Time & location in semester 2, 2024: Fridays

Contact: Thomas Leistner and Marcos Orseli.

*Richard Garner**(**Macquarie University)***Title:****Universal algebra and self-similarity**Date:

Abstract: In this talk I will explain an interesting connection between universal (or general) algebra, and combinatorial examples of self-similarity as exemplified by structures such as the Cuntz C*-algebra or Leavitt path algebras. If time permits I hope also to say something about how this relates to semantics of functional programming languages such as Haskell.**1 November,****11:10pm in Engineering Nth N158 Chapman Lecture Theatre**

*TBA*

*Jesse Gell-Redman**(**University of Melbourne)*

**Title: The Feynman propagator on asymptotically Minkowski spacetimes**Date:

Abstract: The Feynman propagator is a fundamental object in Quantum Field Theory which is also interesting when viewed from the perspective of analysis of linear hyperbolic PDEs. Thanks to the work of Baer-Strohmaier it has also been shown to finally provide an index theory of hyperbolic Dirac operators on some globally hyperbolic Lorentzian manifolds.**18 October,****11:10pm in Engineering Nth N158 Chapman Lecture Theatre**

We will discuss a new construction of the Feynman propagator for the Klein-Gordon (KG) equation on asymptotically Minkowski spacetimes. Our construction generalises existing work, and in particular gives a construction which works for naturally occurring families of Callias-type operators.

In particular, for large classes of potential perturbations, including magnetic potentials in the case of Dirac-Klein-Gordon, we construct global in time solutions to hyperbolic equations whose wavefront sets satisfy a Hadamard condition. Such solutions were shown to exist locally by Duistermaat-Hormander. To accomplish this, we prove a global Fredholm estimate and show that these non-elliptic operators are in fact Fredholm operators between appropriate Sobolev spaces. This is joint work with Dean Baskin and Moritz Doll.

*Michael Albanese**(University of Adelaide)*

**Title: Aspherical Complex Surfaces**Date:

Abstract: The collection of manifolds is vast and diverse. One class that we can hope to understand are those which have contractible universal cover, namely aspherical manifolds. These manifolds are determined up to homotopy equivalence by their fundamental group. There are several conjectures related to the Euler characteristic and signature of aspherical manifolds, namely the Hopf conjecture, the Singer conjecture, and in dimension 4, the Gromov-Luck inequality. We discuss these conjectures in the setting of aspherical complex surfaces. This is joint work with Luca Di Cerbo and Luigi Lombardi.**4 October,****11:10pm in Engineering Nth N158 Chapman Lecture Theatre**

*Sébastien Picard**(University of British Columbia)*

**Title: Non-Kahler Degenerations of Calabi-Yau Threefolds**Date:

Abstract: It was proposed in the works of Clemens, Reid, Friedman to connect Calabi-Yau threefolds with different topologies by a process which degenerates 2-cycles and introduces new 3-cycles. This operation may produce a non-Kahler complex manifold with trivial canonical bundle. In this talk, we will discuss the geometrization of this process by special non-Kahler metrics. This is joint work with T.C. Collins and S.-T. Yau.**6 September,****11:10pm in Engineering Nth N158 Chapman Lecture Theatre**

*Mateo Galdeano**(University of Hamburg)*

**Title: Generalised Einstein metrics on Lie groups**Date:

Abstract: The generalised Einstein condition is the analogue of Ricci flatness in the context of generalised geometry. It appears naturally as one of the equations of motion of certain supergravity theories, and it involves not only a metric but also a closed 3-form and a divergence operator on our manifold. In this talk we will discuss Riemannian and Lorentzian generalised Einstein metrics on Lie groups, including several classification results in four dimensions and beyond. Based on 2407.16562 with Vicente Cortés and Marco Freibert.**30 August,****11:10pm in Engineering Nth N158 Chapman Lecture Theatre**

*Behrouz Taji**(University of University of New South Wales)*

**Title: Boundedness problems in algebraic geometry and their consequences**Date:

Abstract: In 1960’s Shafarevich asked a simple question: Do families of curves of genus at least 2 (over a fixed base) have finite number of deformation classes? Soon after Parshin gave an affirmative answer to this question and furthermore showed that this finiteness property is precisely the geometric underpinning of Mordell’s famous conjecture regarding finiteness of rational points for projective curves of genus at least 2 (over number fields). It was this very observation that led Faltings to his complete proof of Mordell’s conjecture in 1980s. For higher dimensional analogues of curves of high genus, Shafarevich’s problem was settled in 2010 by Kovács and Lieblich, partially thanks to multiple advances in moduli theory of the so-called stable varieties. In this talk I will present this circle of ideas and then focus on our recent solution to Shafarevich’s question for other higher dimensional (non-stable) varieties, e.g. the Calabi-Yau case. This is based on joint work with Kenneth Ascher (UC Irvine).**16 August,****11:10pm in****Barr Smith South 2052**

*Yaping Yang**(University of Melbourne)*

**Title: An application of elliptic cohomology to quantum groups**Date:

Abstract: I will start by reviewing quantum groups (including quantum groups at roots of unity, Yangians, etc) and their representation theory. I will then explain the construction of quantum groups using cohomology theories from topology. The construction uses the so-called cohomological Hall algebra associated to a quiver and an oriented cohomology theory. In examples, we obtain the Yangian, quantum loop algebra and elliptic quantum group, when the cohomology theories are the cohomology, K-theory, and elliptic cohomology respectively. I will explain the application of elliptic cohomology theory in detail.**26 July,****11:10pm in****Barr Smith South 2052**

*Juan Manuel Lorenzo Naveiro**(University of Santiago de Compostela)*

**Title: Nearly Kähler geometry and totally geodesic submanifolds**Date:

Abstract: 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.**31 May,****12:10pm in Engineering Nth N218**

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.

*Yuri Nikolayevsky**(La Trobe University)*

**Title: Killing tensors on symmetric spaces**Date:

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).**17 May,****12:10pm in Engineering Nth N218**

*Michael Eastwood**(University of Adelaide)*

**Title: Killing tensors on complex projective space**Date:

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.**3 May,****12:10pm**

*Zoe Wyatt**(University of Cambridge)*

**Title: Global Stability of Spacetimes with Supersymmetric Compactifications**Date:

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.**5 April,****12:10pm in Engineering Nth N218**

*Peter Hochs (Radboud University)*

**Title: A higher index theorem on finite-volume locally symmetric spaces**Date:

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.**27 March,****15:10pm in Lower Napier LG18 (note the unusual time and location)**

*Finnur Larusson (University of Adelaide)*

**Title: Reconciling dichotomies in holomorphic dynamics**Date:

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".**15 March,****11:10am**