Mathematics Department - Colloquium - Fall 2016

# Colloquium - Fall 2016

## 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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