# Plenary Lectures

Martin Bridson, University of Oxford

**"Finite shadows of infinite groups and the geometry of 3-dimensional manifolds"**

There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects

(equivalently, their profinite completions). But how much understanding can one really gain about an infinite group by examining its finite images? I will sketch some of the

rich history of this problem and describe how input from geometry and low-dimensional topology have transformed the subject in recent years. I shall then sketch recent work,

rooted in the geometry of 3-manifolds, that highlights the importance of finite presentation in this context. And I shall describe some compelling open questions.

Alexandra Carpentier, Universität Potsdam

**"Composite-composite testing and extensions"**

Testing theory is a corner stone of mathematical statistics, with manypractical applications. While asymptotic and parametric testing is relatively well understood, many additional issues arise in non-parametric or high dimensional models. This is particularly the case when the two hypotheses are composite, i.e. do not consist of a single point. In this case, only few theoretical results are available, and seemingly simple questions remain open. In this talk, I will introduce the problem of composite-composite testing, and present some own (together with co-authors) results in this field.

Adrian Constantin, Universität Wien

** "Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere"**

We discuss stationary solutions of Euler's equation on a rotating sphere and their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system. We present some rigidity and stability results.

This is joint work with Pierre Germain (Courant Institute of Mathematical Sciences, New York).

Hugo Duminil-Copin, Institut des Hautes Études Scientifiques (IHÉS)

**"Critical Phenomena Through the Lens of the Ising Model". **

The Ising model is one of the most classical lattice models of statistical physics undergoing a phase transition. Initially imagined as a model for ferromagnetism, it revealed itself as a very rich mathematical object and a powerful theoretical tool to understand cooperative phenomena. Over one hundred years of its history, a profound understanding of its critical phase has been obtained. While integrability and mean-field behavior led to extraordinary breakthroughs in the two-dimensional and high-dimensional cases respectively, the model in three and four dimensions remained mysterious for years. We will present recent progress in these dimensions based on a probabilistic interpretation of the Ising model relating it to percolation models.

Des Higham, University of Edinburgh

**"Should We Be Perturbed About Deep Learning?"**

Many commentators are asking whether current AI solutions are sufficiently robust, resilient, and trustworthy; and how such issues should be quantified and addressed. In an extreme case, it has been shown that a traffic "Stop" sign on the roadside can be misinterpreted by a driverless vehicle as a speed limit sign when minimal graffiti is added. The vulnerability of systems to such adversarial interventions raises questions around security and ethics, and there has been a rapid escalation of heuristic attack and defence strategies. I believe that mathematicians can contribute to this landscape. From a numerical analysis perspective, this is a conditioning issue: how sensitive is the input-output map to perturbations in the input, or to the map itself? After discussing traditional adversarial attacks on input data, I will focus on a class of recently proposed variations where the attacker has complete access to the full AI system. Here, changes to the weights and biases in a deep learning network, or modifications of the underlying architecture, can lead to a perturbed system whose output is (a) unchanged on a large validation set that is hidden from the attacker, but (b) dramatically altered on a specific target input of interest. I will describe a new "network attack" algorithm that (a) can be proved to succeed with high probability under realistic assumptions, and (b) can be seen to operate effectively in practice.

The talk is based on joint work with Alexander Bastounis (Edinburgh), Alexander Gorban (Leicester), Ivan Tyukin (Kings) and Eliyas Woldegeorgis (Leicester)

Fanny Kassel, Institut des Hautes Études Scientifiques (IHÉS)

**"Discrete subgroups of Lie groups in higher rank" **

Discrete subgroups of Lie groups play a fundamental role in several areas of mathematics. In the case of SL(2,R), they are well understood and classified by the geometry of the corresponding hyperbolic surfaces. In the case of SL(n,R) with n>2, they remain more mysterious, beyond the important class of lattices (i.e. discrete subgroups of finite covolume for the Haar measure). These past twenty years, several interesting classes of discrete subgroups have emerged, which are « thinner » than lattices, more flexible, and with remarkable geometric and dynamical properties. We will present some recent developments in the subject.

Alexander Martin, Friedrich-Alexander Universität Erlangen-Nürnberg

**"Mixed Integer Optimization Problems on Networks with PDE Constraints"**

Motivated by challenging questions in the transformation and control of our energy system, we study mixed integer optimization problems on networks with PDE constraints. Control decisions are typically modeled by integer optimization methods, while the physical behavior of water, gas and hydrogen is represented in a continuous nonlinear way, e.g. by partial differential equations (PDEs). The topic of this talk is to discuss mathematical approaches and insights for the efficient coupling of integer and continuous nonlinear optimization in this context. We will also demonstrate the numerical success using examples from gas network optimization within the framework of the SFB/TR 154.

Rahul Pandharipande, ETH Zürich

**"A tour of the geometry of points in affine space"**

The study of the space of d distinct and unordered points in C^{n} is fundamental from several perspectives. In algebraic geometry, the space sits naturally inside the Hilbert scheme of d points – which captures information about the collisions of points. I will explain the remarkable structure of the Hilbert scheme in low dimensional cases (concentrating on n ≤ 3). The geometry is related to many different streams in mathematics: combinatorics, representation theory, knot theory, and gauge theory to name a few. My goal is to present a landscape of connections, known results, and open questions.

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). I will explain some methods that have been used for studying the algebraic topology of this group, and report on a recently obtained conjectural picture of it.

Bernhard Schölkopf, Max Planck Institute for Intelligent Systems Tübingen

**"Intelligent system: symbolic, statistical, and causal"**

We describe basic ideas underlying research to build and understand artificially intelligent systems: from symbolic approaches via statistical learning to interventional models relying on concepts of causality. Some of the hard open problems of machine learning and AI are intrinsically related to causality, and progress may require advances in our understanding of how to model and infer causality from data.

The plenary lectures will take place in the main lecture hall (B.001) of the Department of Chemistry in Arnimallee 22.