Differential Geometry Seminars 2017

School of Mathematical Sciences – The University of Adelaide

~ Next talk ~

• 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.

~ Upcoming talks ~

• 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)
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Friday, 28 April 2017 at 12:10pm in Napier 209

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• Jesse Gell-Redman (University of Melbourne)
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Friday, 5 May 2017 at 12:10pm in Napier 209

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• Aidan Sims (University of Wollongong)
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Friday, 12 May 2017 at 12:10pm in Napier 209

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

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• 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)
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Friday, 26 May 2017 at 12:10pm in Napier 209

- abstract TBA-

• David Roberts (University of Adelaide)
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Friday, 9 June 2017 at 12:10pm in Napier 209

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~ 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.