## Differential Geometry Seminars 2015

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

Stay tuned for Differential Geometry Seminar 2016.

#### ~ Past talks ~

• Simona Paoli (University of Leicester)
Weak globularity in homotopy theory and higher category theory
Thursday, 12 November 2015 at 12:10pm in Ingkarni Wardli B19

Spaces and homotopy theories are fundamental objects of study of algebraic topology. One way to study these objects is to break them into smaller components with the Postnikov decomposition. To describe such decomposition purely algebraically we need higher categorical structures. We describe one approach to modelling these structures based on a new paradigm to build weak higher categories, which is the notion of weak globularity. We describe some of their connections to both homotopy theory and higher category theory.

• Thomas Leistner (The University of Adelaide)
Locally homogeneous pp-waves
Friday, 6 November 2015 at 12:10pm in Ingkarni Wardli B17

For a certain type of Lorentzian manifolds, the so-called pp-waves, we study the conditions implied on the curvature by local homogeneity of the metric. We show that under some mild genericity assumptions, these conditions are quite strong, forcing the pp-wave to be a plane wave, and yielding a classification of homogeneous pp-waves. This also leads to a generalisation of a classical result by Jordan, Ehlers and Kundt about vacuum pp-waves in dimension 4 to arbitrary dimensions. Several examples show that our genericity assumptions are essential. This is joint work with W. Globke.

• Danny Stevenson (The University of Adelaide)
Covariant model structures and simplicial localization
Friday, 30 October 2015 at 12:10pm in Ingkarni Wardli B17

This talk will describe some aspects of the theory of quasi-categories, in particular the notion of left fbration and the allied covariant model structure. If $$B$$ is a simplicial set, then I will describe some Quillen equivalences relating the covariant model structure on simplicial sets over $$B$$ to a certain localization of simplicial presheaves on the simplex category of $$B$$. I will show how this leads to a new description of Lurie's simplicial rigidification functor as a hammock localization and describe some applications to Lurie's theory of straightening and unstraightening functors.

• Anne Thomas (University of Sydney)
Quasi-isometry classification of certain hyperbolic Coxeter groups
Friday, 23 October 2015 at 11:00am in Ingkarni Wardli Conference Room 7.15 (Level 7)

Let $$\Gamma$$ be a finite simple graph with vertex set $$S$$. The associated right-angled Coxeter group $$W$$ is the group with generating set $$S$$, so that $$s^2 = 1$$ for all $$s$$ in $$S$$ and $$st = ts$$ if and only if $$s$$ and $$t$$ are adjacent vertices in Gamma. Moussong proved that the group $$W$$ is hyperbolic in the sense of Gromov if and only if $$\Gamma$$ has no "empty squares". We consider the quasi-isometry classification of such Coxeter groups using the local cut point structure of their visual boundaries. In particular, we find an algorithm for computing Bowditch's JSJ tree for a class of these groups, and prove that two such groups are quasi-isometric if and only if their JSJ trees are the same. This is joint work with Pallavi Dani (Louisiana State University).

• Steve Rosenberg (Boston University)
Chern-Simons classes on loop spaces and diffeomorphism groups
Friday, 16 October 2015 at 12:10pm in Ingkarni Wardli B17

Not much is known about the topology of the diffeomorphism group $$\mathrm{Diff}(M)$$ of manifolds M of dimension four and higher. We'll show that for a class of manifolds of dimension $$4k+1$$, $$\mathrm{Diff}(M)$$ has infinite fundamental group. This is proved by translating the problem into a question about Chern-Simons classes on the tangent bundle to the loop space $$LM$$. To build the CS classes, we use a family of metrics on LM associated to a Riemannian metric on $$M$$. The curvature of these metrics takes values in an algebra of pseudodifferential operators. The main technical step in the CS construction is to replace the ordinary matrix trace in finite dimensions with the Wodzicki residue, the unique trace on this algebra. The moral is that some techniques in finite dimensional Riemannian geometry can be extended to some examples in infinite dimensional geometry.

• Joseph A. Wolf (University of California, Berkeley)
Real Lie Groups and Complex Flag Manifolds
Friday, 9 October 2015 at 12:10pm in Ingkarni Wardli B17

Let $$G$$ be a complex simple direct limit group. Let $$G_{\mathbb{R}}$$ be a real form of $$G$$ that corresponds to an hermitian symmetric space. I'll describe the corresponding bounded symmetric domain in the context of the Borel embedding, Cayley transforms, and the Bergman-Shilov boundary. Let $$Q$$ be a parabolic subgroup of $$G$$. In finite dimensions this means that $$G/Q$$ is a complex projective variety, or equivalently has a Kaehler metric invariant under a maximal compact subgroup of $$G$$. Then I'll show just how the bounded symmetric domains describe cycle spaces for open $$G_{\mathbb{R}}$$ orbits on $$G/Q$$. These cycle spaces include the complex bounded symmetric domains. In finite dimensions they are tightly related to moduli spaces for compact Kaehler manifolds and to representations of semisimple Lie groups; in infinite dimensions there are more problems than answers. Finally, time permitting, I'll indicate how some of this goes over to real and to quaternionic bounded symmetric domains.

• Timur Sadykov (Plekhanov University, Moscow)
Analytic complexity of bivariate holomorphic functions and cluster trees
Friday, 2 October 2015 at 12:10pm in Ingkarni Wardli B17

The Kolmogorov-Arnold theorem yields a representation of a multivariate continuous function in terms of a composition of functions which depend on at most two variables. In the analytic case, understanding the complexity of such a representation naturally leads to the notion of the analytic complexity of (a germ of) a bivariate multi-valued analytic function. According to Beloshapka's local definition, the order of complexity of any univariate function is equal to zero while the $$n$$-th complexity class is defined recursively to consist of functions of the form $$a(b(x,y)+c(x,y))$$, where $$a$$ is a univariate analytic function and $$b$$ and $$c$$ belong to the $$(n-1)$$-th complexity class. Such a represenation is meant to be valid for suitable germs of multi-valued holomorphic functions. A randomly chosen bivariate analytic functions will most likely have infinite analytic complexity. However, for a number of important families of special functions of mathematical physics their complexity is finite and can be computed or estimated. Using this, we introduce the notion of the analytic complexity of a binary tree, in particular, a cluster tree, and investigate its properties.

• Keith Hannabuss (University of Oxford)
T-dual noncommutative principal torus bundles
Friday, 25 September 2015 at 12:10pm in Engineering Maths Building EMG07

Since the work of Mathai and Rosenberg it is known that the T-dual of a principal torus bundle can be described as a noncommutative torus bundle. This talk will look at a simple example of two T-dual bundles both of which are noncommutative. Then it will discuss a strategy for extending this to more general noncommutative bundles.

• Hang Wang (The University of Adelaide)
Base change and K-theory
Friday, 18 September 2015 at 12:10pm in Ingkarni Wardli B17

Tempered representations of an algebraic group can be classified by K-theory of the corresponding group $$C^*$$-algebra. We use Archimedean base change between Langlands parameters of real and complex algebraic groups to compare K-theory of the corresponding $$C^*$$-algebras of groups over different number fields. This is work in progress with K.F. Chao.

• Guo Chuan Thiang (The University of Adelaide)
T-duality and bulk-boundary correspondence
Friday, 11 September 2015 at 12:10pm in Ingkarni Wardli B17

Bulk-boundary correspondences in physics can be modelled as topological boundary homomorphisms in K-theory, associated to an extension of a "bulk algebra" by a "boundary algebra". In joint work with V. Mathai, such bulk-boundary maps are shown to T-dualize into simple restriction maps in a large number of cases, generalizing what the Fourier transform does for ordinary functions. I will give examples, involving both complex and real K-theory, and explain how these results may be used to study topological phases of matter and D-brane charges in string theory.

• Michael Eastwood (The University of Adelaide)
Integrability conditions for the Grushin operators
Friday, 4 September 2015 at 12:10pm in Ingkarni Wardli B17

Fix a non-negative integer k and consider the vector fields in the plane $$X=\frac{d}{dx}$$ and $$Y=x^k \frac{d}{dy}$$. A smooth function $$f(x,y)$$ is locally constant if and only if it is annihilated by the $$k^{\text{th}}$$ Grushin operator $$f\mapsto(Xf,Yf)$$. What about the range of this operator?

• David Baraglia (The University of Adelaide)
Vanishing lattices and moduli spaces
Friday, 28 August 2015 at 12:10pm in Ingkarni Wardli B17

Vanishing lattices are symplectic analogues of root systems. As with roots systems, they admit a classification in terms of certain Dynkin diagrams (not the usual ones from Lie theory). In this talk I will discuss this classification and if there is time I will outline my work (in progress) showing that the monodromy of the $$\mathrm{SL}(n,\mathbb{C})$$ Hitchin fibration is essentially a vanishing lattice.

• Michael Murray (The University of Adelaide)
Equivariant bundle gerbes
Friday, 21 August 2015 at 12:10pm in Ingkarni Wardli B17

I will present the definitions of strong and weak group actions on a bundle gerbe and calculate the strongly equivariant class of the basic bundle gerbe on a unitary group. This is joint work with David Roberts, Danny Stevenson and Raymond Vozzo and forms part of arXiv:1506.07931.

• Melissa Tacy (The University of Adelaide)
Bilinear $$L^p$$ estimates for quasimodes
Friday, 14 August 2015 at 12:10pm in Ingkarni Wardli B17

Understanding the growth of the product of eigenfunctions $$u\cdot{}v$$ $$\Delta{}u=-\lambda^{2}u\quad{}\Delta{}v=-\mu^{2}v$$ is vital to understanding the regularity properties of non-linear PDE such as the non-linear Schrödinger equation. In this talk I will discuss some recent results that I have obtain in collaboration with Zihua Guo and Xiaolong Han which provide a full range of estimates of the form $$||uv||_{L^{p}}\leq{}G(\lambda,\mu)||u||_{L^{2}}||v||_{L^{2}}$$ where $$u$$ and $$v$$ are approximate eigenfunctions of the Laplacian. We obtain these results by re-casting the problem to a more general related semiclassical problem.

• Finnur Larusson (The University of Adelaide)
Gromov's method of convex integration and applications to minimal surfaces
Friday, 7 August 2015 at 12:10pm in Ingkarni Wardli B17

We start by considering an applied problem. You are interested in buying a used car. The price is tempting, but the car has a curious defect, so it is not clear whether you can even take it for a test drive. This problem illustrates the key idea of Gromov's method of convex integration. We introduce the method and some of its many applications, including new applications in the theory of minimal surfaces, and end with a sketch of ongoing joint work with Franc Forstneric.

• Peter Hochs (The University of Adelaide)
Quantising proper actions on Spin-c manifolds
Friday, 31 July 2015 at 11:00am in Ingkarni Wardli Room 7.15

For a proper action by a Lie group on a $$\mathrm{Spin}^{\mathrm{c}}$$ manifold (both of which may be noncompact), we study an index of deformations of the $$\mathrm{Spin}^{\mathrm{c}}$$ Dirac operator, acting on the space of spinors invariant under the group action. When applied to spinors that are square integrable transversally to orbits in a suitable sense, the kernel of this operator turns out to be finite-dimensional, under certain hypotheses of the deformation. This also allows one to show that the index has the quantisation commutes with reduction property (as proved by Meinrenken in the compact symplectic case, and by Paradan-Vergne in the compact $$\mathrm{Spin}^{\mathrm{c}}$$ case), for sufficiently large powers of the determinant line bundle. Furthermore, this result extends to $$\mathrm{Spin}^{\mathrm{c}}$$ Dirac operators twisted by vector bundles. A key ingredient of the arguments is the use of a family of inner products on the Lie algebra, depending on a point in the manifold. This is joint work with Mathai Varghese.

• Yanli Song (University of Toronto)
Dirac operators and Hamiltonian loop group action
Friday, 24 July 2015 at 12:10pm in EM 212

A definition to the geometric quantization for compact Hamiltonian G-spaces is given by Bott, defined as the index of the Spinc-Dirac operator on the manifold. In this talk, I will explain how to generalize this idea to the Hamiltonian LG-spaces. Instead of quantizing infinite-dimensional manifolds directly, we use its equivalent finite-dimensional model, the quasi-Hamiltonian G-spaces. By constructing twisted spinor bundle and twisted pre-quantum bundle on the quasi-Hamiltonian G-space, we define a Dirac operator whose index are given by positive energy representation of loop groups. A key role in the construction will be played by the algebraic cubic Dirac operator for loop algebra. If time permitted, I will also explain how to prove the quantization commutes with reduction theorem for Hamiltonian LG-spaces under this framework.

• Richard Szabo (Heriot-Watt University)
Instantons and Geometric Representation Theory
Friday, 23 July 2015 at 12:10pm in EM 212

We give an overview of the various approaches to studying supersymmetric quiver gauge theories on ALE spaces, and their conjectural connections to two-dimensional conformal field theory via AGT-type dualities. From a mathematical perspective, this is formulated as a relationship between the equivariant cohomology of certain moduli spaces of sheaves on stacks and the representation theory of infinite-dimensional Lie algebras. We introduce an orbifold compactification of the minimal resolution of the A-type toric singularity in four dimensions, and then construct a moduli space of framed sheaves which is conjecturally isomorphic to a Nakajima quiver variety. We apply this construction to derive relations between the equivariant cohomology of these moduli spaces and the representation theory of the affine Lie algebra of type A.

• Tuyen Truong (The University of Adelaide)
Some approaches toward a stronger Jacobian conjecture
Friday, 6 June 2015 at 12:10pm in Napier 144

The Jacobian conjecture states that if a polynomial self-map of $$\mathbb{C}^n$$ has invertible Jacobian, then the map has a polynomial inverse. Is it true, false or simply undecidable? In this talk I will propose a conjecture concerning general square matrices with complex coefficients, whose validity implies the Jacobian conjecture. The conjecture is checked in various cases, in particular it is true for generic matrices. Also, a heuristic argument is provided explaining why the conjecture (and thus, also the Jacobian conjecture) should be true.

• David Baraglia (The University of Adelaide)
Monodromy of the Hitchin system and components of representation varieties
Friday, 29 May 2015 at 12:10pm in Napier 144

Representations of the fundamental group of a compact Riemann surface into a reductive Lie group form a moduli space, called a representation variety. An outstanding problem in topology is to determine the number of components of these varieties. Through a deep result known as non-abelian Hodge theory, representation varieties are homeomorphic to moduli spaces of certain holomorphic objects called Higgs bundles. In this talk I will describe recent joint work with L. Schaposnik computing the monodromy of the Hitchin fibration for Higgs bundle moduli spaces. Our results give a new unified proof of the number of components of several representation varieties.

• Andree Lischewski (The University of Adelaide)
The twistor equation on Lorentzian $$\mathrm{Spin}^{\mathrm{c}}$$ manifolds
Friday, 15 May 2015 at 12:10pm in Napier 144

In this talk I consider a conformally covariant spinor field equation, called the twistor equation, which can be formulated on any Lorentzian $$\mathrm{Spin}^{\mathrm{c}}$$ manifold. Its solutions have become of importance in the study of supersymmetric field theories in recent years and were named "charged conformal Killing spinors". After a short review of conformal $$\mathrm{Spin}^{\mathrm{c}}$$ geometry in Lorentzian signature, I will briefly discuss the emergence of charged conformal Killing spinors in supergravity. I will then focus on special geometric structures related to the twistor equation and use charged conformal Killing spinors in order to establish a link between conformal and CR geometry.

• Koen van den Dungen (Australian National University)
Indefinite spectral triples and foliations of spacetime
Friday, 8 May 2015 at 12:10pm in Napier 144

Motivated by Dirac operators on Lorentzian manifolds, we propose a new framework to deal with non-symmetric and non-elliptic operators in noncommutative geometry. We provide a definition for indefinite spectral triples, which correspond bijectively with certain pairs of spectral triples. Next, we will show how a special case of indefinite spectral triples can be constructed from a family of spectral triples. In particular, this construction provides a convenient setting to study the Dirac operator on a spacetime with a foliation by spacelike hypersurfaces. This talk is based on joint work with Adam Rennie (arXiv:1503.06916).

• Mathai Varghese (The University of Adelaide)
Spherical T-duality: the non-principal case
Friday, 1 May 2015 at 12:10pm in Napier 144

Spherical T-duality is related to M-theory and was introduced in recent joint work with Bouwknegt and Evslin. I will begin by briefly reviewing the case of principal $$\mathrm{SU}(2)$$-bundles with degree 7 flux, and then focus on the non-principal case for most of the talk, ending with the relation to SUGRA/M-theory.

• Anitha Thillaisundaram (Heinrich Heine University of Düsseldorf)
Groups acting on trees
Friday, 10 April 2015 at 12:10pm in Napier 144

From a geometric point of view, branch groups are groups acting spherically transitively on a spherically homogeneous rooted tree. The applications of branch groups reach out to analysis, geometry, combinatorics, and probability. The early construction of branch groups were the Grigorchuk group and the Gupta-Sidki p-groups. Among its many claims to fame, the Grigorchuk group was the first example of a group of intermediate growth (i.e. neither polynomial nor exponential). Here we consider a generalisation of the family of Grigorchuk-Gupta-Sidki groups, and we examine the restricted occurrence of their maximal subgroups.

• Joseph Chan (University of Melbourne)
Higher rank discrete Nahm equations for $$\mathrm{SU}(N)$$ monopoles in hyperbolic space
Wednesday, 8 April 2015 at 11:10am in Engineering & Maths EM213

Braam and Austin in 1990, proved that $$\mathrm{SU}(2)$$ magnetic monopoles in hyperbolic space $$H^3$$ are the same as solutions of the discrete Nahm equations. I apply equivariant K-theory to the ADHM construction of instantons/holomorphic bundles to extend the Braam-Austin result from $$\mathrm{SU}(2)$$ to $$\mathrm{SU}(N)$$. During its evolution, the matrices of the higher rank discrete Nahm equations jump in dimensions and this behaviour has not been observed in discrete evolution equations before. A secondary result is that the monopole field at the boundary of $$H^3$$ determines the monopole.

• Guo Chuan Thiang (The University of Adelaide)
Topological matter and its K-theory
Thursday, 2 April 2015 at 11:10am in Ingkarni Wardli B18

The notion of fundamental particles, as well as phases of condensed matter, evolves as new mathematical tools become available to the physicist. I will explain how K-theory provides a powerful language for describing quantum mechanical symmetries, homotopies of their realisations, and topological insulators. Real K-theory is crucial in this framework, and its rich structure is still being explored both physically and mathematically.

• David Roberts (The University of Adelaide)
Higher homogeneous bundles
Friday, 27 March 2015 at 12:10pm in Napier 144

Historically, homogeneous bundles were among the first examples of principal bundles. This talk will cover a general method that gives rise to many homogeneous principal 2-bundles.

• Pawel Nurowski (University of Warsaw)
Singular Pfaffian systems in dimension 6
Friday, 20 March 2015 at 12:10pm in Napier 144

We consider a pair of rank 3 distributions in dimension 6 with some remarkable properties. They define an analog of the celebrated nearly-Kahler structure on the 6 sphere, with the exceptional simple Lie group $$\mathrm{G}_2$$ as a group of symmetries. In our case the metric associated with the structure is pseudo-Riemannian, of split signature. The 6 manifold has a 5-dimensional boundary with interesting induced geometry. This structure on the boundary has no analog in the Riemannian case.

• Ilya Kossovskiy (University of Vienna)
On the analyticity of CR-diffeomorphisms
Friday, 13 March 2015 at 12:10pm in Engineering North N132

One of the fundamental objects in several complex variables is CR-mappings. CR-mappings naturally occur in complex analysis as boundary values of mappings between domains, and as restrictions of holomorphic mappings onto real submanifolds. It was already observed by Cartan that smooth CR-diffeomorphisms between CR-submanifolds in $$\mathbb{C}^N$$ tend to be very regular, i.e., they are restrictions of holomorphic maps. However, in general smooth CR-mappings form a more restrictive class of mappings. Thus, since the inception of CR-geometry, the following general question has been of fundamental importance for the field: Are CR-equivalent real-analytic CR-structures also equivalent holomorphically? In joint work with Lamel, we answer this question in the negative, in any positive CR-dimension and CR-codimension. Our construction is based on a recent dynamical technique in CR-geometry, developed in my earlier work with Shafikov.

• Jack Hall (Australian National University)
Tannaka duality for stacks
Friday, 6 March 2015 at 12:10pm in Ingkarni Wardli B19

Traditionally, Tannaka duality is used to reconstruct a group from its representations. I will describe a reformulation of this duality for stacks, which is due to Lurie, and briefly touch on some applications.

• Vicente Cortés (University of Hamburg)
Boundary behaviour of Hitchin and hypo flows with left-invariant initial data
Friday, 27 February 2015 at 12:10pm in Ingkarni Wardli B20

Hitchin and hypo flows constitute a system of first order pdes for the construction of Ricci-flat Riemannian mertrics of special holonomy in dimensions 6, 7 and 8. Assuming that the initial geometric structure is left-invariant, we study whether the resulting Ricci-flat manifolds can be extended in a natural way to complete Ricci-flat manifolds. This talk is based on joint work with Florin Belgun, Marco Freibert and Oliver Goertsches, see arXiv:1405.1866.

• Tsuyoshi Kato (Kyoto University)
Nonlinear analysis over infinite dimensional spaces and its applications
Friday, 6 February 2015 at 12:10pm in Ingkarni Wardli B20

In this talk we develop moduli theory of holomorphic curves over infinite dimensional manifolds consisted by sequences of almost Kähler manifolds. Under the assumption of high symmetry, we verify that many mechanisms of the standard moduli theory over closed symplectic manifolds also work over these infinite dimensional spaces. As an application, we study deformation theory of discrete groups acting on trees. There is a canonical way, up to conjugacy to embed such groups into the automorphism group over the infinite projective space. We verify that for some class of Hamiltonian functions, the deformed groups must be always asymptotically infinite.