Veranstaltungen

This talk presents a new approach to the quantization of Poisson manifolds with linearizable transverse Poisson structure through what I call their symplectic avatars: E-symplectic manifolds naturally associated with a given Poisson structure. These avatars provide accessible symplectic models that preserve the core features of the underlying Poisson geometry while allowing the use of powerful symplectic techniques. I will begin with the case of b-symplectic manifolds, showing how their geometric and deformation quantization can be explicitly determined and vividly illustrated in the presence of additional symmetries using the language of Delzant polytopes. This example demonstrates how singular symplectic structures offer a fertile testing ground for quantization beyond the classical smooth setting. I will conclude the talk with a desingularization theorem for linearizable transverse Poisson structures (joint work with Ryszard Nest), establishing that they can be desingularized by suitable E-symplectic avatars. I will discuss how this desingularization principle opens new avenues for both geometric and deformation quantization, effectively bridging Poisson and symplectic worlds.

A classical theorem of Birkhoff (1922) asserts that any essential invariant curve of a symplectic twist map on the cylinder is a Lipschitz graph over the circle. Since then, several extensions of this result to higher-dimensional settings have been obtained (Herman, Katznelson–Ornstein, Siburg, Bialy–Polterovich, among others). A generalization due to Arnaud (2010) in the contangent bundle of any closed manifold, states that an invariant exact Lagrangian, under the flow of a convex Hamiltonian, is a C1 graph over the base. We will try to show that this graph property is not a consequence of invariance itself but rather of recurrence. We will provide an optimal condition on the orbit of an exact Lagrangian submanifold under the Hamiltonian flow, requiring backward and forward convergence (up to subsequences) in a topology controlling the Liouville primitives, which ensures that the submanifold and all its iterates are C1 graphs over the base. We will see that this implies recurrence. The proof combines two complementary viewpoints on weak solutions of the Hamilton–Jacobi equation : the variational viewpoint via graph selectors of exact Lagrangian submanifolds, and the regularity properties of viscosity solutions provided by the weak-KAM approach.

<p>Schrödinger operators and their localization phenomena play a central role in spectral theory and its connection to dynamical systems. In this talk, I will focus on discrete Schrödinger operators defined on large finite graphs and investigate their asymptotic behavior as the graph size tends to infinity. In joint work with A. Avila, we establish a general criterion showing that for any family of graphs without eigenvalues concentration, delocalization of eigenfunctions become asymptotically dense in the space of potentials. Along the way, I will emphasize the interplay with ergodic Schrödinger operators, where the potential is generated by an underlying topological dynamical system.</p>

I will discuss joint work with Philip Engel and Stefan Schreieder, in which we prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class, and that the same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. I will also discuss our most recent result, which shows that on a very general principally polarized abelian 6-fold, the smallest multiple of the minimal curve class which can be represented by an algebraic cycle is exactly 6.

Very little is known about the classification of finite subgroups of Cremona in dimension three. It is natural to start with the case of simple groups, and this step was achieved by Prokhorov in 2009 over the field of complex numbers. In the work I will present, we show that the only non-cyclic finite simple subgroups of the real Cremona group of rank three are A5 and A6. This is a joint project with I. Cheltsov and Y. Prokhorov.

Randomized algorithms are becoming increasingly popular in matrix computations. In fact, randomization is on the verge of replacing existing deterministic techniques for several large-scale linear algebra tasks in scientific computing. The poster child of these developments, randomized SVD, is now one of the state-of-the-art approaches for performing low-rank approximation. In this talk, we will go beyond the randomized SVD and illustrate the great potential of randomization to not only speed up existing algorithms, but to also yield novel and often simple algorithms for solving notoriously difficult problems. Examples covered in this talk include reduced order modeling, acceleration of scientific simulations, joint diagonalization, and large null space computation. A common theme of these developments is that randomization helps to transform linear algebra results that only hold generically into robust and reliable numerical algorithms.

The cutoff phenomenon is a sharp transition in the convergence of high-dimensional Markov chains to equilibrium: the total variation distance remains close to 1 for a long time and then rapidly decreases to almost 0 over a much shorter time window. It was initially discovered in the context of card shuffling by Diaconis and Shahshahani, and since then observed in a variety of different models. In spite of its ubiquity, it is still largely unexplained, and most proofs are model-specific. In this talk, we discuss a high-level approach to establishing cutoff based on transport inequalities, and we illustrate it on a popular algorithm known as the proximal sampler, when the target measure on R^d is log-concave. Based on joint work with Justin Salez.

Classical causal inference typically targets low-dimensional estimands such as the average treatment effect. A richer understanding, however, requires characterizing the full outcome distribution under different treatments. In addition, the ability to simulate counterfactual outcomes is essential for causal model selection and evaluation. Recent advances in distributional learning provide a principled foundation for these goals. In this talk, we build on engression—a distributional learning approach—to develop methods for estimating distributional causal effects and generating data from causal models. We first introduce a distributional method for instrumental variable settings with unobserved confounders, enabling estimation of full interventional distributions from which classical estimands arise as functionals. When additional covariates are observed but marginal causal effects remain the central interest, as is common in clinical trials, we propose a framework that parametrizes the joint observed distribution around the causal margin with no redundancy. This allows for both estimation and simulation under user-specified interventions. Joint work with Anastasiia Holovchak, Sorawit Saengkyongam, Nicolai Meinshausen, Linying Yang, and Robin Evans

We develop a rigorous framework for continuous-time algorithmic trading strategies from the point of view of mathematical finance. To this end, we first establish a universal approximation theorem for neural networks on locally convex spaces with respect to Orlicz-type topologies. When the underlying sigma-algebra is generated by a possibly uncountable family of random variables, we prove that neural networks, through functional representations, can approximate functions in these Orlicz spaces arbitrarily well. We then represent algorithmic strategies as simple predictable processes to establish their approximation capabilities in spaces of stochastic (integral) processes. As applications, we prove that algorithmic strategies can approximate mean-variance optimal hedging strategies arbitrarily well, we establish a no free lunch with vanishing risk condition for algorithmic strategies, and we obtain a 'neural' version of the Bichteler-Dellacherie theorem. This talk is based on joint work with Uwe Schmock (TU Wien).

We explore the zeros of certain Poincaré series P(k,m) of weight k and index m for the full modular group. These are distinguished modular forms, which have played a key role in the analytic theory of modular forms. We study the zeros of P(k,m) when the weight k tends to infinity. The case where the index m is constant was considered by Rankin who showed that in this case almost all of the zeros lie on the unit arc |z|=1. In this talk we will explore the location of the zeros when the index m grows with the weight k, finding a range of different limit laws. Along the way, we also establish a version of Quantum Unique Ergodicity for some ranges.

GKM spaces are important examples in algebraic and symplectic geometry, including smooth toric varieties, homogeneous spaces, smooth Schubert varieties, and non-algebraic examples like the twisted flag manifold. I will quickly introduce the main concepts of GKM theory and explain how the GKM graph completely determines the Gromov-Witten invariants and hence quantum cohomology of its space. This opens up a two-way interaction between the two theories: while GKM theory provides computability, GW theory provides necessary conditions for the Hamiltonian GKM realization problem. I will sketch some first results in both directions and introduce our new software package "GKMtools", which supports explicit computations at this intersection. Joint work with Giosuè Muratore.