Motivated by constructions appearing in mirror symmetry, we consider the problem of finding canonical representatives for a complexified Kähler class on a compact complex manifold. These are complex cohomology classes whose imaginary part is a Kähler class, while the real part is an arbitrary real (1,1)-class. As is often the case in complex geometry, one way to fix a representative of such a class is to impose an elliptic PDE. In this talk, I will explain why a natural choice of PDE is a coupling of the deformed Hermitian Yang-Mills equation and the constant scalar curvature equation. We will then see how to prove the existence of solutions in some special cases and talk about some obstructions to the existence of solutions. Based on arXiv:2209.14157, joint work with Jacopo Stoppa.

Positive Legendrian torus knots in the standard contact 3-sphere were classified by J. Etnyre and K. Honda around 20 years ago via Giroux' Convex Surface Theory. Nowadays, we have tools to deal with convex surfaces “parametrically” so we can attempt to determine the homotopy type of some path-connected component of the space of Legendrian embeddings. In this talk I will explain how to do this for positive Legendrian (p,q)-torus knots with max tb number. This is joint work (in progress) with Hyunki Min.

Fourier restriction inequalities enable the restriction of the Fourier transform to suitable sets of null Lebesgue measure. Fourier restriction to submanifolds is closely related to problems in PDE and in this talk we mainly focus on restrictions to fractals. The class of fractals is extensive and many different Fourier restriction properties can be observed, but necessary conditions are not well understood. In this direction, we show that all Fourier restriction sets avoid a universal set of full Hausdorff dimension.

In this talk I will present the theory of tamed spaces, which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature seen from an Eulerian point of view. The approach is based on the analysis of singular perturbations of Dirichlet forms by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality generalizing the well-known Bakry-Emery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup as well as consequences in terms of functional inequalities.

In the language of $L^\infty$ modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space $(M, m)$ arrying a quasi-regular, strongly local Dirichlet form $E$. Furthermore, we develop a second order calculus if $(M, E, m)$ is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, and Ricci curvature.

For many Hamiltonian PDEs, the long-time evolution can be characterised by the corresponding finite-dimensional dynamical systems. In this talk, we present how this mechanism works through reducibility and almost reducibility in quantum harmonic oscillators.

One of the central questions of random algebraic geometry is to describe the expected behaviour of randomly chosen algebraic varieties. In this talk I wish to explain two results concerning plane algebraic curves randomly chosen with respect to Kostlan distribution. The first of these results is about the expected depth of nested ovals of a random real algebraic plane curve. The second is about the expected area of the amoeba of a random complex algebraic plane curve. Both results are based on joint work with Turgay Bayraktar.

In the 1960s John and Nirenberg introduced the space of bounded mean oscillation functions $BMO$ in connection with differential equations. Since that time, and because of the diverse and direct relationship with other relevant objects in Harmonic Analysis, such as duality of Hardy spaces, upper endpoint estimates of Calderón-Zygmund operators, and the $L^p$ estimates of Commutators of those operators, $BMO$ spaces have been objective of much study. In this talk, we will discuss necessary and sufficient (geometric) conditions in a Banach function space $X$ in such a way that $BMO$ and $BMO_{X}$ are equivalent spaces. The new results that we will present in this talk are based on joint works with E. Lorist and A. Lerner.

I will present a geometric approach to the theory of integrability by hydrodynamic reductions to establish an equivalence, for a large class of quasilinear systems, between hydrodynamic integrability and the existence of nets compatible with the geometry induced on the codomain of the system. This unifies and extends known results for three subclasses of such systems. The generalization is obtained by studying the algebraic geometry of the characteristic correspondence of the system, and by introducing a generalized notion of conjugate nets.

The talk will deal with the key challenge of creating prediction sets in the functional data framework. Starting from the investigation of the literature concerning this topic, we propose an innovative approach building on top of Conformal Prediction able to overcome the main drawbacks characterizing the existing approaches. We will show how the new proposed nonparametric method is able to construct finite-sample either valid or exact prediction bands under minimal distributional assumptions. Different specifications of the method will be compared in terms of efficiency in some simulated and real case scenarios.