Schedule
Wednesday February 3, 8pm-9pm: Colloquium talk
Prof. Martin Hairer
Title: Taming infinities
Abstract: Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! What's worse, this doesn't just happen for some exotic theories, but in the standard theories describing some of the most fundamental aspects of nature. Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will dip our toes into some of the conceptual and mathematical aspects of these techniques and we will see how they have recently been used in probability theory to study equations whose meaning was not even clear until recently.
Thursday February 4, 8pm-9pm: Survey of research directions
Prof. Martin Hairer
Title: Open problems and conjectures in SPDE theory
Abstract: We will survey a number of open problems and conjectures both within SPDE theory and linking SPDE theory to other areas of mathematics.
Tuesday February 9, 1.30-2:30pm: Students activity
A set of Honours student level video lectures on regularity structures, designed by Zihan Zhang (Honours student at ANU, supervised by P. Portal), will be made available to student participants. They will be complemented by discussions over Zoom, led by Zihan and Pierre. This activity will prepare students so that they can fully benefit from Professor Hairer's mini course.
Wednesday February 10, 11-12pm: Talks by Australian mathematicians
A/Prof. Pierre Portal
Title: Paradifferential calculus and (S)PDE
Abstract: We will present an approach to singular (S)PDE given in a celebrated paper of Gubinelli, Imkeller, and Perkowski entitled ``Paracontrolled distributions and singular PDEs" (Forum Math. Pi 3 2015). This approach can be seen as a harmonic analytic cousin of Martin Hairer's regularity structures.
Thursday February 11, 11-12pm: Talks by Australian mathematicians
Dr. Xi Geng
Title: Rough Path Theory and Its Applications in Stochastic Analysis
Abstract: The theory of rough paths, which was originally developed by T. Lyons in 1998, provides an analytical approach to study differential equations driven by irregular paths that goes beyond Ito's classical framework of stochastic calculus. To some extent, it is a one-dimensional version of Martin Hairer's theory of regularity structures and provides some important insights towards the development of this more general theory. In this talk, we outline the essential ideas of rough path theory and survey its applications in the study of stochastic differential equations driven by Gaussian rough paths. Some of the fundamental works, such as ergodicity and smoothness of density, were due to Martin Hairer and his collaborators.
Friday February 12, 1:30-2:30pm: Talks by Australian mathematicians
Prof. Jan De Gier
Title: Computation of KPZ distribution functions in lattice models based on quantum groups and Hecke algebras
Abstract: I will discuss how KPZ distribution functions such as the Tracy-Widom distribution from random matrix theory arise in integrable stochastic processes such as the asymmetric exclusion process. I will illustrate how transition probabilities can be computed exactly in stochastic Markov chains that have an underlying Hecke algebra or quantum group, and how these expression can be analysed asymptotically in a long time limit. If time permits I will discuss a relationship to Macdonald polynomials and the construction of stochastic dualities.
Wednesday February 17, 8pm-9pm: Mini-course Part I
Prof. Martin Hairer
Title: Introduction to regularity structures I
Thursday February 18, 8pm-9pm: Mini-course Part II
Prof. Martin Hairer
Title: Introduction to regularity structures II