I will discuss several results, mostly obtained in collaborations with several authors, dealing with the optimal design problems, both in the framework of linear and nonlinear elasticity. Among the several models presented I will focus also on the case in which the hyperelastic energy density has nonstandard, possibly variable, growth. The case of thin structures will also be considered.

In this talk we aim at establishing large deviation estimates for the probability that a simple random walk on the Euclidean lattice (d>2) covers a substantial fraction of a macroscopic body. It turns out that, when such rare event happens, the random walk is locally well approximated by random interlacements with a specific intensity, which can be used as a pivotal tool to obtain precise exponential rates. Random interlacements have been introduced by Sznitman in 2007 in order to describe the local picture left by the trace of a random walk on a large discrete torus when it runs up to times proportional to the volume of the torus, and has been since a popular object of study. In the first part of the talk we introduce random interlacements and give a brief account of some results surrounding this object. In the second part of the talk we study the event that random interlacements cover a substantial fraction of a macroscopic body. This allows to obtain an upper bound on the probability of the corresponding event for the random walk. Finally, by constructing a near-optimal strategy for the random walk to cover a macroscopic body, we discuss a matching large deviation lower bound. The talk is based on ongoing work with M. Nitzschner (NYU Courant).

Smoluchowski’s coagulation equation, an integro-differential equation of kinetic type, is a classical mean-field model for mass aggregation phenomena. The solutions of the equation exhibit rich behavior depending on the rate of coagulation considered, such as gelation (formation of particles with infinite mass in finite time) or self-similarity (preservation of the shape over time). In this talk I will first discuss some fundamental properties of the Smoluchowski equation. I will then present some recent results on the problem of existence or non-existence of stationary solutions, both for single and multi-component systems, under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. The most striking feature of these stationary solutions is that, whenever they exist, the solutions to multi-component systems exhibit an unusual “spontaneous localization” phenomenon: they localize along a line in the composition space as the total size of the particles increase. This localization is a universal property of multicomponent systems and it has also been recently proved to occur in time dependent solutions to mass conserving coagulation equations. (Based on joint works with M.Ferreira, J.Lukkarinen and J. Velázquez)

Consider $\mathbb R^d\times \mathbb R^m$ with the group structure of a $2$-step Carnot Lie group and natural parabolic dilations. The maximal operator originally introduced by Nevo and Thangavelu in the setting of the Heisenberg groups is generated by (noncommutative) convolution associated with measures on spheres or generalized spheres in $\mathbb R^d$. We discuss a number of approaches that have been taken to prove $L^p$ boundedness and then talk about recent work with Jaehyeon Ryu in which we drop the nondegeneracy condition in the known results on Métivier groups. The new results have the sharp $L^p$ boundedness range for all two step Carnot groups with $d\ge 3$.

We study open symplectic manifolds with pseudoholomorphic $\mathbb C^*$-actions whose $S^1$-part is Hamiltonian, and construct their associated symplectic cohomology. From this construction, we obtain a filtration on quantum/ordinary cohomology that depends on the choice of the $\mathbb C^*$-action. One should think about this filtration as a Floer-theoretic analogue of the Atiyah-Bott filtration. We construct filtration functional on the Floer chain complex, allowing us to compute the aforementioned filtration via Morse-Bott spectral sequence that converges to symplectic cohomology, which is readily computable in examples. We compare our filtration with known ones from algebraic geometry/representation theory literature. Time-allowing, I may present the $S^1$-equivariant picture as well. This is joint work with Alexander Ritter.

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.