Seminars from until

Following A. Beauville, a complex algebraic variety $X$ is said to be symplectic if it admits a holomorphic symplectic form $\omega$ on its smooth locus such that, for every resolution $\pi: Y \to X$, $\pi^*\omega$ extends to a holomorphic $2$-form on $Y$. When this extension is actually non-degenerate (a de facto symplectic form) on $Y$, we call $\pi$ a symplectic (or crepant) resolution.

Let $G$ be a complex reductive group and $A$ an abelian variety of dimension $d$. The aim of this talk is to show that all moduli spaces of $G$-Higgs bundles over $A$ are symplectic varieties, and that, for $G=\mathrm{GL}(n,\mathbb C)$, the canonical Hilbert-Chow morphism is a symplectic resolution if and only if $d=1$.

Moreover, using a little representation theory, we can obtain explicit expressions for the Poincaré polynomials of all Hilbert-Chow resolutions (either $d=1$, all $n$; or $n=1,2,3$ and all $d$). This is joint work with I. Biswas and A. Nozad.

Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x". Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique is useful, including an application to the Simple Exclusion Process with reservoirs.

The question whether a symplectic manifold embeds into another is central in symplectic topology. Since Gromov nonsqueezing theorem, it is known that this is a different problem from volume preserving embedding. There are several nice results about symplectic embeddings between open subsets of $\mathbb R^{2n}$ showing that even for those examples the question can be completely nontrivial. The problem is substantially more well understood when the manifolds are toric domains and have dimension $4$, mostly because of obstructions coming from embedded contact homology (ECH). In this talk we are going to discuss symplectic embedding problems in which the target manifold is the disk cotangent bundle of a two-dimensional sphere, i.e., the set consisting of the covectors with norm less than $1$ over a Riemannian sphere. We shall talk about some tools such as ECH capacities and action angle coordinates. Much of this talk is based on joint works with Vinicius Ramos and Alejandro Vicente.

In this talk, I will present a nonasymptotic process level control between the so-called telegraph process (a.k.a. Goldstein–Kac equation) and a diffusion process with suitable (explicit) diffusivity constant via a transportation Wasserstein path-distance with quadratic average cost.

We stress that the telegraph process solves a partial linear differential equation of the hyperbolic type for which explicit computations can be carried by in terms of Bessel functions. In the present talk, I will discuss a coupling approach, which is a robust technique that in principle can be used for more general PDEs. The proof is done via the interplay of the following couplings: coin-flip coupling, synchronous coupling and the celebrated Komlós–Major–Tusnády coupling. In addition, nonasymptotic estimates for the corresponding $L^p$ time average are given explicitly.

The talk is based on joint work with Jani Lukkarinen, University of Helsinki, Finland.

We discuss a general principle of perturbing higher order operators with lower order derivatives in order to restore the maximum principle in the framework in which it is well known to fail. This is somehow delicate and the main ingredient is a new Harnack-type inequality. We first prove De Giorgi type level estimates for functions in $W^{1,t}$, with $t>2$. This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi's classes as obtained by Di Benedetto-Trudinger for functions in $W^{1,2}$. As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains and in any dimension, provided either lower order derivatives or inertial effects are taken into account.

Studying metrics with special curvature properties on compact Kähler manifolds is a fundamental problem in Kähler geometry. In this talk, I will focus on the existence and uniqueness of singular Kähler-Einstein metrics whose singular behavior is prescribed. These results are based on a series of joint works with T. Darvas and C. Lu.

A new generation of large aperture and large field of view telescopes is allowing the exploration of large volumes of the Universe in an unprecedented fashion. In order to take advantage of these new telescopes, notably the Vera C. Rubin Observatory, a new time domain ecosystem is developing. Among the tools required are fast machine learning aided discovery and classification algorithms, interoperable tools to allow for an effective communication with the community and follow-up telescopes, and new models and tools to extract the most physical knowledge from these observations. In this talk I will review the challenges and progress of building one of these systems: the Automatic Learning for the Rapid Classification of Events (ALeRCE) astronomical alert broker. ALeRCE is an alert annotation and classification system led by an interdisciplinary and interinstitutional group of scientists from Chile since 2019. ALeRCE is focused around three scientific cases: transients, variable stars and active galactic nuclei. Thanks to its state-of-the-art machine learning models, ALeRCE has become the 3rd group to report most transient candidates to the Transient Name Server, and it is enabling new science with different astrophysical objects, e.g. AGN science. I will discuss some of the challenges associated with the problem of alert classification, including the ingestion of multiple alert streams, annotation, database management, training set building, feature computation and distributed processing, machine learning classification and visualization, or the challenges of working in large interdisciplinary teams. I will also show some results based on the real‐time ingestion and classification using the Zwicky Transient Facility (ZTF) alert stream as input, as well as some of the tools available.