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The principles of Relativity and Quantum Mechanics form the pillars of our understanding of Nature, and are extremely constraining when we compute observables in fundamental physics. But over the past few decades, there has been growing evidence that this standard physical picture obscures astonishing simplicity and hidden symmetries seen only at the very end of complicated calculations. This has led some physicists to seek radically different ways of conceptualizing physics, that leads much more directly to the final answer, involving the discovery of interesting new mathematical structures. In this talk I will describe some emerging ideas along these lines, and present a new formulation of some very basic physics — fundamental to particle scattering and to cosmology — not following from quantum evolution in space-time, but arising from new ideas in combinatorics, algebra and geometry.

The geometry, topology and intersection theory of moduli spaces of stable vector bundles on curves have been topics of interest for more than 50 years. In the 90s, Jeffrey and Kirwan managed to prove a formula proposed by Witten for the intersection numbers of tautological classes on such moduli spaces. In this talk, I will explain a different way to calculate those numbers and, more generally, intersection numbers on moduli of parabolic bundles. Enriching the problem with a parabolic structure gives access to powerful tools, such as wall-crossing, Hecke transforms and Weyl symmetry. If time allows, I will explain how this approach gives a new proof of (a generalization to the parabolic setting of) a vanishing result conjectured by Newstead and proven by Earl and Kirwan.

We present some recent results on the asymptotic behavior of almost periodic solutions to stochastic conservation laws and, more generally, degenerate parabolic-hyperbolic equations. Two types if stochastic perturbations are considered: forcing and rough-flux. The part concerning the forcing stochastic source is from joint works with Claudia Espitia and Daniel Marroquin. The part concerning stochastic rough-flux is from a joint project with Rui Jin Yachun Li and João Nariyoshi.

The study of geometric properties of solutions of Partial Differential Equations is of great interest. One of the most celebrated results in this topic is the one obtained by B. Gidas, W. M. Ni and L. Nirenberg in [1], where the authors establish radial symmetry to nonnegative solutions of the problem

\[
\begin{cases}-\Delta u=f(u) & \text { on } B \\ u=0 & \text { on } \partial B\end{cases}
\]

where $B \subset \mathbb{R}^N$ (with $N \geqslant 2$ ) is a ball. Later on, several authors went on to generalize this result in several directions.

In this talk, I present a quantitative version of a Gidas-Ni-Nirenberg-type symmetry result involving the $p$-Laplacian. Quantitative stability is achieved here via integral identities based on the proof of rigidity established by J. Serra in 2013, which extended to general dimension and the $p$-Laplacian operator an argument proposed by P. L. Lions in dimension 2 for the classical Laplacian.

In passing, we obtain a quantitative estimate for the measure of the singular set and an explicit uniform gradient bound.

This work was done in collaboration with S. Dipierro, G. Poggesi, E. Valdinoci.

References:
[1] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209-243.

The Formality Theorem of Deligne-Griffits-Morgan-Sullivan states that the rational homotopy type of a compact Kähler manifold is entirely determined by its cohomology ring. This provides homotopical obstructions for a compact complex manifold to admit a Kähler metric. On the other hand, by results of Deligne and Morgan, the rational homotopy groups of simply connected Kähler manifolds carry natural mixed Hodge structures. However, there exist diffeomorphic compact Kähler manifolds with the same pure Hodge structures on cohomology but with different mixed Hodge structures on rational homotopy groups. To better understand this phenomenon, in this talk I will introduce a "mixed Hodge formality" theory and provide complex geometric invariants obstructing this stronger notion of formality.

Quantum Signal Processing (QSP) is an algorithmic process by which one represents a function $f$ on the unit interval as the upper left entry of a product of $SU(2)$ matrices parametrized by the input variable $x \in [0,1]$ and some “phase factors” $\{\psi_k\}_{k \geq 0}$ depending on $f$. We show that, after a change of variables, QSP is actually the $SU(2)$-valued nonlinear Fourier transform, and the phase factors correspond to the nonlinear Fourier coefficients. By exploiting a nonlinear Plancherel identity and using some basic spectral theory, we extend QSP to represent any function $f$ satisfying a mild $\log$ integrability condition.

In this talk the derivation and dynamics of some 1D models for the propagation of internal waves are reviewed. From the starting point of the corresponding Euler equations and under certain physical hypotheses, Boussinesq-type systems are derived. Then a numerical analysis of the models, based on the approximation with spectral methods and efficient time integrators, is developed. This will be finally used to study, by computational means, some issues of their dynamics, mainly focused on the solitary wave solutions.