Veranstaltungen

I will talk about my recent work with E. Breuillard establishing limit theorems for random walks on nilpotent Lie groups. Most previous works assumed the law of increment to be centered in the abelianization of the group. Our major contribution is to allow the law of increment to be non-centered. In this case, new phenomena appear: the large scale geometry of the walk depends on the increment average, and the limiting measure in the central limit theorem may not have full support in the group.

In contrast to density property or volume density property, there are few symplectic Stein manifolds known to admit symplectic holomorphic density property. One example is $\mathbb{C}^{2n}$. In this talk we first introduce the symplectic holomorphic density property with the corresponding version of Anders{\'e}n--Lempert theory. Then we outline a proof for this property for the Calogero--Moser space $\mathcal{C}_n$ of $n$ particles, and describe its group of holomorphic symplectic automorphisms. This is joint work with Rafael B. Andrist.

A classic result from optimal stopping theory asks when one should stop and accept a value from a sequence of observed values, presented one by one, if one knows the distributions that these values are coming from and wants to select the maximum. The catch is that, if a value is rejected, it can never be selected again. This setting, which combines online decision making with stochastic uncertainty, is called the prophet inequality. Over the past decade, there have been two very exciting discoveries, connecting auction theory and online combinatorial optimization with prophet inequality-type problems. In this talk, we cover some of the standard ideas and results on prophet inequalities and online combinatorial optimization. Afterwards, we present the first results in a new direction that studies prophet inequalities for minimization instead, a setting which resents rich and qualitatively different results from the maximization case.

The generating functions of genus g Gromov-Witten invariants of Calabi-Yau threefolds lead to factorially divergent series, when the genus is large. Finding the precise asymptotics of these series is an interesting question. The theory of resurgence of Jean Ecalle associates a very rich structure to a factorially divergent series, involving a collection of “trans-series”, which in particular makes it possible to obtain the precise asymptotics. Determining the resurgent structure of generating functions in Gromov-Witten theory is a difficult problem, and in this talk I will present some partial results and conjectures on this structure, based on trans-series solutions of the holomorphic anomaly equations. As a consequence of these results, I will formulate precise conjectures for the asymptotics of the generating functions of the quintic CalabiYau, and for the asymptotics of fixed degree orbifold Gromov-Witten invariants of C^3/Z_3.

If one is interested in finitely generated residually finite groups, it is natural to ask to what extent such a group is determined by all its finite quotients. The finite quotients of a group are encoded in the group's profinite completion. Perhaps the boldest question in this area is that attributed to Remeslennikov: "If G has the same profinite completion as a free group of finite rank, must G be isomorphic to that free group?" The answer to this still seems remote, given current knowledge, but some success has been found on variants of it, restricting the type of group G is allowed to be. Another natural variant of Remeslennikov's question is whether a free factor of a group G can be detected from its profinite completion: if G has a subgroup H, whose closure in the profinite completion of G is a profinite free factor, must H be a free factor of G? This question is still hard, with positive known answer only when G itself is a free group. I will report on joint work with A. Jaikin in which we provide a new proof of the above positive answer and extend it to the case when G is virtually free. The methods may be extended to other classes of groups if some interesting questions are answered on their completed group algebras. No knowledge of profinite groups will be assumed for the talk.

Reduced order modeling (ROM) techniques, such as the reduced basis method, provide nowadays an essential toolbox for the efficient approximation of parametrized differential problems, whenever they must be solved either in real-time, or in several different scenarios. These tasks arise in several contexts like, e.g., uncertainty quantification, control and monitoring, as well as data assimilation, ultimately representing key aspects in view of designing predictive digital twins in engineering or medicine. On the other hand, in the last decade deep learning algorithms have witnessed a dramatic blossoming in several fields, ranging from image and signal processing to predictive data-driven models. More recently, deep neural networks have also been exploited for the numerical approximation of differential problems yielding powerful physics-informed surrogate models. In this talk we will explore different contexts in which deep neural networks (DNNs) can enhance the efficiency of ROM techniques, ultimately allowing the real-time simulation of large-scale nonlinear time-dependent problems. We show how to exploit DNNs to build ROMs for parametrized PDEs in a fully non-intrusive way, exploiting deep autoencoders as main building block, ultimately yielding deep learning-based ROMs (DL-ROMs) and their further extension to POD-enhanced DL-ROMs (POD-DL-ROMs). In particular, we will provide some guidelines for the design of deep autoencoders, showing the interplay between their minimal latent dimension and some topological properties of the solution manifold, and illustrating some theoretical results on the approximation errors entailed by the proposed approach, as well as more recent investigations on the use of deep convolutional autoencoders. Other examples of ROM strategies enhanced by deep learning include the use of DNNs for (i) learning nonlinear ROM operators, thus yielding hyper-reduced order models enhanced by deep neural networks (Deep-HyROMnets), or (ii) enhancing the accuracy of low-fidelity ROMs through a multi-fidelity neural network regression technique for the sake of input/output evaluations. Through a set of applications from engineering including, e.g., structural mechanics and fluid dynamics problems, we will highlight the opportunities provided by deep learning in the context of ROMs for parametrized PDEs, as well as those challenges that are still open.

We consider supercritical bond percolation on Z^d and study the chemical distance, i.e., the graph distance on the infinite cluster. It is well-known that there exists a deterministic constant μ(x) such that the chemical distance D(0,nx) between two connected points 0 and nx grows like nμ(x). We prove the existence of the rate function for the upper tail large deviation event {D(0,nx)>nμ(x)(1+ϵ),0↔nx} for d>=3. Joint work with Shuta Nakajima.

Stable commutator length (scl) is a measure of homological complexity in groups that has many surprising connections with various topics in geometric topology and group theory. We will introduce scl as well as some of those connections, and discuss the (hard) problem of computing scl. Time permitting, we will explain some of the ideas behind rationality theorems, which provide algorithms to compute scl in free groups and some generalisations.

An affine cubic surface is defined by an irreducible cubic polynomial f with integer coefficients in three variables, and its integral points correspond to integral roots of this polynomial. These solutions tend to be sparser and much harder to find than in cases of higher dimension or lower degree, as witnessed by the search for representations of integers k as sums of three cubes — that is, the case f = x³ + y³ + z³ - k – whose existence or nonexistence has only recently been settled for the first one hundred integers. To shed new light on this kind of problem, we develop a Hardy–Littlewood heuristic for the number of integral points on affine cubic surfaces, arriving at a prediction that looks similar to Manin's conjecture on rational points and related problems. We compare this heuristic to Heath-Brown's prediction for sums of three cubes, to Zagier's count on the Markoff surface and to Baragar's and Umeda's work on variants of it, as well as to numerical data. This is joint work with Tim Browning.

Ensemble weather forecasts based on multiple runs of numerical weather prediction models typically show systematic errors and require post-processing to obtain reliable forecasts. Accurately modeling multivariate dependencies is crucial in many practical applications, and various approaches to multivariate post-processing have been proposed where ensemble predictions are first post-processed separately in each margin and multivariate dependencies are then restored via copulas. These two-step methods share common key limitations, in particular the difficulty to include additional predictors in modeling the dependencies. We propose a novel multivariate post-processing method based on generative machine learning to address these challenges. In this new class of nonparametric data-driven distributional regression models, samples from the multivariate forecast distribution are directly obtained as output of a generative neural network. The generative model is trained by optimizing a proper scoring rule which measures the discrepancy between the generated and observed data, conditional on exogenous input variables. Our method does not require parametric assumptions on univariate distributions or multivariate dependencies and allows for incorporating arbitrary predictors. In two case studies on multivariate temperature and wind speed forecasting at weather stations over Germany, our generative model shows significant improvements over state-of-the-art methods and particularly improves the representation of spatial dependencies. A preprint is available at https://arxiv.org/abs/2211.01345.

I will explain some new results about the problem of counting pointed curves of fixed complex structure in blow-ups of projective space at general points. The geometric and virtual Gromov-Witten counts in genus 0 and in higher genus for large degree agree in the Fano (and some (-K)-nef) examples, but not in general. For toric blow-ups, geometric counts can be expressed in terms of integrals on products of Jacobians and symmetric products of the domain curves, and evaluated explicitly in genus 0 and in the case of Bl_q(P^r). Formulas for the the virtual counts in the case of the blow-up of P^r at one point can also be computed via the quantum cohomology ring.