Differential Geometry Seminars 2021

School of Mathematical Sciences – The University of Adelaide

Previous years


























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.

~ Upcoming talks ~

~ Past talks in 2021 ~

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”. 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.

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. 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. 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.

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, 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.