Topics in Mathematics 2023/2024

The talk is intended to provide a pedagogical introduction to optimal control theory in continuous time and to its connections to PDEs. I will present the main ideas of the Dynamic Programming approach for a family of optimal control problems, both in the deterministic and in the stochastic framework. Some applications in economic growth theory and epidemiological models will be illustrated.

An elementary argument (for sure well-known to the operator theory community) allows to compare the orthogonal projection of a Hilbert space H onto a given closed subspace of H, with (any) bounded non-orthogonal projection acting among the same spaces: this yields an operator identity that is valid in the Hilbert space H. This paradigm has deep implications in analysis, at least in two settings: -in the specific context where the Hilbert space consists of the square-integrable functions along the boundary of a rectifiable domain D in Euclidean space, taken with with respect to, say, induced Lebesgue measure ds (the Lebesgue space L^2(bD)), and the closed subspace is the holomorphic Hardy space H^2(D). In this context the orthogonal projection is the Szego projection, and the non-orthogonal projection is the Cauchy transform (for planar D), or a so-called Cauchy-Fantappie’ transform (for D in C^n with n¥geq 2). -in the specific context where the Hilbert space is the space of square-integrable functions on a domain D in Euclidean space taken with respect to Lebesgue measure dV, and the closed subspace is the Bergman space of functions holomorphic in D that are square-integrable on D. Here the orthogonal projection is the Bergman projection, and the non-orthogonal projection is some ``solid’’ analog of the Cauchy (or Cauchy-Fantappie’) transform. A prototypical problem in both of these settings is the so-called ``L^p-regularity problem’’ for the orthogonal projection where p¥neq 2. This is because the Szego and Bergman projections, which are trivially bounded in L^2 (by orthogonality), are also meaningful in L^p, p¥neq 2 but proving their regularity in L^p is in general a very difficult problem which is of great interest in the theory of singular integral operators (harmonic analysis). Three threads emerge from all this: (1) a link between the (geometric and/or analytic) regularity of the ambient domain and the regularity properties of these projection operators. (2) applications to the numerical solution of a number of boundary value problems on a planar domain D that model phenomena in fluid dynamics. For a few of these problems there can be no representation formula for the solution: numerical methods are all there is. (3) the effect of dimension: for planar D the projection operators are essentially two and can be studied either directly or indirectly via conformal mapping (allowing for a great variety of treatable domains); as is well known, in higher dimensional Euclidean space there is no Riemann mapping theorem: conformal mapping is no longer a useful tool. On the other hand the basic identity in L^2 (see above) is still meaningful but geometric obstructions arise (the notion of pseudoconvexity) that must be reckoned with.

In any context of life, human beings aim to achieve the best possible result with minimal effort. In this talk, we discuss how to implement this general principle to the numerical approximation of partial differential equations, where the aim is to obtain accurate approximations at low computational costs. Using the approximation of the Poisson equation by standard finite element methods as a prototypical example, we show how adaptive algorithms based on rigorous a posteriori error estimation lead to approximations that are, in a certain sense, optimal.

As generative AI technologies are revolutionizing industries and our daily lives, what is going to happen to the role of the mathematician? In this talk, I will highlight recent breakthroughs in deep learning and AI and explore how current and future advancements might alter the way we do mathematics.

The theory of machine learning has been stimulated, in recent years, by a series of empirical observations that challenged the standard knowledge inherited by the classical statistical theory. In the first part of the talk, I will review some results on simple mean-field models, that allowed statisticians and physicists to understand some of these unexpected behaviors, e.g., the double-descent phenomenon or the effectiveness of ensembling. The models rely on some simplifying assumptions, one of them related to some kind of "Gaussianity of the dataset". In the second part of the talk, I will present therefore two recent works in which we characterized regression and classification tasks on fat-tailed datasets. We showed how Gaussian universality can break down and how non-Gaussianity can affect the generalization performances, for example, the generalization rates, the existence of an MLE in a classification task, and the robustness of a Huber estimator.

Diophantine approximation in number theory is to approximate a given irrational number with rational numbers. From a geometric perspective, it quantifies the rate at which a given geodesic flow approaches the cusp on the fundamental domain of the modular group within hyperbolic space. In this talk, we will discuss various Diophantine approximations on the real line and also on the complex plane. In the context of the complex plane, we approximate a complex number using elements from a specified imaginary quadratic field. Lastly, we will consider Diophantine approximation on circles and spheres and study intriguing examples.