Optimal transport theory provides a powerful mathematical and computational framework for problems in data science and economics. This talk explores two key contributions in this field: inferring unknown cost functions in noisy optimal transport plans and leveraging deep learning to infer trading barriers in international commodity trade. We start by discussing the classic optimal transportation problems studied by Gaspard Monge and Leonid Kantorovich, before focusing on the respective inverse problem, so-called inverse optimal transport. Hereby we wish to infer the underlying transportation cost from solutions that are corrupted by noise. Then we generalize this approach to identify transport costs in global food and agricultural trade. Our analysis reveals that the global South suffered disproportionately from the war in Ukraine's impact on wheat markets. Additionally, it examines the effects of free-trade agreements, trade disputes with China, and Brexit's impact on British-European trade, uncovering hidden patterns not evident from trade volumes alone.

We discuss the algebraic geometry of low rank symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories.

Looking around us, many surfaces including the Earth are no plain Euclidean domains but special cases of Riemannian manifolds. Uncertain physical phenomena on these surfaces can for example be described by random fields and stochastic partial differential equations. In this talk, I will introduce how to compute random samples and approximations of solutions to such equations and give convergence results to characterize their quality. Furthermore, I will show how these samples can be used to generate random shapes and time-evolving stochastic manifolds.

We shall review a number of mathematical areas in which it is not reasonable to expect a classification of all the objects of study. We shall then explain how to attack such questions if we take more structure into account and if we think geometrically. This will lead us to the foothills of Tensor-Triangular Geometry.

"Künstliche Intelligenz*innen'' sind bei den Aufgaben der Mathematik-Olympiaden besser als gute menschliche Kandidat*innen. Sie sind u.a. auch bei den Interviews für Stipendien für Master-Programme in Mathematik-basierten Fächern besser als die meisten Bewerber*innen. Wozu sollen wir da die nächste Generation noch mit Mathematik quälen? Sollen wir die wertvolle Zeit nicht besser für die Förderung "kreativer'' Fähigkeiten nutzen? – Es ist die innere Logik und das Ziel der Mathematik, sich selbst zu trivialisieren ("algorithmisieren''). Das noch nicht Algorithmisierte wächst aber schneller als das schon Algorithmisierte. Das ist die kreative Unlimitierbarkeit der Mathematik und die Attraktivität der Mathematik für die besten Köpfe jeder Generation und die Quelle höchster gesellschaftlicher, politischer und immer mehr globaler Relevanz der Mathematik im jeweiligen historischen Moment. – Im Vortrag werden diese Gedanken für die mathematische und nicht-mathematische Öffentlichkeit im Detail entwickelt.

A map is a topological surface obtained by identifying in pairs the edges of a finite collection of polygons. We consider maps as abstract combinatorial objects rather than discretization of surfaces embedded in the space. With this point of view, the countable collection of maps of a given topology can be seen as a set of discrete metrics defined on a fixed topological surface, for instance, by endowing the set of vertices of a map with the combinatorial graph distance. Therefore, considering models of random maps provides a natural setting for studying random metrics, and in this context, a question of interest concerns the asymptotic geometry of such a random object as n goes to infinity. In this talk, we will review some aspects of the very rich enumerative theory of maps, and discuss how it allows to define limiting objects that are, in a sense, canonical models of random 2-dimensional metric spaces.

Inverse Problems are at the core of many measurement processes in the sciences and engineering. In these situations one can often only access a system through indirect, non-invasive measurements. This gives rise to fundamental challenges such as severe ill-posedness. The Calderón Problem is a prototypical example of an inverse problem: one seeks to reconstruct the unknown conductivity properties of a conducting body through voltage and current measurements at its boundary. In this talk, I will discuss some of the challenges associated with the Calderón problem and investigate a nonlocal variant of it, the fractional Calderón problem. I will outline how nonlocality in the fractional Calderon problem provides novel structures, which even allow one to study highly anisotropic settings, and how it can offer new perspectives on the classical local Calderón problem.

I'll discuss commuting varieties and a new upper bound for the density of pairs of commuting n x n matrices with integer entries. Our approach uses Fourier analysis and reduction modulo a suitably chosen prime, together with a result about the flatness of the commutator Lie bracket, which we also solve. This is joint work with Will Sawin and Victor Wang.

When combinatorialists discover two families of objects that are counted by the same formula, they usually aim to prove this by constructing an explicit bijection. Such proofs often bring more clarity to a statement, may lead to interesting generalizations, and are usually aesthetically pleasing. Alternating sign matrices and plane partitions are classical objects in enumerative combinatorics that offer a variety of equinumerosity phenomena. However, for none of them there is a simple bijective proof so far, despite people have been searching for such proofs for more than 40 years already. The Littlewood identity, on the other hand, admits a bijective proof based on the famous Robinson-Schensted-Knuth correspondence, which is a fundamental algorithm underlying many bijective proofs. Recently, Littlewood-type identities related to alternating sign matrices and plane partitions have been discovered, bringing new perspectives to the previously hopeless search for bijections in this field.

Extremely sparse random binary matrices tend to be singular, due to the likely presence of "local combinatorial dependencies" such as all-zero columns or pairs of identical columns. We discuss this phenomenon, and some results showing that these kinds of combinatorial dependencies are in some sense the "only" causes of singularity. This is joint work with Asaf Ferber, Margalit Glasgow, Ashwin Sah and Mehtaab Sawhney.

Pay-as-you-go (PAYG) pension systems are particularly vulnerable to demographic risks. This raises serious concerns about the long-term sustainability of public pension systems that rely on a PAYG financing model, where current contributions fund current pension expenditures. In this talk, we explore two potential approaches to reforming such systems. The first approach assumes that any deficit in the PAYG scheme is immediately covered by the state. In return, individuals are required to invest a specified amount into a fund. This investment is structured such that individuals are expected, with a certain probability, to repay the state for the deficit incurred by the PAYG scheme. We compare the outcomes of this approach with those of the traditional direct contribution model and identify the conditions under which the proposed scheme would be advantageous. The second scenario introduces a mixed pension framework (PAYG and funding) tailored for economies with a declining working-age population, operating under a defined benefit assumption. In this setup, we analyse the consequences of guaranteeing a zero return on the investments made in the funded component. Although this mixed system does not provide a hedge against demographic risks, it serves to delay the need for immediate pension reforms.

Der Vortrag reflektiert das Aufkommen internationaler Vergleichsstudien (insbesondere TIMSS und PISA) und deren Unterrichts- und Fachdidaktik-bezogene Auswirkungen auf nationale Bildungssysteme, insbesondere jenes in Österreich. Es wird dargestellt, welche Impulse die Initiative IMST (Innovations in Mathematics and Science Teaching, später Innovationen Machen Schulen Top) auf lokaler, regionaler, nationaler und internationaler Ebene setzen wollte und (teilweise) konnte. An Beispielen wird gezeigt, was dabei für die Unterrichtspraxis relevant wurde. Derzeit liegt der Fokus von IMST vor allem auf der Begleitung und Erforschung eines Schulversuchs, an welchem 57 Mittelschulen mit MINT-Schwerpunkt teilnehmen, getragen von einem Netzwerk von Partnerinstitutionen und erfahrenen Praktiker:innen.

The abelian sandpile has been introduced in physics by Bak-Tang-Wiesenfeld in 1987 as a model based on simple rules of mass distribution which exhibits peculiar properties and beautiful dynamics, and which is from the mathematical point of view hardly understood. I will introduce the model and state some of its main properties, and show how to understand it as a random walk on a finite group. As such, once can ask about stationary distributions and mixing times, and these are some of the questions that I will address during the talk. Moreover, I will also speak about stabilization/explosion of sandpiles on state spaces of fractal nature and I will state several open questions. The talk is based on joint recent works with Robin Kaiser and Nico Heizmann.

Wie "funktioniert'' Mathematik? Wie arbeiten Mathematiker:innen? Welches Bild habe ich von Mathematik und welches Bild vermittle ich (bewusst oder unbewusst) meinen Schüler:innen? Im Vortrag wird Forschung zu Einstellungen und Überzeugungen (engl.: "Beliefs'') zur Mathematik und zum Betreiben von Mathematik vorgestellt, wobei sowohl auf Beliefs von Lehrkräften, als auch von Schüler:innen eingegangen wird. Anschließend werden empirisch festgestellte Zusammenhänge solcher Beliefs mit dem Lehren von Lernen von Mathematik aufgezeigt.