## Differential Geometry Seminars 2017

### School of Mathematical Sciences – The University of Adelaide

#### ~ Next talk ~

• Vicente Cortés (Universität Hamburg)
Quaternionic Kähler manifolds of co-homogeneity one
Friday, 16 June 2017 at 12:10pm in Ligertwood 231

Quaternionic Kähler manifolds form an important class of Riemannian manifolds of special holonomy. They provide examples of Einstein manifolds of non-zero scalar curvature. I will show how to construct explicit examples of complete quaternionic Kähler manifolds of negative scalar curvature beyond homogeneous spaces. In particular, I will present a series of examples of co-homogeneity one, based on arXiv:1701.07882.

#### ~ Upcoming talks ~

• Mike Eastwood (University of Adelaide)
- title TBA-
Friday, 28 July 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• James McCoy (University of Wollongong)
- title TBA-
Friday, 4 August 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• Tuyen Truong (University of Adelaide)
- title TBA-
Friday, 11 August 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• (availible slot)
- title TBA-
Friday, 18 August 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• Guo Chuan Thiang (University of Adelaide)
- title TBA-
Friday, 25 August 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• (availible slot)
- title TBA-
Friday, 1 September 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• Alex Chi-Kwong Fok (University of Adelaide)
- title TBA-
Friday, 8 September 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• (11-15 Spetember: IGA workshop: String Geometry and Duality)
• Marcy Robertson (University of Melbourne)
- title TBA-
Friday, 22 September 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• Malabika Pramanik (University of British Columbia)
- title TBA-
Wednesday, 27 September 2017 at 11:10am in Engineering & Math EM213 (tentative)

- abstract TBA-

• Leandro Arosio (University of Rome)
- title TBA-
Friday, 29 September 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• (availible slot)
- title TBA-
Friday, 6 October 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• (availible slot)
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Friday, 13 October 2017 at 12:10pm in Engineering Sth S111

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• (availible slot)
- title TBA-
Friday, 20 October 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• Michael Hallam (University of Adelaide)
- title TBA-
Friday, 27 October 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

• Glen Wheeler (University of Wollongong)
- title TBA-
Friday, 3 November 2017 at 12:10pm in Engineering Sth S111

- abstract TBA-

#### ~ Past talks ~

• Finnur Larusson (University of Adelaide)
An equivariant parametric Oka principle for bundles of homogeneous spaces
Friday, 3 March 2017 at 12:10pm in Napier 209

I will report on new joint work with Frank Kutzschebauch and Gerald Schwarz (arXiv:1612.07372). Under certain conditions, every continuous section of a holomorphic fibre bundle can be deformed to a holomorphic section. In fact, the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. What if a complex Lie group acts on the bundle and its sections? We have proved an analogous result for equivariant sections. The result has a wide scope. If time permits, I will describe some interesting special cases and mention two applications.

• Diarmuid Crowley (University of Melbourne)
Diffeomorphisms of discs, harmonic spinors and positive scalar curvature
Friday, 17 March 2017 at 11:10am in Engineering Nth N218

Let Diff(D^k) be the space of diffeomorphisms of the k-disc fixing the boundary point wise. In this talk I will show for k > 5, that the homotopy groups \pi_*Diff(D^k) have non-zero 8-periodic 2-torsion detected in real K-theory. I will then discuss applications for spin manifolds M of dimension 6 or greater: 1) Our results input to arguments of Hitchin which now show that M admits a metric with a harmonic spinor. 2) If non-empty, space of positive scalar curvature metrics on M has non-zero 8-periodic 2-torsion in its homotopy groups which is detected in real K-theory. This is part of joint work with Thomas Schick and Wolfgang Steimle.

• Antonio Alarcon (University of Granada)
Minimal surfaces and complex analysis
Friday, 24 March 2017 at 12:10pm in Napier 209

A surface in the Euclidean space R^3 is said to be minimal if it is locally area-minimizing, meaning that every point in the surface admits a compact neighborhood with the least area among all the surfaces with the same boundary. Although the origin of minimal surfaces is in physics, since they can be realized locally as soap films, this family of surfaces lies in the intersection of many fields of mathematics. In particular, complex analysis in one and several variables plays a fundamental role in the theory. In this lecture we will discuss the influence of complex analysis in the study of minimal surfaces.

• Nicholas Buchdahl (University of Adelaide)
Geometric structures on moduli spaces
Friday, 31 March 2017 at 12:10pm in Napier 209

Moduli spaces are used to classify various kinds of objects, often arising from solutions of certain differential equations on manifolds; for example, the complex structures on a compact surface or the anti-self-dual Yang-Mills equations on an oriented smooth 4-manifold. Sometimes these moduli spaces carry important information about the underlying manifold, manifested most clearly in the results of Donaldson and others on the topology of smooth 4-manifolds. It is also the case that these moduli spaces themselves carry interesting geometric structures; for example, the Weil-Petersson metric on moduli spaces of compact Riemann surfaces, exploited to great effect by Maryam Mirzakhani. In this talk, I shall elaborate on the theme of geometric structures on moduli spaces, with particular focus on some recent-ish work done in conjunction with Georg Schumacher.

• Peter Hochs (University of Adelaide)
K-types of tempered representations
Friday, 7 April 2017 at 12:10pm in Napier 209

Tempered representations of a reductive Lie group G are the irreducible unitary representations one needs in the Plancherel decomposition of L^2(G). They are relevant to harmonic analysis because of this, and also occur in the Langlands classification of the larger class of admissible representations. If K < G is a maximal compact subgroup, then there is a considerable amount of information in the restriction of a tempered representation to K. In joint work with Yanli Song and Shilin Yu, we give a geometric expression for the decomposition of such a restriction into irreducibles. The multiplicities of these irreducibles are expressed as indices of Dirac operators on reduced spaces of a coadjoint orbit of G corresponding to the representation. These reduced spaces are Spin-c analogues of reduced spaces in symplectic geometry, defined in terms of moment maps that represent conserved quantities. This result involves a Spin-c version of the quantisation commutes with reduction principle for noncompact manifolds. For discrete series representations, this was done by Paradan in 2003.

• Ctirad Klimcik (Aix-Marseille Université, Marseille)
Poisson-Lie T-duality and integrability
Thursday, 13 April 2017 at 11:10am in Engineering & Math EM213

The Poisson-Lie T-duality relates sigma-models with target spaces symmetric with respect to mutually dual Poisson-Lie groups. In the special case if the Poisson-Lie symmetry reduces to the standard non-Abelian symmetry one of the corresponding mutually dual sigma-models is the standard principal chiral model which is known to enjoy the property of integrability. A natural question whether this non-Abelian integrability can be lifted to integrability of sigma model dualizable with respect to the general Poisson-Lie symmetry has been answered in the affirmative by myself in 2008. The corresponding Poisson-Lie symmetric and integrable model is a one-parameter deformation of the principal chiral model and features a remarkable explicit appearance of the standard Yang-Baxter operator in the target space geometry. Several distinct integrable deformations of the Yang-Baxter sigma model have been then subsequently uncovered which turn out to be related by the Poisson-Lie T-duality to the so called lambda-deformed sigma models. My talk gives a review of these developments some of which found applications in string theory in the framework of the AdS/CFT correspondence.

• Jessica Purcell (Monash University)
Geometric limits of knot complements
Friday, 28 April 2017 at 12:10pm in Napier 209

The complement of a knot often admits a hyperbolic metric: a metric with constant curvature -1. In this talk, we will investigate sequences of hyperbolic knots, and the possible spaces they converge to as a geometric limit. In particular, we show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. This is joint work with Autumn Kent.

• Jesse Gell-Redman (University of Melbourne)
Hodge theory on the moduli space of Riemann surfaces
Friday, 5 May 2017 at 12:10pm in Napier 209

The Hodge theorem on a closed Riemannian manifold identifies the deRham cohomology with the space of harmonic differential forms. Although there are various extensions of the Hodge theorem to singular or complete but non-compact spaces, when there is an identification of L^2 Harmonic forms with a topological feature of the underlying space, it is highly dependent on the nature of infinity (in the non-compact case) or the locus of incompleteness; no unifying theorem treats all cases. We will discuss work toward extending the Hodge theorem to singular Riemannian manifolds where the singular locus is an incomplete cusp edge. These can be pictured locally as a bundle of horns, and they provide a model for the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Joint with J. Swoboda and R. Melrose.

• Aidan Sims (University of Wollongong)
Graded K-theory and C*-algebras
Friday, 12 May 2017 at 11:10am in Engineering Nth N218

C*-algebras can be regarded, in a very natural way, as “noncommutative” algebras of continuous functions on topological spaces. The analogy is strong enough that topological K-theory in terms of formal differences of vector bundles has a direct analogue for C*-algebras. There is by now a substantial array of tools out there for computing C*-algebraic K-theory. However, when we want to model physical phenomena, like topological phases of matter, we need to take into account various physical symmetries, some of which are encoded by gradings of C*-algebras by the two-element group. Even the definition of graded C*-algebraic K-theory is not entirely settled, and there are relatively few computational tools out there. I will try to outline what a C*-algebra (and a graded C*-algebra is), indicate what graded K-theory ought to look like, and discuss recent work with Alex Kumjian and David Pask linking this with the deep and powerful work of Kasparov, and using this to develop computational tools.

• Michael Murray (University of Adelaide)
Real bundle gerbes
Friday, 19 May 2017 at 12:10pm in Napier 209

Bundle gerbe modules, via the notion of bundle gerbe K-theory provide a realisation of twisted K-theory. I will discuss the existence or Real bundle gerbes which are the corresponding objects required to construct Real twisted K-theory in the sense of Atiyah. This is joint work with Richard Szabo (Heriot-Watt), Pedram Hekmati (Auckland) and Raymond Vozzo which appeared in arXiv:1608.06466.

• Hiep Tuan Dang (National centre for theoretical sciences, Taiwan)
Schubert Calculus on Lagrangian Grassmannians
Tuesday, 23 May 2017 at 12:10pm in EM 213 (unusual date and venue)

The Lagrangian Grassmannian $LG = LG(n,2n)$ is the projective complex manifold which parametrizes Lagrangian (i.e. maximal isotropic) subspaces in a symplective vector space of dimension $2n$. This talk is mainly devoted to Schubert calculus on $LG$. We first recall the definition of Schubert classes in this context. Then we present basic results which are similar to the classical formulas due to Pieri and Giambelli. These lead to a presentation of the cohomology ring of $LG$. Finally, we will discuss recent results related to the Schubert structure constants and Gromov-Witten invariants of $LG$.

• Franc Forstneric (University of Ljubljana, Slovenia)
Holomorphic Legendrian curves
Friday, 26 May 2017 at 12:10pm in Napier 209

I will present recent results on the existence and behaviour of noncompact holomorphic Legendrian curves in complex contact manifolds. We show that these curves are ubiquitous in \C^{2n+1} with the standard holomorphic contact form \alpha=dz+\sum_{j=1}^n x_jdy_j; in particular, every open Riemann surface embeds into \C^3 as a proper holomorphic Legendrian curves. On the other hand, for any integer n>= 1 there exist Kobayashi hyperbolic complex contact structures on \C^{2n+1} which do not admit any nonconstant Legendrian complex lines. Furthermore, we construct a holomorphic Darboux chart around any noncompact holomorphic Legendrian curve in an arbitrary complex contact manifold. As an application, we show that every bordered holomorphic Legendrian curve can be uniformly approximated by complete bounded Legendrian curves.

• David Roberts (University of Adelaide)
Constructing differential string structures
Wednesday, 7 June 2017 at 2:10pm in EM213

String structures on a manifold are analogous to spin structures, except instead of lifting the structure group through the extension Spin(n)\to SO(n) of Lie groups, we need to lift through the extension String(n)\to Spin(n) of Lie *2-groups*. Such a thing exists if the first fractional Pontryagin class (1/2)p_1 vanishes in cohomology. A differential string structure also lifts connection data, but this is rather complicated, involving a number of locally defined differential forms satisfying cocycle-like conditions. This is an expansion of the geometric string structures of Stolz and Redden, which is, for a given connection A, merely a 3-form R on the frame bundle such that dR = tr(F^2) for F the curvature of A; in other words a trivialisation of the de Rham class of (1/2)p_1. I will present work in progress on a framework (and specific results) that allows explicit calculation of the differential string structure for a large class of homogeneous spaces, which also yields formulas for the Stolz-Redden form. I will comment on the application to verifying the refined Stolz conjecture for our particular class of homogeneous spaces. Joint work with Ray Vozzo