Events

In 1988 Yoccoz proved that the size of the stability domain (Siegel disk) around an irrationally indifferent fixed point in the complex plane is given by a purely arithmetic function--called Brjuno's function--up to a more regular (L^\infty) correction. The Hölder interpolation conjecture (aka Marmi-Moussa-Yoccoz conjecture) states that for quadratic polynomials this correction is in fact 1/2-Hölder continuous. An analogous version of the conjecture stands also for other dynamical systems, including the standard family. Hölder continuity seems to be the relevant regularity for these problems also since it measures the difference between formulations of the arithmetical function corresponding to different continued fraction algorithms (Gauss, nearest integer, by-excess, ...) Surprisingly finally, very similar functions are used to study the convergence of trigonometric sums involving the divisor function, as discovered by Wilton almost a century ago, and also for the study of the differentiability properties of integrals of modular forms. The talk will include recent work in collaboration with Seul Bee Lee, Izabela Petrykiewicz and Tanja Schindler: https://arxiv.org/abs/2111.13553 and https://arxiv.org/abs/2111.10807

I will report on recent progress on the problem of building a stochastic process that admits the hypothetical Yang-Mills measure as its invariant measure. One interesting feature of our construction is that it preserves gauge-covariance in the limit even though it is broken by the regularisation procedure. This is based on joint work with Ajay Chandra, Ilya Chevyrev, and Hao Shen.

For n>=1, the Cremona group is the group of birational transformations of the projective space of dimension n. Algebraically, it is the Galois group of a purely transcendental extension. I will present the group and speak about the following question: which groups are quotients of this groups? I will try to present in a geometric way the answer to this question, that is very different in dimension 1,2,3 and 4.

We consider the 2-dimensional Stochastic Heat Equation (SHE), which falls outside the scope of existing solution theories for singular stochastic PDEs. When we regularise the SHE by discretising space-time, the solution can be identified with the partition function of a statistical mechanics model, the so-called directed polymer in random environment. We prove that as the discretisation is removed and the noise strength is rescaled in a critical way, the solution converges to a unique continuum limit: a universal process of random measures on R^2, which we call the critical 2d Stochastic Heat Flow. We investigate its features, showing in particular that it cannot be the exponential of a generalised Gaussian field. Based on joint work with R. Sun and N. Zygouras.

The problem of recovering a high-dimensional low-rank matrix from a limited set of random measurements has enjoyed various applications and gained a detailed theoretical foundation over the last 15 years. An instance of particular interest is the matrix completion problem where the measurements are entry observations. The first rirgorous recovery guarantees for this problem were derived for the nuclear norm minimization approach, a convex proxy for the NP-hard problem of constrained rank minimization. For matrices whose entries are ”spread out” well enough, this convex problem admits a unique solution which corresponds to the ground truth. In the presence of random measurement noise, the reconstruction performance is also well-studied, but the performance for adversarial noise remains less understood. While some error bounds have been derived for both convex and nonconvex approaches, these bounds exhibit a gap to information-theoretic lower bounds and provable performance for Gaussian measurements. However, a recent analysis of the problem suggests that under small-scale adversarsial noise, the reconstruction error can be significantly amplified. In this talk, we investigate this amplification quantitatively and provide new reconstruction bounds for both small and large noise levels that suggest a quadratic dependence between the reconstruction error and the noise level. This is joint work with Julia Kostin (TUM/ETH) and Dominik Stöger (KU Eichstätt-Ingolstadt).

A classical notion of curvature is the sectional curvature of a Riemannian manifold. Alexandrov and Gromov generalized the Riemannian curvature notion to metric spaces. In 2006 Januszkiewicz and Świątkowski introduced systolicity as a non-positive curvature criterion for simplicial complexes. I will define systolic complexes and give some examples. I will also mention some properties of these spaces and of groups acting on them. We will also see how this notion compares to other notions on non-positive curvature.

In this seminar we outline a recent example of a <i>turbulent</i> divergence free velocity field \(u \in C^\alpha ([0,1 ] \times \T^2)\), with \(\alpha < 1\), having the \textit{non-selection} property. The latter is defined as follows: consider the sequence \(\{ \theta_\nu \}_{\nu >0}\) of solutions to the associated advection diffusion equation with viscosity parameter \(\nu>0\) and fixed initial datum \(\theta_{\text{in}} \in C^\infty\). Then, at least two distinct limiting solutions of the advection equation in the weak* topology arise from the sequence \(\{\theta_\nu\}_{\nu >0}\) as \(\nu \to 0\). Finally, we also mention a recent result of <i>anomalous dissipation</i>, at the level of the forced Navier--Stokes equations in the sharp regularity class \(L^3_t C^{1/3-}_x\) based on the previous <i>turbulent</i>} velocity field, which in particular implies the failure of the energy balance in the forced Euler equations. These are joint works with Elia Bru\'e, Maria Colombo, Gianluca Crippa and Camillo De Lellis.

We study the asymptotic behavior of normalized maxima of real-valued particles with mean-field drift interaction. Our main result establishes propagation of chaos: in the large population limit, the normalized maxima behave as those arising in an i.i.d. system where each particle follows the associated McKean-Vlasov limiting dynamics. Because the maximum depends on all particles, our result does not follow from classical propagation of chaos, where convergence to an i.i.d. limit holds for any fixed number of particles but not all particles simultaneously. This is joint work with Nikos Kolliopoulos and Zeyu Zhang.