Title: Looking at Euler flows through a contact mirror: From universality to Turing completeness

Abstract: The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier-Stokes equations based on the concept of universality. Inspired by this proposal, in this course we show that the stationary Euler equations exhibit several universality features, In the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Sullivan, Etnyre and Ghrist more than two decades ago. Another application of this contact mirror concerns the study of singular periodic orbits (including escape orbits in Celestial mechanics joint with Cédric Oms) that we briefly discuss. We end up this minicourse addressing an apparently different question: What kind of physics might be non-computational? The universality result above yields the Turing completeness of the steady Euler flows on a 17-dimensional sphere, But, can this result be improved? We will give a positive answer in dimension three, discuss some applications and new constructions using billiard-type maps

Plan of the course:

Session 1 (Wednesday 8): First part: A primer on Euler flows, Navier-Stokes and Beltrami Fields by Daniel Peralta (45 minutes). Second part: The contact mirror and the h-principle in Geometry by EvaMiranda (45 minutes). This first session will introduce the audience to the basic definitions and tools (analytic and geometrical) of the course.

Session 2: (Friday 10) Title of the session: Universality and Turing completeness of Euler flow (or when the 29000 rubber ducks were lost in the ocean in 1992). This will be a 1 hour and 30 minutes talk interlacing Daniel and Eva.This second session follows several joint works with Robert Cardona and Fran Presas and a recent joint article with Robert Cardona.

Title: The supersymmetric nonlinear sigma model as a geometric variational problem

Abstract: The supersymmetric nonlinear sigma model is an important model in modern quantum field theory whose action functional is fixed by the invariance under various symmetries. In the physics literature it is usually formulated in terms of supergeometry. However, when abandoning the invariance under supersymmetry transformations, it can also be studied using well-established tools from the geometric calculus of variation. Following this approach one obtains an action functional that involves a map between two manifolds and a vector spinor defined along that map.

In the case of a Riemannian domain its critical points couple the harmonic map equation to spinor fields, these became known as Dirac-harmonic maps and variants thereof in the mathematics literature. If the domain manifold is Lorentzian, the critical points couple the wave map equation to spinor fields.

In this talk we will present various geometric and analytic results on Dirac-harmonic maps and their extensions. Moreover, we will discuss the difficulties that arise when trying to prove existence results and in which cases these can be overcome.

Title: Elementary geometric quantities and gravitational energy

Abstract: Gravity manifests itself as curvature of spacetime. Curvature’s strength can be measured by considering the variations of the basic geometric quantities (area, volume, radius) of small balls with respect to their counterparts in flat spacetime. These variations are directly related, via the Einstein field equations, to the energy density of matter at the ball’s centre. In this talk I consider what happens when the matter energy density vanishes. Elementary geometric measurements still feel the effect of pure gravity, and the resulting changes should still be related to the gravitational strength or, in simple words, to the gravitational energy density. This leads to a novel prescription for the quasi-local energy of the pure gravitational field.

Title: Universal Constraint for Relaxation Rates for Quantum Dynamical Semigroup

Abstract: A general property of relaxation rates in open quantum systems is discussed. We propose a constraint for relaxation rates that universally holds in fairly large classes of quantum dynamics. It is conjectured that this constraint is universal, i.e., it is valid for all quantum dynamical semigroups. The conjecture is supported by numerical analysis. Moreover, we show that the conjectured constraint is tight by providing a concrete model that saturates the bound. This constraint provides, therefore, a physical manifestation of complete positivity. Our conjecture also has two important implications: it provides (i) a universal constraint for the spectra of quantum channels and (ii) a necessary condition to decide whether a given channel is consistent with Markovian evolution.

Title: Geometric quantization and Noncommutative algebraic geometry

Abstract: The quantization of the integer multiples of a symplectic structure via the choice of a Kahler polarization lead us to the homogeneous coordinate rings familiar in algebraic geometry. I will explain how this framework may be deformed, making contact with noncommutative algebraic geometry. The main tool will be the deformation of the Kahler polarization to a generalized Kahler polarization. This is joint work with Francis Bischoff (arXiv:2108.01658).