2021 Differential Geometry Seminars

School of Mathematical Sciences – The University of Adelaide.

Due to COVID-19 safety precautions, the seminar will be held  in hybrid mode.
Time & location: Fridays 12:10pm in EMG06 and on Zoom.
Contact:  Ben McMillan and Thomas Leistner.

2021 Seminars

• Ramiro Lafuente (University of Queensland)

  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.

• Stuart Teisseire (University of Adelaide)

  Title: Conformal group actions on Cahen-Wallach spaces
  Friday, 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.

• Elliot Herrington (University of Adelaide)
  Title: Homogeneous Kobayashi-hyperbolic manifolds

  Friday, 8 October 2021 at 12:10pm in Eng Math EMG06, Zoom link by e-mail

  Download slides
  Abstract: 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 Kobayashi-hyperbolic manifolds.'

• Tim Moy (University of Adelaide)
  Title: Legendrean Contact geometry
  Friday, 24 September 2021 at 12:10pm in Eng Math EMG06, Zoom link by e-mail
  
  Download slides

  Abstract: Darboux's 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 structures'.        

• Finnur Larusson (University of Adelaide)
  Title: Gromov’s Oka principle for equivariant maps
  Friday, 10 September 2021 at 12:10pm at Eng Sth S111 Lecture Theatre and on Zoom (coordinates by e-mail)

  
  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.

• Speaker: Lachlan MacDonald (University of Adelaide)
  Title: Chern-Weil theory for singular foliations
  Friday 18th June from 12:10pm-1pm via Zoom

  
  Abstract: Chern-Weil theory describes a procedure for constructing characteristic classes for smooth manifolds from geometric data (such as a Riemannian metric). In the 1970s and 1980s, Chern-Weil theory was successfully adapted by R. Bott to describe the characteristic classes of the leaf space of any regular foliation, including the so-called secondary classes such as the Godbillon-Vey invariant. The extension of this theory to singular foliations has, however, re- mained elusive. In this talk, I will describe recent, joint work with Benjamin McMillan which gives a Chern-Weil homomorphism for a family of singular foliations whose singularities are not “too big”.

• Speaker: Jarah Evslin (Chinese Academy of Sciences)
  Title: An Introduction to Schrodinger Wave Functionals
  Friday 4th June from 12:10pm-1pm via Zoom

  
  Abstract: Each configuration in a quantum field theory corresponds to a map from a space X of functions or bundles with sections to the space of complex numbers. These maps are called Schrodinger wave functionals. They generalize wave functions in quantum mechanics, which are maps from a finite-dimensional manifold to the complex numbers. We review the main properties of wave functions and wave functionals in a series of examples. We describe an embedding of X into this quantum configuration space and argue that perturbative quantum field theory only probes a tubular neighborhood of its image, but that the poorly understood global properties of the quantum configuration space are relevant to the confinement problem in supersymmetric QCD.

• Speaker: Yi Huang (Yau Mathematical Science Centre, Tsinghua University)
  Title: Simple closed curves on surfaces
  Friday 14th May from 12:10pm-1pm, online via zoom.

  
  Abstract: The qualitative and quantitative behaviour of simple closed curves on surfaces can reveal a great deal of geometric information about the underlying surface. We look at three theories within this theme, all pertaining to hyperbolic surfaces: Birman and Series’s geodesic sparsity theorem, McShane and Rivin’s simple length spectrum growth rate asymptotics (as well as later improvements by Mirzakhani), and McShane identities. I hope to give a feel for why these results hold, as well as my input in extending these results to more general types of surfaces.

• Speaker: Melissa Tacy (University of Auckland)
  Title: Adapting analysis/synthesis pairs to pseudodifferential operators
  Friday 30th of April from 12:10pm-1pm, online via zoom.

  
  Abstract: Many problems in harmonic analysis are resolved by producing an analysis/synthesis of function spaces. For example the Fourier or wavelet decompositions. In this talk I will discuss how to use Fourier integral operators to adapt analysis/synthesis pairs (developed for the constant coefficient PDE case) to the pseudodifferential setting. I will demonstrate how adapting a wavelet decomposition can be used to prove Lp bounds for joint eigenfunctions.

• Speaker: Hang Wang (East China Normal University)
  Title: A K-theoretic approach to semisimple Lie groups and their lattices
  Friday 16/04/21 from 12:10pm-1pm, online via Zoom.

  
  Abstract: Semisimple Lie groups and their lattices are of interest in many areas such as differ- ential geometry, number theory, ergodic theory and geometric group theory. In this talk, we propose an operator K-theory framework to study the groups and their K-theoretic functoriality, also motivated by the Baum-Connes conjecture for such pairs of groups. In the case of a uniform lattice, we find a cohomological interpretation of the trace formula involving the K-theory of the maximal group C∗-algebras of a semisimple Lie group and its lattice. As an application, this implies the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici, in this special case of a uniform lattice. This is joint work with Bram Mesland and Mehmet Haluk Sengun.

• Speaker: Finnur Larusson (University of Adelaide)
  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.

• Speaker: David Roberts (University of Adelaide)
  Title: Local extension operators for nonlinear function spaces, extending Whitney

  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.