Seminars from until

The symplectic version of the problem of packing K balls into a ball in the densest way possible (in 4 dimensions) can be extended to that of symplectically embedding an ellipsoid into a ball as small as possible. A classic result due to McDuff and Schlenk asserts that the function that encodes this problem has a remarkable structure: its graph has infinitely many corners, determined by Fibonacci numbers, that fit together to form an infinite staircase.

This ellipsoid embedding function can be equally defined for other targets, and this talk will be about other targets for which the function has and does not have an infinite staircase. Firstly we will see how in the case when these targets have lattice moment polygons, the targets with infinite staircases seem to be exactly those whose polygon is reflexive (i.e., has one interior lattice point). Secondly, we will look at the family of one-point blowups of $CP^2$, where the answer involves self-similar behaviour akin to the Cantor set.

This talk is based on various projects, joint with Dan Cristofaro-Gardiner, Tara Holm, Alessia Mandini, Maria Bertozzi, Tara Holm, Emily Maw, Dusa McDuff, Grace Mwakyoma, Morgan Weiler, and Nicki Magill.

In the framework of the functional analysis, this talk aims to illustrate the research we have developed during the last years: from the study of geometric and algebraic properties of triple structures to a one-to-one correspondence with TKK Lie algebras. We shall highlight the last novelties achieved in the Lie setting.

The classical contraction principle is one of those basic results in Analysis with many fundamental applications. In this talk, we will examine a variational interpretation of it which turns out to be more flexible. In particular, it can be used to deal with situations where existence and uniqueness of solutions is known or expected. Though many classical situations can be treated, due to time restrictions we will focus on two representative examples: that of initial-value Cauchy problems for autonomous ODE systems, and the case of non-linear, non-variational monotone PDE equations in divergence form. It remains to be seen if this perspective could help in new situations.

The tone of the talk will be elementary. No specialized background is required.

In quantum physics, and more specifically in quantum optics, several notions of ``classical’’ and hence ``nonclassical’’ state are in use. They rely on the positivity of quasi-probability distributions, specifically the Glauber-Sudarshan or Wigner functions of the state. Characterizing the classical states is in general a difficult task, involving interesting questions of functional and harmonic analyss. In this talk, we will, after reviewing the subject, report on some recent progress.

In this talk we explore the application of central extensions of Lie groups and Lie algebras to derive the viscous quasi-geostrophic (QGS) equations, with and without Rayleigh friction term, on the torus as critical points of a stochastic Lagrangian. We begin by introducing central extensions and proving the integrability of the Roger Lie algebra cocycle $\omega_\alpha$, which is used to model the QGS on the torus. Incorporating stochastic perturbations, we formulate two specific semi-martingales on the central extension and study the stochastic Euler-Poincaré reduction. Specifically, we add stochastic perturbations to the $\mathfrak{g}$ part of the extended Lie algebra $\widehat{\mathfrak{g}} = \mathfrak{g} \rtimes_{\omega_\alpha} \mathbb{R}$ and prove that the resulting critical points of the stochastic right-invariant Lagrangian solve the viscous QGS equation, with and without Rayleigh friction term.

When M is a Fano variety and D is an anticanonical divisor in M, mirror symmetry suggests that the quantum cohomology of M should be a deformation of the symplectic cohomology of M \ D. We prove that this holds under even weaker hypotheses on D (although not in general), and explain the consequences for mirror symmetry. We also explain how our methods give rise to interesting symplectic rigidity results for subsets of M. Along the way we hope to give a brief introduction to Varolgunes’ relative symplectic cohomology, which is the key technical tool used to prove our symplectic rigidity results, but which is of far broader significance in symplectic topology and mirror symmetry as it makes the computation of quantum cohomology “local”. This is joint work with Strom Borman, Mohamed El Alami, and Umut Varolgunes.