Martin Hairer: Stochastic Quantisation of Yang-Mills

We report on recent progress on the problem of building a stochastic process that admits the hypothetical Yang-Mills measure as its invariant measure. One interesting feature of our construction is that it preserves gauge-covariance in the limit even though it is broken by our UV regularisation. This is based on joint work with Ajay Chandra, Ilya Chevyrev, and Hao Shen.

Chair: Christoph Thiele

Irene Fonseca (Carnegie Mellon University): From Phase Separation in Heterogeneous Media to Learning Training Schemes for Image Denoising

What do these two themes have in common? Both are treated variationally, both deal with energies of different dimensionalities, and concepts of geometric measure theory prevail in both.

Phase Separation in Heterogeneous Media

Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations using a variational approach based on the gradient theory of phase transitions, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed, now leading to an isotropic interfacial energy. In the subcritical case with moving wells, where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains. This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).

Learning Training Schemes for Image Denoising

Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image training pairs. Such methods with spacedependent parameters which are piecewise constant on dyadic grids are considered, with the grid itself being part of the minimization. Existence of minimizers for discontinuous parameters is established, and it is shown that box constraints for the values of the parameters lead to existence of finite optimal partitions. Improved performance on some representative test images when compared with constant optimized parameters is demonstrated. This is joint work with Elisa Davoli (TU Wien, Austria), Jose Iglesias (U. Twente, The Netherlands) and Rita Ferreira (KAUST, Saudi Arabia)

Chair: Margherita Disertori

Stephen Morris (MIT): The Almost Common Knowledge Topology on Information Structures

Two information structures are said to be close if, with high probability, there is approximate common knowledge that interim beliefs are close under the two information structures. We define an “almost common knowledge” topology reflecting this notion of closeness. We show that it is coarsest topology generating continuity of equilibrium outcomes.

Joint work with Dirk Bergemann and Rafael Veiel

Chair: Sven Rady

Karen E. Willcox (The University of Texas in Austin): The Mathematics of Digital Twins

A digital twin is a computational model or set of coupled models that evolves over time to persistently represent the structure, behavior, and context of a unique physical system, process or biological entity. A bidirectional feedback flow between virtual and physical is central to a digital twin. Digital twins have the potential to enable safer and more efficient engineering systems, a greater understanding of the natural world around us, and better medical outcomes for all of us as individuals. Mathematics plays a central role in digital twins, from the mathematical models that form the core of a digital twin’s ability to represent complex phenomena, to the inverse theory that underpins the assimilation of data, to the control theory that underpins digital twin-enabled decision-making.

Chair: Jürgen Dölz

Tristan Buckmaster (New York University): Singularities in Fluid: Self-similar Analysis, Computer Assisted Proofs and Neural Networks

In this presentation, I will provide an overview of how techniques involving self-similar analysis, computer assisted proofs and neural networks can be employed to investigate singularity formation in the context of fluids.

Chair: Sergio Conti

Welcome addresses:

Gonca Türkeli-Dehnert, Vice-Minister for Culture and Science of the State North Rhine-Westphalia
Joachim Escher, President of the German Mathematical Society DMV
Michael Hoch, Rector of the University of Bonn

Akshay Venkatesh (IAS Princeton): Where Stands Arithmetic Topology?

There is a striking parallel, not readily formalized, between certain aspects of number theory and the geometry of 3-dimensional manifolds. I will talk through a few instances of this parallel as it appears in old and new mathematics, but I will not shed any light on its ultimate origin – that remains a fundamental mystery!

Chair: Daniel Huybrechts

Stéphane Mallat (Collège de France): Mathematical Mysteries of Deep Neural Networks

Deep neural networks have spectacular applications to classification, regression or generation of images, sounds, texts and most types of data. They are responsible of the renewal of artificial intelligence. They also provide new computational tools and models in all sciences including physics. Yet, we mostly do not understand the underlying mathematics, allowing them to avoid the curse of dimensionality. By taking examples and inspiration from statistical physics, this talk will emphasize the role of concentration phenomena, multiscale interactions, random matrices and invariants. Applications will be shown in physics and image classification problems.

Chair: Jochen Garcke

Sylvie Méléard (École Polytechnique): Multiscale Eco-evolutionary Models: from Individuals to Populations

Going back to Darwin and Galton-Watson, there is a long tradition of new probabilistic concepts arising from biological discoveries. In recent years, single cell measurement techniques have developed tremendously, thanks to high-performance microscopes and the development of microfluidic observations. Motivated by recent biological experiments, we emphasize the effects of small and random populations on long time population dynamics. We will quantify such effects on macroscopic approximations. The individual behaviors are described by the mean of a stochastic processes. We study different long time asymptotic behaviors depending on mutation size and frequency and on horizontal transmission rate. In some cases, simulations indicate that these models should exhibit surprising asymptotic behaviors such as cyclic behaviors. We explore these behaviors when population sizes and time are on a log-scale, leading in some cases to Hamilton-Jacobi equations.

Chair: Anton Bovier

Ola Svensson (EPFL): Polyhedral Techniques in Combinatorial Optimization: Matchings and Tours

We overview recent progress on two of the most classic problems in combinatorial optimization: the matching problem and the traveling salesman problem. We focus on deterministic parallel algorithms for the perfect matching problem and the first constant-factor approximation algorithm for the asymmetric traveling salesman problem. While these questions pose seemingly different challenges, recent progress has been achieved using similar polyhedral techniques. In particular, for both problems, we will explain the use linear programming formulations, even exponential-sized ones, to extract structure from problem instances to guide the design of better algorithms.

Chair: Vera Traub

Laura DeMarco (Harvard University): Intersection Theory and the Mandelbrot Set

One of the most famous – and still not fully understood – objects in mathematics is the Mandelbrot set. It is defined as the set of complex numbers c for which the polynomial f_c(z)=z^2+c has a connected Julia set. But the Mandelbrot set turns out to be related to many different areas of mathematics. Inspired by recent results in arithmetic geometry, I will describe how the tools ofarithmetic intersection theory can be applied in the setting of these complex dynamical systems to give new information about the Mandelbrot set. This is joint work with Myrto Mavraki.

Chair: Ursula Hamenstädt

Mohammed Abouzaid (Stanford University): Bordism and Floer theory

In the late 1980’s Andreas Floer revolutionized low dimensional and symplectic topology by discovering the existence an extension of Morse theory to an infinite dimensional setting where the standard methods of variational calculus fail. While he foresaw that his theory should be able to encompass generalised homology theory (bordism, K-theory, ...), severe foundational difficultiesprevented any significant progress on this question until two years ago. I will explain the advances that have been made on two fronts: (I) defining concrete models, in terms of equivariant vector bundles, for the moduli spaces that appear in Floer theory, and (II) understanding the geometric consequences of lifting Floer homology to generalised homology theories. I will end by formulating how the notion of derived orbifold bordism provides a universal receptacle for Floer’s invariants, and its descendants.

Chair: Jessica Fintzen

Alessio Figalli (ETH Zürich): Generic Regularity of Free Boundaries for the Obstacle Problem

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. Explicit examples show that the singular set could be, in general, as large as the regular set.In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has codimension 3 inside the free boundary, solving a conjecture of Schaeffer in dimension n≤4. The aim of this talk is to give an overview of these results.

Chair: Martin Rumpf

Annalisa Buffa (EPFL): Design and simulation: The Challenge of Geometry Simplification

Complex geometrical models are created and processed using computer-aided design tools (CAD) within the realm of computer-aided engineering. Complex geometries, such as those found in components like the parts of the engine of a car, are often rich in intricate details spanning multiple scales. Additive manufacturing significantly amplifies this complexity.
Defeaturing involves simplifying geometrical models by eliminating the geometric features deemed irrelevant from the geometric standpoint. This removal of features and simplification of the geometric models facilitates the creation of a mesh, which is essentially a partition of the geometric model into simple geometric entities, such as simplexes. This, in turn, leads to fast simulations for engineering analysis problems modelled as partial differential equations (PDEs) on the simplified computational domain.
The impact of defeaturing on the accuracy of the computed PDE solution is typically disregarded, and as of today, there are very few strategies available for quantitatively assessing such an impact. Our approach involves formalizing the defeaturing process and developing indicators capable of providing quantitative measures of the accuracy loss resulting from geometric simplification.

Chair: Angkana Rüland

Oscar Randal-Williams (University of Cambridge): Homeomorphisms of Euclidean Space

The topological group of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology, closely related to understanding the difference between diffeomorphisms and homeomorphisms of all d-dimensional manifolds (except when d=4). Over the last few years a great deal of progress has been made in understanding the algebraic topology of this group. I will report on some of the methods involved, and an emerging conjectural picture.

Chair: Markus Hausmann

Éva Tardos (Cornell University): Stability and Learning in Strategic Games

Over the last two decades we have developed good understanding how to quantify the impact of strategic user behavior on outcomes in many games (including traffic routing and online auctions) and showed that the resulting bounds extend to repeated games assuming players use a form of no-regret learning to adapt to the environment. We will review how this area evolved since its early days, and also discuss some of the new frontiers, including when repeated interactions have carry-over effects between rounds: when outcomes in one round effect the game in the future, as is the case in many applications. In this talk, we study this phenomenon in the context of a game modeling queuing systems: routers compete for servers, where packets that do not get served need to be resent, resulting in a system where the number of packets at each round depends on the success of the routers in the previous rounds. In joint work with Jason Gaitonde, we analyze the resulting highly dependent random process, and show bounds on the excess server capacity needed to guarantee that all packets get served despite the selfish (myopic) behavior of the queues.

Chair: Jens Vygen

Jason Miller (University of Cambridge): Conformal Removability of SLEκ for κ∈[4,8)

We consider the Schramm-Loewner evolution (SLEκ) with κ=4, the critical value of κ>0 at or below which SLEκ is a simple curve and above which it is self-intersecting. We show that the range of an SLE_4 curve is a.s. conformally removable. Such curves arise as the conformal welding of a pair of independent critical (γ=2) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set X⊆C to be conformally removable which applies in the case that X is not necessarily the boundary of a simply connected domain. We will also describe how this theorem can be applied to obtain the conformal removability of the SLEκ curves for κ∈(4,8) in the case that the adjacency graph of connected components of the complement is a.s. connected. This talk will assume no prior knowledge of SLE or LQG.

Chair: Andreas Eberle

Geordie Williamson (University of Sydney): A Panoramic View of Modular Representation Theory

I will try to give a glimpse of exciting developments in representation theory over the last two decades. A central focus will be on the representations of symmetric groups over the complex numbers and fields of positive characteristic. Over the complex numbers our understanding is very good, however the case of positive characteristic fields has turned out to be more complicated than (I suspect) the pioneers would have ever imagined. Remarkably, there appears to be a way forward which combines ideas which emerged in the Langlands program with techniques from mod p algebraic topology (Smith theory).

Chair: Catharina Stroppel

Concert at the Arithmeum

Valentin Blomer & Akos Quartet