10:15-10:45

 

 

Title: A few of my favourite harmonic oscillators.

 

Abstract: A famous quote attributed to Sidney Coleman states that "the career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction". In this talk, I will describe properties of a few incarnations of the harmonic oscillator that I have recently studied. The emphasis will be on appreciating the importance of multiple perspectives: analytic, algebraic, probabilistic, and quantum mechanical. I will point out how representation theory helps the analysis, but also where analysis questions distinguish between equivalent representations. I will also show how quantisation arises from a probabilist point of view. Perhaps not coincidently, these perspectives are closely tied to the history of the ANU, starting with the work of Joe Moyal. The talk will be in the style of a colloquium, hiding technical aspects, and focusing on heuristics.

 

10:50-11:20

 

 

Title: Classical conformal mapping from a probabilistic point of view

 

Abstract:

The conformal invariance of Brownian motion up to a time-change, first proved by Paul Lévy, is a powerful result that can be used to reinterpret classical results from complex analysis and conformal mapping. The resulting set-up gives a more explicit construction than classical extremal techniques and lends itself to some interesting variations. In this talk, I will show how the properties of Brownian motion permit concrete, short proofs of such results, starting from the Riemann mapping theorem and the Loewner differential equation. I will then examine the effect of different Brownian motion boundary conditions including orthogonal reflection and excursion-reflected Brownian motion, which was proved by Drenning and Lawler to realise canonical slit domain mappings from multiply connected domains.

 

10:25-11:55

 

 

Title: Optimal curves, their action functionals, and uniformization

Abstract: What is a canonical random curve, or graph, embedded in a two-dimensional space? From Loewner's theory of slit mappings, one can define a natural model of random curves, often termed Schramm-Loewner evolution. It is, in a sense, a "quantum" model of fractal curves, that depends on a parameter (\kappa) measuring the fractality -- and its semiclassical, aka large deviations limit (\kappa \to 0) gives rise to interesting structures in complex geometry, Teichmüller theory, enumerative geometry, mathematical physics, and beyond. I will focus on the problem of finding a meaning for the "optimal curves" which minimize the action functional (energy) associated to this model.

 

Lunch 12:00-1:00

 

1:00-1:30

 

 

Title: The Segal-Neretin semigroup of annuli

 

Abstract: The semigroup of annuli consists of Riemann surfaces homeomorphic to annuli, with the operation of gluing (‘conformal welding’). This semigroup plays a crucial role in the mathematics of two-dimensional conformal field theory. In this talk I will introduce the semigroup, discuss a certain compactification of the semigroup recently constructed in joint work with Henriques, and explain the connection to a question in classical complex function theory that has bothered me for the past decade.

 

1:35-2:05

 

 

Title: Quantum Ergodicity: an introduction.

 

2:10-2:40

 

 

Title: What is quantum Teichmuller theory?