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New Perspectives in Complex Analysis

A workshop on New Perspectives in Complex Analysis.

schedule Date & time
Date/time
4 Feb 2025 10:00am - 4 Feb 2025 4:00pm
contact_support Contact
Ian Le (Australian National University)
Senior Lecturer
Tony Martin (Australian National University)
Administrative Coordinator (Marketing, Conferences and Events)

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Description

This workshop surveys recent tools and developments in complex analysis, including harmonic functions, the heat equation, Brownian motion, Schramm-Loewner Evolution (SLE), conformal field theory and quantization.

Speakers

Six 30-minute talks from:

Laurence Field (Australian National University)

Andrew Hassell (Australian National University)

Ian Le (Australian National University)

Eveliina Peltola (Universitat Bonn)

Pierre Portal (Australian National University)

James Tener (Australian National University)

Lunch

A light lunch will be provided, please register your attendance.

Program

10:15-10:45
 
Pierre Portal
 
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
 
Laurence Field
 
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
 
Eveliina Peltola
 
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
 
James Tener
 
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
 
Andrew Hassell
 
Title: Quantum Ergodicity: an introduction.
 
2:10-2:40
 
Ian Le
 
Title: What is quantum Teichmuller theory?

General information for visitors

Information

Location

Mathematical Sciences Institute

ANU College of Science 

Seminar Rooms 1.33 & 1.37

Hanna Neumann Building #145, Science Road

The Australian National University

Canberra ACT 2600

-35.275389387895, 149.11926090717

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