model is a special case of a large class of Ising models with long range order. This model has interesting properties as for example the existence of Dobrushin interfaces. Other interesting properties for nonlocal models are for example to study critical shapes via isoperimetric inequalities. We will give a definition of such a nonlocal perimeter and prove under which conditions the optimal shape in a two-dimensional model. Finally we will connect these optimal shapes to metastability problems. This talk is based on joint work with V. Jacquier (UU) and C. Spitoni (UU).
25th of March - Dan Hill (Oxford)
Title: Think Global, Act Local: Inducing Fully Localised 2D Patterns via Spatial Heterogeneity
Abstract: The existence of localised two-dimensional patterns has been observed and studied in numerous experiments and simulations: ranging from optical solitons, to patches of desert vegetation, to fluid convection. And yet, our mathematical understanding of these emerging structures remains extremely limited beyond one-dimensional examples.
In this talk I will discuss how adding a compact region of spatial heterogeneity to a PDE model can not only induce the emergence of fully localised 2D patterns, but also allows us to rigorously prove and characterise their bifurcation. The idea is inspired by experimental and numerical studies of magnetic fluids and tornados, where our compact heterogeneity corresponds to a local spike in the magnetic field and temperature gradient, respectively. In particular, we obtain local bifurcation results for fully localised patterns both with and without radial or dihedral symmetry, and rigorously continue these solutions to large amplitude. Notably, the initial bifurcating solution (which can be stable at bifurcation) varies between a radially-symmetric spot and a 'dipole' solution as the width of the spatial heterogeneity increases.This work is in collaboration with David J.B. Lloyd and Matthew R. Turner (both University of Surrey).
29th of April - Tim van Erven (UvA)
Title: The Risks of Recourse in Explainable Machine Learning
Abstract: Algorithmic recourse provides explanations that help users overturn an unfavorable decision by a machine learning system. For instance, customers whose loan application is denied might want to know how they can get a loan in the future. Instead of considering the effects of recourse on individual users, as is typical in the literature, we study the effects at the population level. Surprisingly, we find that the effect is typically negative, because providing recourse tends to reduce classification accuracy. In the case of loan applications, this would lead to many extra customers who end up defaulting. We further study whether the party deploying the classifier has an incentive to strategize in anticipation of having to provide recourse, and we find that sometimes they do, to the detriment of their users. Providing algorithmic recourse may therefore also be harmful at the systemic level. All in all, we conclude that the current concept of algorithmic recourse is not reliably beneficial, and therefore requires rethinking. This talk is based on: H. Fokkema, D. Garreau and T. van Erven. The Risks of Recourse in Binary Classification. AIStats 2024.