Differential Geometry Seminars 2024

15 March 2024
11:10 am
Lower Napier LG19

Finnur Lárusson
University of Adelaide

Reconciling dichotomies in holomorphic dynamics

I will describe recent joint work with Leandro Arosio (University of Rome, Tor Vergata) in holomorphic dynamics. We study the iteration of endomorphisms of complex manifolds of a fairly general kind, as well as the iteration of automorphisms for a class that is somewhat smaller, but still includes almost all linear algebraic groups. The main goal of our work is to reconcile two fundamental dynamical dichotomies in these settings, "attracted vs. recurrent" and "calm vs. wild".

27 March 2024

3:10pm 

Lower Napier LG18

Peter Hochs

Radboud University

A higher index theorem on finite-volume locally symmetric spaces

Let G be a (connected, real, semisimple, real rank one) Lie group, and K a maximal compact subgroup. Let Gamma be a torsion-free, discrete subgroup of G. If the double-coset space X = Gamma\G/K is compact, then we can do index theory on it, both in the classical Atiyah-Singer sense and in the sense of higher index theory with values in the K-theory of the C*-algebra of Gamma. But in many relevant cases, X has finite volume, but is noncompact. This includes the case where G = SL(2,R), K = SO(2) and Gamma = SL(2,Z). Then Moscovici constructed an index of Dirac operators on X, and Barbasch and Moscovici computed it using the (Arthur-)Selberg trace formula. In ongoing work with Hao Guo and Hang Wang, we upgrade this to a higher index with values in a relevant K-theory group. This talk is a relatively gentle introduction, focusing mainly on the classical case.

5 April 2024 

12:10pm

Engineering North N218

Zoe Wyatt

University of Cambridge 

Global Stability of Spacetimes with Supersymmetric Compactifications

Compact spaces with special holonomy, such as Calabi-Yau manifolds, play an important role in supergravity and string theory. In this talk, I will present a recent result showing the stability of a spacetime which is the cartesian product of Minkowski spacetime and a compact special holonomy space. I will also explain how this stability result relates to conjectures of Penrose and Witten. This is based on joint work with L. Andersson, P. Blue and S-T. Yau.

3 May 2024

12:10pm

Engineering North N218

Michael Eastwood

University of Adelaide 

Killing tensors on complex projective space

The Killing tensors on the round sphere are well understood. In particular, these are finite-dimensional vector spaces with very nice formulae for their dimensions. What about the corresponding story for complex projective space? Again, these are finite-dimensional spaces and their dimensions can be computed. Formulae for their dimensions, however, are simultaneously nice but seriously puzzling! If time permits: what about quaternionic projective space and so on? Some of this was discussed in my TOPS series last September but I’ll start again from scratch and, in particular, explain what are Killing tensors and why they are useful.

17 May 2024

12:10pm

Engineering Nth N218

Yuri Nikolayevsky

La Trobe University

Killing tensors on symmetric spaces

I will present some recent results on the structure of the algebra of Killing tensors on Riemannian symmetric spaces. The fundamental question is whether any Killing tensor field on a Riemannian symmetric space is a polynomial in (a symmetric product of) Killing vector fields. For spaces of constant curvature, the answer is in the positive (as has been known for quite some time). The same is true for the complex projective space (Eastwood, 2023). Surprisingly, for other rank one symmetric spaces (quaternionic projective space and Cayley projective plane), the answer is almost always in the negative (Matveev-Nikolayevsky, 2024). If time permits we’ll also discuss the case of higher rank and some related results. This is a joint project with V. Matveev (University of Jena, Germany).

31 May 2024

12:10pm

Engineering Nth N218

Juan Manuel Lorenzo Naveiro

University of Santiago de Compostela

Nearly Kähler geometry and totally geodesic submanifolds

A theorem of Butruille asserts that the (simply connected, homogeneous) Riemannian manifolds of dimension six admitting a strict nearly Kähler metric are the round sphere S6, the space F(C3) of full flags in C3, the complex projective space CP3 and the almost product S3 x S3. These spaces belong to the general class of naturally reductive homogeneous spaces, whose geometry can be understood in purely Lie-algebraic terms.

The aim of this talk is to describe a joint work with Alberto Rodríguez-Vázquez (KU Leuven) in which we classify the totally geodesic submanifolds of the aforementioned spaces. To this end, we will develop the algebraic tools needed to work with naturally reductive homogeneous spaces and to attack our problem, and later on we will exhibit the examples that appear in each case.