The Distinguished Lecture Series

Together with the Pascal and Collingwood Lecture Series, the Distinguished Lecture Series stands among the Department’s most esteemed academic traditions. Each year, we welcome one to two leading mathematicians to deliver a lecture designed to engage a broad and diverse audience. This series has brought internationally recognised and prominent scholars to the Department, whose contributions have enriched our academic community and inspired wide interest in contemporary mathematics. Previous Distinguished Lectures include:

Optimal transport methods in risk management

Professor Nizar Touzi, New York University

Abstract: We review several optimal transport problems motivated by risk management in financial engineering and optimal incentive theory in economics, with interesting connections to the Skorohod Embedding Problem in probability, mean field games in optimal control theory and generative methods in artificial intelligence. We finally discuss important applications to model risk measurement and management.

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Hilbert’s 19th problem and discrete De Giorgi–Nash–Moser theory: analysis and applications

Professor Endre Süli, University of Oxford

Abstract: Mathematical models of non-Newtonian fluids play an important role in science and engineering, and their analysis has been an active field of research over the past decade. This talk is concerned with the mathematical analysis of numerical methods for the approximate solution of systems of nonlinear elliptic partial differential equations that arise in models of chemically reacting viscous incompressible non-Newtonian fluids, such as the synovial fluid found in the cavities of synovial joints. The synovial fluid consists of an ultra filtrate of blood plasma that contains hyaluronic acid, whose concentration influences the shear-thinning property and helps to maintain a high viscosity; its function is to reduce friction during movement. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power law-exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove the convergence of the sequence of numerical approximations to a solution of this coupled system of nonlinear partial differential equations, a uniform Hölder norm bound needs to be derived for the sequence of numerical approximations to the concentration in a setting, where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely an $L^{\infty}$ function. This necessitates the development of a discrete counterpart of the De Giorgi–Nash–Moser theory. Motivated by an early paper by Aguilera and Caffarelli (1986) in the simpler setting of Laplace’s equation, we derive such uniform Hölder norm bounds on the sequence of continuous piecewise linear finite element approximations to the concentration. We then use these to deduce the convergence of the sequence of approximations to a weak solution of the coupled system of nonlinear partial differential equations under consideration.

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Equivalence relations, topology and C^{*}-algebras

Professor Lisa Orloff Clark, Victoria University of Wellington

Abstract: C^{*}-algebras provide the mathematical underpinnings of quantum mechanics. They have a rich theory that has been developed over the last century. In 1943, Gelfand and Naimark showed that every C^{*}-algebra is isomorphic to a subalgebra of operators on a Hilbert space. So, in the finite dimensional setting, the study of C^{*}-algebras is essentially the study of matrices. Yet even with this powerful theorem, an abstract C^{*}-algebra can be a complicated beast and understanding basic properties can be difficult. In this talk we describe how to build a C^{*}-algebra from an equivalence relation so that one can see properties of the algebra by looking at properties of the equivalence relation. To get a robust class of C^{*}-algebras in this way, we will consider equivalence relations on topological spaces. This talk will include several examples and assumes no C^{*}-algebraic background.

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Progress and Challenges in the Theory of Hyperbolic Conservation Laws

Professor Constantine Dafermos, Brown University

Abstract: This lecture will provide a survey of the state of the art in the theory of hyperbolic conservation laws emphasizing both recent achievements and future challenges.

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