Mathematics Department - Colloquium - Fall 2016

# Colloquium - Fall 2016

## Friday, November 4th

### "Symplectic topology away from the large volume limit "

Time: 4:00 PM
Location: Hill 705
Abstract: Symplectic topology is unique within geometry, in that the deeper structure of the spaces under consideration appears only after non-local "instanton corrections" have been taken into account. This is most readily apparent from a string theory motivation, but it also has a direct impact on classical problems from Hamiltonian mechanics. In the theory, the instanton corrections are set up as small perturbations, which corresponds to thinking of the target space as having infinitely large size (the "large volume limit"). Mirror symmetry suggests that it would be interesting to keep the size finite. Attempting to do that has seemingly paradoxical consequences, which one can sometimes get a handle on by changing the space involved. The talk will give an introduction to this problem, based on simple examples, and explain a little of what is known or expected.

## Friday, November 11th

### "An overview of recent progress on the formation of shock singularities "

Time: 4:00 PM
Location: Hill 705
Abstract: A fundamental issue permeating the study of quasilinear hyperbolic PDEs is that, aside from equations with special structure, initially smooth solutions are expected to often form shocks in finite time. Roughly, a shock is a singularity such that the solution remains bounded but its derivatives blow up. Although many such results have been proved in one spatial dimension, there are very few rigorous results in higher dimensions. In this colloquium, I will provide an overview of recent progress on the formation of shocks in two and three spatial dimensions. I will start by describing prior contributions from many researchers including B. Riemann, P. Lax, F. John, S. Alinhac, and especially D. Christodoulou, whose remarkable 2007 monograph yielded a sharp description of shock formation in vorticity-free small-data solutions to the relativistic Euler equations in three spatial dimensions. I will then describe some of my recent work, some of it joint with J. Luk, G. Holzegel, S. Klainerman, and W. Wong, in which we extended Christodoulou's framework to prove similar results for general classes of equations and new types of initial conditions. I will especially focus on my work with J. Luk on the compressible Euler equations, in which we obtained the first constructive result on the long-time behavior of the vorticity up to the first singularity: for an open set of initial conditions, generic first derivatives of the velocity blow up but the vorticity remains bounded! The proof relies on a new formulation of the equations exhibiting surprisingly good structures, reminiscent of the type found in equations that admit global solutions. Remarkably, the good structures are a key ingredient in proving that a singularity forms. Throughout the talk, I will highlight some of the main ideas behind the analysis including the critical role played by geometric decompositions adapted to characteristic hypersurfaces.

## Friday, December 2nd

### "TBA"

Time: 4:00 PM
Location: Hill 705

## Friday, December 9th

### "TBA"

Time: 4:00 PM
Location: Hill 705

## Friday, October 21st

Special Colloquium

### " Quantitative transversality in symplectic geometry"

Time: 2:00 PM
Location: Hill 705
Abstract: I will survey some applications of Donaldson's technique of quantitative transversality of "approximately holomorphic" functions in symplectic geometry. I will explain the basic terms and present the main ideas of the technique. Donaldson used it to show that the Poincare dual of any sufficiently large multiple of an integral symplectic form is represented by a symplectic submanifold.

Another application is joint work with E. Giroux in which we prove the existence of Lefschetz fibrations on certain symplectic manifolds.

## Friday, October 14th

### "Singularity formation in black hole interiors "

Time: 4:00 PM
Location: Hill 705
Abstract: The prediction that solutions of the Einstein equations in the interior of black holes must always terminate at a singularity was originally conceived by Penrose in 1969, under the name of "strong cosmic censorship hypothesis." The nature of this break-down (i.e. the asymptotic properties of the space-time metric as one approaches the terminal singularity) is not predicted, and remains a very hotly debated question to this day. One key question is the causal nature of the singularity (space-like, vs null for example). Another is the rate of blow-up of natural physical/geometric quantities at the singularity. Mutually contradicting predictions abound in this topic. Much work has been done under the assumption of spherical symmetry (for various matter models). We present recent developments (due to the speaker and G. Fournodavlos) which go well beyond this restrictive class. A key role is played by the axial symmetry reduction of the Einstein equations, where a wave map structure appears.

## Friday, October 7th

### "Heat rises: 100 Years of Rayleigh-Benard convection"

Time: 4:00 PM
Location: Hill 705
Abstract: Buoyancy forces result from density variations, often due to temperature variations, in the presence of gravity. Buoyancy-driven fluid flows shape the weather, ocean dynamics and climate, and the structure of the earth and stars. In 1916 Lord Rayleigh published a paper entitled "On Convection Currents in a Horizontal Layer of Fluid, when the Higher Temperature is on the Under Side" that introduced a minimal mathematical model of buoyancy-driven fluid flows now known as "Rayleigh-Benard convection" that has served for a century as one of the primary paradigms for nonlinear science, dynamical pattern formation, chaos and turbulence. In this presentation, following an introduction to and history of Rayleigh's model and review of some applications of convection, we describe recent progress and open challenges for mathematical analysis in the strongly nonlinear regime of turbulent convection.

## Friday, September 30th

### "On the geometric semantics of algebraic quantum mechanics "

Time: 4:00 PM
Location: Hill 705
Abstract: We approach the formalism of quantum mechanics from the logician point of view and treat the canonical commutation relations and the conventional calculus based on it as an algebraic syntax of quantum mechanics. We then aim to establish a geometric semantics of this syntax. This leads us to a geometric model, the space of states with the action of time evolution operators, which is a limit of finite models. The finitary nature of the space allows us to give a precise meaning and calculate various classical quantum mechanical quantities.

This talk is based on my paper "The semantics of the canonical commutation relation" arxiv.org/abs/1604.07745

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