Friday, 5 November 2021 at 09:50am
Title: Non-compact Einstein manifolds with symmetry
Friday, 29 October 2021 at 12:10pm on Zoom, link by e-mail,
and in Eng Math EMG07 (note the room change!)
Abstract: In this talk we will discuss recent joint work in collaboration with Christoph Böhm in which we obtain structure results for non-compact Einstein manifolds admitting a cocompact isometric action of a connected Lie group. As an application, we prove the Alekseevskii conjecture (1975): any connected homogeneous Einstein space of negative scalar curvature is diffeomorphic to a Euclidean space.
Title: Conformal group actions on Cahen-Wallach spacesFriday, 22 October 2021 at 12:10pm in Eng Math EMG06, Zoom link by e-mail
Abstract: The Lichnerowicz conjecture is an open question of compact Lorentzian manifolds, asking when a manifold's conformal group is the same as the isometry group for some rescaled metric. This talk explains the conjecture, then explores the case of quotients of Cahen-Wallach spaces, finishing with an outline of two partial negative results. This talk is based on work in my thesis of the same name.
Friday, 8 October 2021 at 12:10pm in Eng Math EMG06, Zoom link by e-mail
Kobayashi-hyperbolic manifolds are an important and
well-studied class of complex manifolds defined by the
property that the Kobayashi pseudodistance is in fact a true
distance. Such manifolds that have a sufficiently large
automorphism group can be classified up to biholomorphism,
and the goal of this project is to continue the
classification of homogeneous Kobayashi-hyperbolic manifolds
started by Alexander Isaev in the early 2000s. We proceed in
this endeavour by analysing the Lie algebra of the
automorphism groups that act on such manifolds. This talk
will begin with a discussion and definition of a
Kobayashi-hyperbolic manifold, before outlining some of the
work involved in the classification. This talk is based on
work in my PhD thesis 'Highly symmetric homogeneous
theorem shows there are no local invariants in contact
geometry. Now add some some extra structure: a splitting of
the contact distribution into two integrable subbundles of
equal rank on which the canonical skew-form vanishes. This
is called a Legendrean contact structure on M and it is not
true that all such arrangements are locally equivalent. In
this talk I will explain how to construct a canonical vector
bundle and connection on M which is flat if and only if M is
locally isomorphic to the canonical model: the space of
lines inside hyperplanes in R^2n. This talk is based on work
in my thesis 'Legendrean and G_2 contact
Abstract: Differential Geometry Seminar attendees have heard me talk about Oka theory many times over the years. In this talk I will discuss how to adapt Oka theory to the presence of a group action. In recent work, Frank Kutzschebauch, Gerald Schwarz, and I took the first steps in the development of an equivariant version of Oka theory. I will describe the equivariant versions of the basic concepts of the theory, our main results, and some interesting examples.
Title: Dynamics of generic endomorphisms of Oka-Stein manifolds
Friday 09/04/21 from 12:10pm-1pm, N158 Chapman Lecture Theatre.
Abstract: I will describe joint work with Leandro Arosio (University of Rome Tor Vergata) on the dynamics of a generic endomorphism of an Oka-Stein manifold. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. The family of endomorphisms of an Oka-Stein manifold is so large and diverse that little can be said about its dynamics without restricting the analysis to suitable subfamilies that are usually taken to be quite small. We have shown that many interesting dynamical properties are generic with respect to the compact-open topology, which is the only natural topology in this context. Hence, somewhat surprisingly, strong dynamical theorems hold for very large subfamilies of endomorphisms of Oka-Stein manifolds. Even in the very special and much studied case of X = Cn most of our results are new.
Title: Local extension operators for nonlinear function spaces,
Friday 26/03/21 from 12:10pm-1pm, N158 Chapman Lecture Theatre.
Abstract: Whitney’s extension theorem tells us that for data encoding a ’smooth’ function on an arbitrary closed set in the reals, there is a smooth function on all of the real line extending it, and this extension can be specified by a continuous linear operator on function spaces. This reflects the fact the restriction operator is a split surjection of Fr ́echet spaces. One can study this problem in higher dimensions and for varying regularity, and this has been solved for (Banach spaces of) Ck real-valued functions on closed sets in Euclidean space by Fefferman, and partially so for smooth real-valued functions with sufficient conditions on the closed set by Frerick—the boundary cannot be too rough as there are known counterexamples. In joint work with Alexander Schmeding, we have proved an analogue of Frerick’s work for manifolds of smooth functions on a suitable closed ”submanifold with rough boundary” of a given manifold, with values in another manifold, where the analogues of extension operators are local sections of a submersion of infinite-dimensional manifolds. As well as covering this and related results, I will also indicate some applications to constructions in higher geometry.