Seminars from until

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.

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.