Structure recovery is at the heart of modern Data Science. Roughly put, the goal in structure recovery problems is to identify (or at least approximate) an unknown object using limited, random information – e.g., a random sample of the unknown object.

As it happens, key questions on recovery are fundamental (open) problems in Asymptotic Geometric Analysis and High Dimensional Probability. In this talk I will give one example (out of many) that exhibits the rather surprising ties between those seemingly unrelated areas.

I will explain why noise-free recovery is dictated by the geometry of natural random sets: for a class of functions 𝐹 and n i.i.d random variables 𝜎 = (𝑋_1,…,𝑋_n), the random sets are 𝑃_𝜎 (𝐹) = { (𝑓(𝑋_1),….,𝑓(𝑋_n)) : 𝑓 ∈ 𝐹 }.

I will outline a (sharp) estimate on the structure of a typical 𝑃_𝜎 (𝐹) that leads to the solution of the noise-free recovery problem under minimal assumptions. I will explain why the same estimate resolves various questions in high dimensional probability (e.g., the smallest singular values of certain random matrices) and high dimensional geometry (e.g., the Gelfand width of a convex body).

The optimality of the solution is implied by a exposing a “hidden extremal structure” contained in 𝑃_𝜎 (𝐹), which in turn is based on a complete answer to Talagrand’s celebrated entropy problem. _____________________________________________

Invited by Prof. Holger Rauhut.

See https://www.lmu.de/ai-hub/en/news-events/all-events/event/munich-ai-lectures-prof.-dr.-jean-luc-starck.html

Matrix Concentration inequalities such as Matrix Bernstein inequality (Oliveira and Tropp) have played an important role in many areas of pure and applied mathematics. These inequalities are intimately related to the celebrated noncommutative Khintchine inequality of Lust-Piquard and Pisier. While these tend to be optimal when the underlying matrices are commutative, they are known to be sub-optimal in several other instances. Recently, we have leveraged ideas from Free Probability to fully remove the sub-optimal dimensional dependencies in these inequalities in a range of instances, yielding sharp bounds in many settings of interest. In this talk I will describe these results, some of the recent and ongoing work that it has sparked, and several open problems. Includes joint work with: March Boedihardjo (MSU); Ramon van Handel and Giorgio Cipolloni (Princeton); Petar Nizic-Nikolac, Anastasia Kireeva, Kevin Lucca, and Dominik Schroder (ETH); Xinmeng Zeng (Stanford); Dustin Mixon (OSU); Dmitriy Kunisky (Johns Hopkins); Thomas Rothvoss (U Washington); Haotian Jiang (U Chicago); Sivakanth Gopi (MSR). ______________________

Invited by Prof. Holger Rauhut

Thanks to their high degree of control and tunability, ultracold atomic gases provide a rich platform for the study of quantum many-body effects. Ultracold gases of highly magnetic atoms exhibit unique interaction properties that lead to striking behaviors, both at the mean-field level and beyond [1]. A decade ago, a universal stabilization mechanism driven by quantum fluctuations was discovered in these gases. This mechanism prevents the systems from collapsing when the mean-field interactions become attractive, and instead allows exotic states of matter to arise, including ultradilute quantum droplets, crystallized quantum states, and especially the so-called supersolids [2]. In my colloquium, I will present the seminal observations of these states and how they emerged from the long-standing progress in the field. I will discuss the theoretical description of these systems via an effective mean-field theory, including the effect of quantum fluctuations via a higher-order effective interaction. I will outline our current understanding of the properties of these states and highlight open questions.

[1] L. Chomaz & al, Dipolar physics: a review of experiments with magnetic quantum gases, Reports on Progress in Physics 86, 026401 (2023).

[2] L. Chomaz, Quantum-stabilized states in magnetic dipolar quantum gases, arXiv preprint 2504.06221 (2025)

_____________________

Invited by Prof. Arnaud Triay

We study finite-player dynamic stochastic games with heterogeneous interactions and non-Markovian linear-quadratic objective functionals. We derive the Nash equilibrium explicitly by converting the first-order conditions into a coupled system of stochastic Fredholm equations, which we solve in terms of operator resolvents. When the agents' interactions are modeled by a weighted graph, we formulate the corresponding non-Markovian continuum-agent game, where interactions are modeled by a graphon. We also derive the Nash equilibrium of the graphon game explicitly by first reducing the first-order conditions to an infinite-dimensional coupled system of stochastic Fredholm equations, then decoupling it using the spectral decomposition of the graphon operator, and finally solving it in terms of operator resolvents. Moreover, we show that the Nash equilibria of finite-player games on graphs converge to those of the graphon game as the number of agents increases. This holds both when a given graph sequence converges to the graphon in the cut norm and when the graph sequence is sampled from the graphon. We also bound the convergence rate, which depends on the cut norm in the former case and on the sampling method in the latter. Finally, we apply our results to various stochastic games with heterogeneous interactions, including systemic risk models with delays and stochastic network games. ______________________________

Invited by Prof. Alexander Kalinin

Das Beweisen ist für die Mathematik als Disziplin von zentraler Bedeutung und spielt daher auch in der mathematischen Ausbildung eine wichtige Rolle. Lernende sollen die Mathematik als deduktives System begreifen, die Art der Absicherung mathematischer Ergebnisse verstehen, argumentative Herausforderungen erfolgreich bewältigen können und so ein adäquates Verständnis von mathematischen Beweisen aufbauen. Ausgehend von einem theoretischen Rahmenmodell zum mathematischen Beweisverständnis werden Ergebnisse empirischer Studien vorgestellt, die das Beweisverständnis von Lernenden in unterschiedlichen Phasen der mathematischen Ausbildung untersuchen und Möglichkeiten der Förderung aufzeigen. ______________________

Invited by Prof. Stefan Ufer

TBA

_____________________

Invited by Prof. Christian Hainzl

Quantum mechanics, now about a century old, is a very successful physical theory of matter on a small scale. From its first description until today, it has surprised scientists and laypersons alike by the strange behaviour it attributes to particles, atoms, and molecules. This behaviour can be characterized by the keywords Uncertainty, Superposition, and Entanglement. It took about sixty years before it was realized that these three characteristics do not just express a certain vagueness and strangeness of matter on a small scale but can actually be USEd to our advantage. In 1994 Peter Shor made this idea concrete by devising an algorithm that would enable large arrays of quantum systems to perform specific calculations (factoring large integers), which are impossible to do in practice on any classical device. With this algorithm, present-day cryptographic schemes can be broken, provided such "quantum computers” can be made to work. Starting from a discussion of the "two-slit experiment”, we sketch the working of Shor's algorithm and discuss the possibilities of future quantum computers.

About the speaker: Hans Maassen is a dutch mathematical physicist and emeritus professor specializing in quantum probability and quantum information theory. Standing out among his discoveries is the entropic uncertainty relation, named after himself and Jos Uffink, a fundamental  inequality in quantum mechanics.

This talk is open to the general public and all interested persons, and is presented by the SFB TRR352 "Mathematics of Many-Body Quantum systems and their collective phenomena" in cooperation with the TUM-IAS Workshop "Beyond IID in Information Theory".