Differential Geometry Seminars: Semester 1

Title: A Hecke module structure on the KK-theory of arithmetic groups
When: Friday, 2 March 2018 at 1:10pm in Barr Smith South Polygon Lec theatre 
Speaker: Bram Mesland (Unversität Bonn)

Abstract: Let $G$ be a locally compact group, $\Gamma$ a discrete subgroup and $C_{G}(\Gamma)$ the commensurator of $\Gamma$ in $G$. The cohomology of $\Gamma$ is a module over the Shimura Hecke ring of the pair $(\Gamma,C_G(\Gamma))$. This construction recovers the action of the Hecke operators on modular forms for $SL(2,\mathbb{Z})$ as a particular case. In this talk I will discuss how the Shimura Hecke ring of a pair $(\Gamma, C_{G}(\Gamma))$ maps into the $KK$-ring associated to an arbitrary $\Gamma$-C*-algebra. From this we obtain a variety of $K$-theoretic Hecke modules. In the case of manifolds the Chern character provides a Hecke equivariant transformation into cohomology, which is an isomorphism in low dimensions. We discuss Hecke equivariant exact sequences arising from possibly noncommutative compactifications of $\Gamma$-spaces. Examples include the Borel-Serre and geodesic compactifications of the universal cover of an arithmetic manifold, and the totally disconnected boundary of the Bruhat-Tits tree of $SL(2,\mathbb{Z})$. This is joint work with M.H. Sengun (Sheffield).

Title: Radial Toeplitz operators on bounded symmetric domains
When: Friday, 9 March 2018 at 1:10pm in Lower Napier LG11
Speaker: Raul Quiroga-Barranco (CIMAT, Guanajuato, Mexico) 

Abstract: The Bergman spaces on a complex domain are defined as the space of holomorphic square-integrable functions on the domain. These carry interesting structures both for analysis and representation theory in the case of bounded symmetric domains. On the other hand, these spaces have some bounded operators obtained as the composition of a multiplier operator and a projection. These operators are highly noncommuting between each other. However, there exist large commutative C*-algebras generated by some of these Toeplitz operators very much related to Lie groups. I will construct an example of such C*-algebras and provide a fairly explicit simultaneous diagonalization of the generating Toeplitz operators.

Title: Quantum Airy structures and topological recursion
When: Wednesday, 14 March 2018 at 1:10pm in Ingkarni Wardli B17
Speaker: Gaetan Borot

Abstract: Quantum Airy structures are Lie algebras of quadratic differential operators — their classical limit describes Lagrangian subvarieties in symplectic vector spaces which are tangent to the zero section and cut out by quadratic equations. Their partition function — which is the function annihilated by the collection of differential operators — can be computed by the topological recursion. I will explain how to obtain quantum Airy structures from spectral curves, and explain how we can retrieve from them correlation functions of semi-simple cohomological field theories, by exploiting the symmetries. This is based on joint work with Andersen, Chekhov and Orantin.

 Title: Family gauge theory and characteristic classes of bundles of 4-manifolds
When: Friday, 16 March 2018 at 1:10pm in Barr Smith South Polygon Lec theatre
Speaker: Hokuto Konno (University of Tokyo)

Abstract: I will define a non-trivial characteristic class of bundles of 4-manifolds using families of Seiberg-Witten equations. The basic idea of the construction is to consider an infinite dimensional analogue of the Euler class used in the usual theory of characteristic classes. I will also explain how to prove the non-triviality of this characteristic class. If time permits, I will mention a relation between our characteristic class and positive scalar curvature metrics. 

Title: Computing trisections of 4-manifolds
When: Friday, 23 March 2018 at 1:10pm in Barr Smith South Polygon Lec theatre
Speaker: Stephen Tillmann

Abstract: Gay and Kirby recently generalised Heegaard splittings of 3-manifolds to  trisections of 4-manifolds. A trisection describes a 4-dimensional manifold as a union of three 4–dimensional handlebodies. The complexity of the 4–manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The minimal genus of such a surface is the trisection genus of the 4-manifold. After defining trisections and giving key examples and applications, I will describe an algorithm to compute trisections of 4–manifolds using arbitrary triangulations as input. This results in the first explicit complexity bounds for the trisection genus of a 4–manifold in terms of the number of pentachora (4–simplices) in a triangulation. This is joint work with Mark Bell, Joel Hass and Hyam Rubinstein. I will also describe joint work with Jonathan Spreer that determines the trisection genus for each of the standard simply connected PL 4-manifolds.

Title: Chaos in higher-dimensional complex dynamics
When: Friday, 20 April 2018 at 1:10pm in Barr Smith South Polygon Lec theatre
Speaker: Finnur Larusson

Abstract: I will report on new joint work with Leandro Arosio (University of Rome, Tor Vergata). Complex manifolds can be thought of as laid out across a spectrum characterised by rigidity at one end and flexibility at the other.  On the rigid side, Kobayashi-hyperbolic manifolds have at most a finite-dimensional group of symmetries.  On the flexible side, there are manifolds with an extremely large group of holomorphic automorphisms, the prototypes being the affine spaces $\mathbb C^n$ for $n \geq 2$.  From a dynamical point of view, hyperbolicity does not permit chaos.  An endomorphism of a Kobayashi-hyperbolic manifold is non-expansive with respect to the Kobayashi distance, so every family of endomorphisms is equicontinuous.  We show that not only does flexibility allow chaos: under a strong anti-hyperbolicity assumption, chaotic automorphisms are generic.  A special case of our main result is that if $G$ is a connected complex linear algebraic group of dimension at least 2, not semisimple, then chaotic automorphisms are generic among all holomorphic automorphisms of $G$ that preserve a left- or right-invariant Haar form.  For $G=\mathbb C^n$, this result was proved (although not explicitly stated) some 20 years ago by Fornaess and Sibony.  Our generalisation follows their approach.  I will give plenty of context and background, as well as some details of the proof of the main result.

Title: Index of Equivariant Callias-Type Operators
When: Friday, 27 April 2018 at 1:10pm in Barr Smith South Polygon Lec theatre
Speaker: Hao Guo (University of Adelaide)

Abstract: Suppose M is a smooth Riemannian manifold on which a Lie group G acts properly and isometrically. In this talk I will explore properties of a particular class of G-invariant operators on M, called G-Callias-type operators. These are Dirac operators that have been given an additional Z_2-grading and a perturbation so as to be “invertible outside of a cocompact set in M”. It turns out that G-Callias-type operators are equivariantly Fredholm and so have an index in the K-theory of the maximal group C*-algebra of G. This index can be expressed as a KK-product of a class in K-homology and a class in the K-theory of the Higson G-corona. In fact, one can show that the K-theory of the Higson G-corona is highly non-trivial, and thus the index theory of G-Callias-type operators is not obviously trivial. As an application of the index theory of G-Callias-type operators, I will mention an obstruction to the existence of G-invariant metrics of positive scalar curvature on M.

Title: Index of Equivariant Callias-Type Operators
When: Friday, 4 May 2018 at 1:10pm in Barr Smith South Polygon Lec theatre
Speaker: Tony Licata (Australian National University)

Abstract: The Artin braid group arise in a number of different parts of mathematics.  The goal of this talk will be to explain how basic group-theoretic questions about the Artin braid group can be answered using some modern tools of linear and homological algebra, with an eye toward proving some open conjectures about other groups.

Title: Hitchin’s Projectively Flat Connection for the Moduli Space of Higgs Bundles
When: Friday, 15 June 2018 at 1:10pm in Barr Smith South Polygon Lec theatre
Speaker: John McCarthy (University of Adelaide)

Abstract: In this talk I will discuss the problem of geometrically quantizing the moduli space of Higgs bundles on a compact Riemann surface using Kahler polarisations. I will begin by introducing geometric quantization via Kahler polarisations for compact manifolds, leading up to the definition of a Hitchin connection as stated by Andersen. I will then describe the moduli spaces of stable bundles and Higgs bundles over a compact Riemann surface, and discuss their properties. The problem of geometrically quantizing the moduli space of stables bundles, a compact space, was solved independently by Hitchin and Axelrod, Del PIetra, and Witten. The Higgs moduli space is non-compact and therefore the techniques used do not apply, but carries an action of C*. I will finish the talk by discussing the problem of finding a Hitchin connection that preserves this C* action. Such a connection exists in the case of Higgs line bundles, and I will comment on the difficulties in higher rank.

Title: Comparison Theorems under Weak Assumptions
When: Friday, 29 June 2018 at 11.10am in EMG06
Speaker: Kwok Kun Kwong (National Cheng Kung University)

Abstract: The classical volume comparison states that under a lower bound on the Ricci curvature, the volume of the geodesic ball is bounded from above by that of the geodesic ball with the same radius in the model space. On the other hand, counterexamples show the assumption on the Ricci curvature cannot be weakened to a lower bound on the scalar curvature, which is the average of the Ricci curvature. In this talk, I will show that a lower bound on a weighted average of the Ricci curvature is sufficient to ensure volume comparison. In the course I will also prove a sharp volume estimate and an integral version of the Laplacian comparison theorem. If time allows, I will also present the Kahler version of the theorem.


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