Monday, 25 January 2021, 3-4pm Zoom

• Speaker: Simon Donaldson (Imperial and Stony Brook)
  
  Title: Fibred $G_{2}$-manifolds and deformations of singular sets.
  
  Abstract: In the first part of the talk I will review some general background material about the exceptional Lie group $G_{2}$, and its role as a holonomy group in 7-dimensional Riemannian geometry. Starting with work of Joyce in the 1990's, there are now many thousands of examples of deformation classes of manifolds with holonomy $G_[{2}$ and there are good reasons for thinking that many of these have fibrations with general fibre diffeomorphic to a particular 4-manifold: the K3 manifold. These fibrations are analogues of Lefschetz fibrations in algebraic geometry. In the second part of the talk we will discuss some more specific analysis questions which arise in the study of these fibrations and their "adiabatic limits". These brings up PDE problems, involving the discrimant set of the fibration, which are analogous to free boundary problems.

Monday, 8 March 2021, 3-4pm Zoom

• Speaker: Endre Suli (Oxford)
  
  Title: Navier-Stokes-Fokker-Planck Systems: Modelling, Analysis, Approximation, Computation
  
  Abstract: Statistical physics is a fertile source of high-dimensional partial differential equations. We shall survey recent developments concerning a system of nonlinear partial differential equations, which involves the Navier-Stokes system coupled with a high-dimensional parabolic Fokker-Planck equation describing the motion of polymer molecules in a viscous fluid occupying a bounded spatial domain. The model arises in the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where polymer molecules are idealised as bead-spring chains with finitely or infinitely extensible nonlinear elastic spring potentials, and has been the subject of active research over the past decade. We shall report recent results concerning the existence of large-data global weak solutions to this high-dimensional system. We shall also highlight a number of nontrivial open questions concerning the mathematical analysis, approximation and numerical analysis of high-dimensional Navier-Stokes-Fokker-Planck systems.

Monday, 22 March 2021, 3-4pm Zoom

• Speaker: Giulia Sacca (Columbia)
  
  Title: Compact Hyper-K\"ahlers and Fano Manifolds
  
  Abstract: Projective hyper-K\"ahler (HK) manifolds are among the building blocks of projective manifolds with trivial first Chern class. Fano manifolds are projective manifolds with positive first Chern class. Despite the fact that these two classes of algebraic varieties are very different (HK manifolds have a holomorphic symplectic form which governs all of its geometry, Fano manifolds have no holomorphic forms) their geometries have some strong ties. For example, starting from some special Fano manifolds one can sometimes construct HK manifolds as parameter spaces of objects on the Fano. In this talk I will explain this circle of ideas and focus on some recent work exploring the converse: given a projective HK manifold, how to recover a Fano manifold?

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You'll find the old colloquium schedules here.