Weekly Bulletin

Floer homology and the related invariants of 4-manifolds has given us deep insight in smooth differential topology in dimensions 3 and particularly 4. The theory has yielded insights like existence of exotic differentiable structures on 4 dimensional euclidean space, complex curves minimize genus in complex projective space, killing the Hauptvermuntung, there even appear to be connection to the 4 color map theorem. This course will build up Floer homology of three manifolds from scratch. The focus will be on Instanton Floer homology but we will mention other versions and develop applications as the course goes on.

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 course we will investigate questions of weak turbulence theory by using as explicit example of wave interactions the solutions to periodic and nonlinear Schrödinger equations. We will start with Strichartz estimates on periodic setting, then we will move to well-posedness. We will then present two different ways of introducing the evolution of the energy spectrum. We will first work on a method proposed by Bourgain and involving the growth of high Sobolev norms. Then, we will give some ideas of how to derive rigorously the effective dynamics of the energy spectrum itself (wave kinetic equation), when one considers weakly nonlinear dispersive equations.

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 rapid development of quantum simulators in the past decades has opened unprecedented possibilities to study quantum many-body systems. A wide range of Hamiltonians has been directly engineered in analog simulators, and provided crucial insights into quantum phases of matter, as well as into dynamical phenomena. This progress was accompanied by recent breakthroughs in digital quantum devices, promising to grant computational capacities far beyond the reach of classical architectures. This talk illustrates the opportunities and outstanding challenges presented by this rapidly developing field. In the first part, we demonstrate the power of combining theoretical approaches with experiments in a trapped ion quantum simulator, allowing us to identify and observe signatures of universal transport in a long-range interacting quantum magnet. We then turn to the challenges of protecting quantum coherence against environmental noise. In contrast to the general expectation that in an open system coherent information is quickly destroyed by the dissipative environment, we show that this is not necessarily the case, in particular, intrinsic chaotic quantum dynamics can efficiently protect quantum coherence against generic boundary noise. We comment on the implications of these results for designing robust quantum devices.

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar/Izabella-Lovas.htmlcall_made

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 investigate the properties of a family of polytopes that naturally arise in connection with a problem in distributed data storage, namely service rate region polytopes. We make use of polytope theory to bring deeper understanding to the service rate problem.

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.

The MinRank Problem is the computational problem of finding a low rank linear combination of a set of matrices. There are several cryptographic attacks that can be reduced to MinRank. During this talk I will introduce a recent approach to the problem: the Support Minors Modeling. We will discuss the modeling itself and the heuristics that the authors made in order to estimate the complexity.

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/

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.