Title: Killing tensors on complex projective space
Date: 3 May, 12:10pm
Abstract: 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.
Title: Killing tensors on symmetric spaces
Date: 17 May, 12:10pm in Engineering Nth N218
Abstract: 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).
Title: TBA
Date: 26 July, 12:10pm in Engineering Nth N218
Abstract: TBA
Title: TBA
Date: 16 August, 12:10pm in Engineering Nth N218
Abstract: TBA
Title: TBA
Date: 30 August, 12:10pm in Engineering Nth N218
Abstract: TBA
Title: Global Stability of Spacetimes with Supersymmetric Compactifications
Date: 5 April, 12:10pm in Engineering Nth N218
Abstract: 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.
Title: A higher index theorem on finite-volume locally symmetric spaces
Date: 27 March, 15:10pm in Lower Napier LG18 (note the unusual time and location)
Abstract: 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.
Title: Reconciling dichotomies in holomorphic dynamics
Date: 15 March, 11:10am
Abstract: 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".