Veranstaltungen

More than twenty years ago, Etnyre and Ghrist established a connection between Reeb fields and a class of stationary solutions to the 3D Euler equations for ideal fluids. In this talk, we present a new framework that allows assigning contact/symplectic invariants to large sets of time-dependent solutions to the Euler equations on any three-manifold with an arbitrary fixed Riemannian metric, thus broadening the scope of contact topological methods in hydrodynamics. We use it to prove a general non-mixing result for the infinite-dimensional dynamical system defined by the equation and to construct new conserved quantities obtained from embedded contact homology spectral invariants. This is joint work with Francisco Torres de Lizaur.

In this talk, we will explore the inner workings of interactive proof assistants such as Lean4 and discover a profound connection between mathematical logic and computer programs known as the Curry-Howard Isomorphism. At the heart of this correspondence lies type theory, the formal study of type systems---the same kind you know from strongly-type programming languages like C++. Lean4 is a functional programming language that is also strongly typed. But it's special in sense that it's equipped with a type system so powerful, that it is capable of expressing any mathematical statement (as a type) and formally verifying their proofs (by constructing the type). In this way it's possible to construct the entirety of mathematics within Lean4. People are actually doing this by creating a library called Mathlib! There's even a project that tries to rewrite every theorem and proof of the undergraduate math curriculum of Imperial College London in Lean. Famous mathematicians like Terrance Tao are also using Lean to formalize their complex conjectures. The most impressive real-life application of Lean4 to me is the use of it in AI-driven mathematics. For instance, Google DeepMind's AlphaProof, built with Lean4, recently solved an International Math Olympiad problem at the level of a Silver Medalist! This breakthrough shows how proof assistants have the potential to transform mathematics, paving the way for automated mathematical superintelligence.

More information: https://zucmap.ethz.ch/call_made

The question of whether solutions exist globally in time or if they develop singularities in finite time is a question of great importance in the study of incompressible fluids. In this talk, I will discuss some of the recent results of the field, with a special emphasis on the 3D incompressible Euler equations.

The Kontsevich-Zorich (KZ) cocycle is a key dynamical system that is closely related to the derivative cocycle of the Teichmüller geodesic flow. We will state and sketch a proof of a central limit theorem for the KZ cocycle, and explain some of the motivations, especially towards the goal of proving the existence of large fluctuations of the Hodge norm of the parallel transport of vectors along Teichmüller horocycles. Such fluctuations were leveraged by Chaika-Khalil-Smillie in their work on the ergodic measures of the Teichmüller horocycle flow. Our work is joint with Giovanni Forni.

We present an unbiased method for Bayesian posterior means based on kinetic Langevin dynamics that combines advanced splitting methods with enhanced gradient approximations. Our approach avoids Metropolis correction by coupling Markov chains at different discretization levels in a multilevel Monte Carlo approach. Theoretical analysis demonstrates that our proposed estimator is unbiased, attains finite variance, and satisfies a central limit theorem. We prove similar results using both approximate and stochastic gradients and show that our method's computational cost scales independently of the size of the dataset. Our numerical experiments demonstrate that our unbiased algorithm outperforms the "gold-standard" randomized Hamiltonian Monte Carlo.

Causal reasoning is a cornerstone of human intelligence and a critical capability for artificial systems aiming to achieve advanced understanding and decision-making. While large language models (LLMs) excel on many tasks, a key question still remains: How can these models reason better about causality? Causal questions that humans can pose span a wide range of fields, from Newton’s fundamental question, “Why do apples fall?” which LLMs can now retrieve from standard textbook knowledge, to complex inquiries such as, “What are the causal effects of minimum wage introduction?”—a topic recognized with the 2021 Nobel Prize in Economics. My research focuses on automating causal reasoning across all types of questions. To achieve this, I explore the causal reasoning capabilities that have emerged in state-of-the-art LLMs, and enhance their ability to perform causal inference by guiding them through structured, formal steps. Finally, I will outline a future research agenda for building the next generation of LLMs capable of scientific-level causal reasoning. https://zhijing-jin.com/fantasy/about/

Can hyperbolic groups act on spaces exhibiting nonpositive curvature in a noncoarse way? The quest to understand this has led to the study of injective metric spaces. In this talk, we will discuss some basic properties of such spaces and explore their connection to nonpositive curvature. Then, we will shift to combinatorial dimension, a closely related notion which finds application in describing locally elliptic actions.

Der Einsatz von Computern in der Vektorgeometrie führt zu neuen Algorithmen. Traditionell sind die Aufgaben in Lehrbüchern für Berechnungen von Hand ausgerichtet. Verwendet man Computer, so muss nicht mehr darauf geachtet werden, dass die Rechnungen einfach und möglichst mit ganzen Zahlen durchgeführt werden können. Wir zeigen, dass die Anwendung von Givensrotationen (Drehmatrizen) zu neuen, oft einfacheren Algorithmen führt. Diese Givensrotationen sind zwar geometrisch anschaulich, aber ungeeignet für Handrechnungen, weil zu aufwändig. Dem Computer ist es aber egal, ob er mit ganzen Zahlen oder mit 10-stelligen Dezimalbrüchen rechnet! <br /><br /> <a href="https://www.math.ethz.ch/content/dam/ethz/special-interest/math/math-ausbildung-dam/documents/kolloquium/Gander_Ellipsen.pdf">Präsentation Ellipsen als PDF</a><br /> <br /> <a href="https://www.math.ethz.ch/content/dam/ethz/special-interest/math/math-ausbildung-dam/documents/kolloquium/Gander_Kolloquium.pdf">Präsentation Kolloquium als PDF</a><br />

We study the expected utility maximization problem of a large investor who is allowed to make transactions on tradable assets in an incomplete financial market with endogenous permanent market impacts. The asset prices are assumed to follow a nonlinear price curve quoted in the market as the utility indifference curve of a representative liquidity supplier. We show that optimality can be fully characterized via a system of coupled forward-backward stochastic differential equations (FBSDEs) which corresponds to a non-linear backward stochastic partial differential equation (BSPDE). We show existence of solutions to the optimal investment problem and the FBSDEs in the case where the driver function of the representative market maker grows at least quadratically or the utility function of the large investor falls faster than quadratically or is exponential. Furthermore, we derive smoothness results for the existence of solutions of BSPDEs. Examples are provided when the market is complete, the utility function is exponential or the driver is positively homogeneous.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $K$ be an imaginary quadratic field. Assume that $N=N^+ N^-$ with $N^+$ split in $K$ and $N^-$ squarefree and inert in $K$. Under this "generalised" Heegner hypothesis, in the last thirty years there have been many works building $K$-Heegner points on $E$ by studying the arithmetic of Eichler orders of level $N^+$ inside the quaternion algebra of discriminant $N^-$ over $\mathbb{Q}$. The existence of nontrivial systems of Heegner points has always deep consequences, leading to rank one results for $E(K)$ and to control on the arithmetic of $E$ over anticyclotomic $p$-extensions of $K$. Much less is known when $N^-$ is not squarefree. In this talk, I will explain how one can use the arithmetic of Pizer orders to build Heegner points in this setting, building on a recent work of Longo, Rotger and de Vera-Piquero. I will then show how their work could be generalized to study other Galois representations, mainly focusing on Hida families of modular forms. This is a joint work with Luca Dall'Ava.

Tetrahedron instantons were recently introduced by Pomoni-Yan-Zhang in string theory, as a way to describe systems of D0-D6 branes with defects. We propose a rigorous geometric interpretation of their work by the point of view of Donaldson-Thomas theory. We will explain how to naturally construct the moduli space of tetrahedron instantons as a Quot scheme, parametrizing quotients of a torsion sheaf over a certain singular threefold, and how to construct a virtual fundamental class in this setting using quiver representations and the recent machinery of Oh-Thomas (which is in principle designed for moduli spaces of sheaves on Calabi-Yau 4-folds). Furthermore, we will show how to formalize mathematically the invariants considered by Pomoni-Yan-Zhang (initially defined via supersymmetric localization in Physics) and how to rigorously compute them, solving some open conjectures. Joint work with Nadir Fasola.