We are delighted to welcome Nalini Joshi, Payne-Scott Professor of Mathematics & the Chair of Applied Mathematics at the University of Sydney, who will deliver the next Distinguished Lecture Series (DLS) on 3 – 5 March 2025.
Nalini will be hosting three talks during her visit, including a colloquium-style lecture on Monday 3rd March which will be followed by a drinks reception. Talk titles and abstracts can be found below.
Registration is necessary, so please ensure to complete the registration form to secure your place.
Lecture 1: Monday 3 March 2025 at 16:15 [Colloquium] – followed by drinks reception.
Venue: Seminar Room 2.04, School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG
Dynamics on and off elliptic curves (I)
Felix Klein said that the study of new transcendental functions defined by differential equations was “the central problem of the whole of modern mathematics”. The beginnings of this study lay in elliptic functions, which were generalised by the Painlevé transcendents. The analytic theory of their governing differential equations has a counterpart in the theory of the algebraic curves. The most famous examples are elliptic curves. What is not widely known is that the evolution of a solution of a Painlevé equation changes the underlying elliptic curve and points move from one such curve to another under its time evolution. I will give an introductory overview of how this connects with their asymptotic behaviours and a simple model to describe how Boutroux (1913) initiated such asymptotic descriptions.
Lecture 2: Tuesday 4 March 2025 at 15:00
Venue: Seminar Room 2.04, School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG
Dynamics on and off elliptic curves (II)
Starting with the simple model from the last lecture, I will describe behaviours in the limit as the independent variable approaches infinity, which are analogous to those seen in the Painlevé equations. Next, we give an overview of asymptotic results for the first Painlevé equation before describing how we deduced global results from a geometric description of the regularized projective space of initial values. The latter were carried out in collaboration with many co-authors: Duistermaat and Joshi (2011), Howes and Joshi (2014), Joshi and Radnovic (2016-2019), and Heu, Joshi and Radnovic (2023).
Lecture 3: Wednesday 5 March 2025 at 15:00
Venue: Seminar Room 2.04, School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG
On q-difference Painlevé equations and their Riemann-Hilbert problems
A widely used method of studying such transcendental functions is through their formulation as Riemann-Hilbert problems, i.e., given functions in certain domains and jumps across their common boundaries, the problem of finding a global function that agrees with the given information. The formulation of a Riemann problem for difference equations was initiated by Birkhoff in 1913. In this talk, I will outline some recent results for q-Riemann-Hilbert problems and their ramifications for special functions that solve q-difference Painlevé equations.
About the Speaker: Nalini Joshi received her PhD from Princeton University with Martin Kruskal as her advisor. Her research focuses on integrable systems, including the Painlevé equations, lattices and geometric asymptotics. Nalini was elected to the Australian Academy of Science as a Fellow in 2008, awarded a Georgina Sweet Australian Laureate Fellowship in 2012 and appointed Officer of the Order of Australia in 2016. She was President of the Australian Mathematical Society, Vice-President of the International Mathematical Union and is currently a member of its Executive Committee.
In the 2016 Queen’s Birthday honours, Nalini was appointed an Officer of the Order of Australia for distinguished service to mathematical science and tertiary education as an academic, author and researcher, to professional societies, and as a role model and mentor of young mathematicians.
The Heilbronn Annual Conference is the Institute’s flagship event. It takes place over two days and it covers a broad range of mathematics, including algebra, combinatorics, data science, geometry, number theory, probability, quantum information. It brings together members of the Institute, distinguished visiting speakers, and other members of the UK mathematical community. This year we welcome eight distinguished speakers, to deliver lectures intended to be accessible to a general audience of mathematicians.
