Events

General Relativity modelizes gravity as an elastic behavior of spacetime described in a Lagrangian fashion by the Einstein-Hilbert energy. Applying principles of Noether's Theorem one can then exploit symmetries of a given model to obtain conserved quantities measuring  how far an Einsteinian spacetime (or an interesting spaceslice of it) is from the model. Among those, the mass of an Asymptotically Euclidean manifold (seen as the conserved quantity induced by the time translational symmetry of the Euclidean slice in Minkowski spacetime) stands out for its application to Riemannian geometry with the Positive Mass Theorem. After explaining the principles of construction and trying to give a feel of the nature of such conserved quantities, we will apply them to a fourth order gravitational theory and explore the Riemannian consequences.

<p>Forcing is arguably the most versatile and widely used technique for proving consistency and independence results in set theory and in mathematics more generally. The aim of this talk is to introduce this method and to use it to prove that the continuum hypothesis is independent of ZFC. No prior knowledge is required.</p>

The celebrated Keen collar lemma guarantees that a simple closed geodesic on a hyperbolic Riemann surface has a collar (tubular neighborhood) whose width only depends on its length. Viewing a Riemann surface as the quotient of the unit ball in the complex plane, a natural generalization is to ball quotients in higher dimensions where the Poincaré metric is replaced by the Bergman metric (also known as the complex hyperbolic metric). Such ball quotients are called complex hyperbolic manifolds. The focus of this talk will be on embedded complex geodesics in complex hyperbolic 2-manifolds; a complex geodesic has complex codimension one in the quotient complex 2-manifold. We prove a tubular neighborhood theorem for such a complex geodesic where the width of the tube depends only on the Euler characteristic of the embedded complex geodesic. We also derive an explicit estimate for this width. After giving a short history of the collar lemma generalizations and discussing the basics of complex hyperbolic geometry, we will discuss the ideas leading to the proof of this tubular neighborhood theorem. This is joint work with Youngju Kim.

We present a mathematical and numerical framework for modeling the dynamics of observables in complex systems, termed Flow Map Learning (FML). Rather than approximating governing equations, FML constructs discrete-time evolution operators directly from data, enabling accurate prediction of the observables even when the underlying full system is unknown. We establish that, for a broad class of systems, observable dynamics admit finite-dimensional delay representations, leading to closed evolution equations with minimal memory. This perspective provides a foundation for incorporating temporal dependence in data-driven models and fast predictions of observable dynamics. We then present various numerical examples to demonstrate the efficacy of FML for long-time prediction of observable dynamics.

The Weil-Petersson metric on the moduli space of hyperbolic surfaces with $g$ genus and $n$ cusps is of finite volume, and hence induce a probability measure and a model of random cusped surfaces. The study of nearly optimal spectral gap for random closed hyperbolic surfaces is a great breakthrough in recent years. In this talk, we consider the Weil-Petersson random cusped hyperbolic surfaces, and show a uniform lower bound of the spectral gap when $n=o(\sqrt{g})$. We will carefully introduce the knowledge of Weil-Petersson random hyperbolic surfaces and the spectrum in the beginning. The talk is based on the joint work with Yuxin He and Yunhui Wu.

Quantum chaos is the study of spectral data of classically chaotic systems. Key objects of study are the eigenvalues and eigenfunctions of the laplacian on hyperbolic surfaces. Recently, large-scale quantum chaos-type results have garnered considerable interest. In this talk, we aim to give an introduction to this field, and some of its tools. In ongoing work, we provide a new insight into the shapes of short geodesic loops on surfaces, which, in particular, can be applied to the study of eigenfunctions in the large-scale regime.

We introduce a framework for stochastic games on large sparse graphs, covering continuous-time and discrete-time dynamic games as well as static games. Players are indexed by the vertices of simple, locally finite graphs, allowing both finite and countably infinite populations, with asymptotics described through local weak convergence of marked graphs. The framework allows path-dependent utility functionals that may be heterogeneous across players. Under a contraction condition, we prove existence and uniqueness of Nash equilibria and establish exponential decay of correlations with graph distance. We further show that global equilibria can be approximated by truncated local games, and can even be reconstructed exactly on subgraphs given information on their boundary. Finally, we prove convergence of Nash equilibria along locally weakly convergent graph sequences, including sequences sampled from hyperfinite unimodular random graphs. This is a joint work with Sturmius Tuschmann.

I will report on joint work in progress with Joaquín Rodrigues Jacinto and Gautier Ponsinet concerning Iwasawa cohomology of vector bundles on the Fargues--Fontaine curve along a (deeply ramified) p-adic Lie extension $K_\infty/K$. By working with derived analytic vectors we obtain a reinterpretation of classical results for bundles of slope 0 for any p-adic Lie extension. For general slopes we obtain a reinterpretation in terms of the (dual of the) \psi=1 Eigenspace of $(\varphi,\Gamma)$-modules in the case that $K_\infty/K$ is the cyclotomic extension. For general $p$-adic Lie extensions the latter is badly behaved while the former can still be defined and is expected to be well-behaved. I will focus on the Lubin--Tate case in which we can prove this expectation. If time permits I will mention some partial results for SL_2 and difficulties in the general case.