# Geometry

Research in pure mathematics at the University of Adelaide is about geometry and its applications. The combined focus on both the most fundamental questions in geometry and their relevance to other areas makes our research group unique in Australian mathematics.

Many pure mathematics researchers at the University of Adelaide work in various areas in differential geometry. These include complex, symplectic, Riemannian and pseudo-Riemannian, and Spin geometry. Researchers also explore links to other kinds of geometry, like algebraic geometry, noncommutative geometry and finite geometry.

## Applications

Physics is an important area of applications of the geometry research at the University of Adelaide. Some of the areas impacted by our work are string theory, topological phases of matter, Einstein's general relativity theory, and the relation between classical and quantum mechanics.

Researchers also investigate the role of geometry within mathematics. If a mathematical problem can be viewed from a geometric angle, we can use geometric intuition to solve it in unexpected ways. For example, this applies to analysis, cryptography and representation theory.

The University of Adelaide is home to some of the world's top researchers in these areas. In the most recent Excellence in Research for Australia, pure mathematics at the University of Adelaide received the highest possible score of 5, ranking it well above world standard.

In recent years, our HDR students have received numerous awards, such as Dean's commendations, scholarships for further study, and the Bernhard Neumann Prize for the best student presentation at the meeting of the Australian Mathematics Society.

David Baraglia | Differential geometry, Higgs bundles and their moduli spaces, generalised complex geometry, mathematical physics |

Susan Barwick | Finite projective geometry |

Nicholas Buchdahl | Complex analysis, complex analytic and algebraic geometry, differential geometry, Gauge theory, mathematical physics |

Michael Eastwood | Differential geometry, Integral geometry, Several complex variables, Lie groups, and Invariant theory. |

Wolfgang Globke | Lie theory, discrete transformation groups, pseudo-Riemannian geometry |

Peter Hochs | Geometric quantisation (in the noncompact setting), operator algebras, index theory, differential geometry and geometric analysis, Lie theory, links between K-theory, K-homology and representation theory |

Wen-Ai Jackson | Finite projective geometry |

Finnur Larusson | Complex analysis, complex analytic and algebraic geometry, applied homotopy theory |

Thomas Leistner | Differential geometry, Lorentzian and semi-Riemannian geometry, conformal geometry, holonomy theory and Lie groups |

Michael Murray | Differential geometry, Gauge theory, mathematical physics, mathematics of string theory |

Danny Stevenson | Algebraic topology: K-theory, abstract homotopy theory and higher category theory, higher geometric structures: stacks and gerbes |

Guo Chuan Thiang | Applications of K-theory, operator algebras, and noncommutative geometry to mathematical physics, especially T-duality in string theory and topological matter. |

Tuyen Truong | Algebraic and analytic geometry, Oka manifolds |

Mathai Varghese | Differential geometry; Index theory; Secondary Index invariants; K-theory; Geometric quantisation; noncommutative geometry; classical phase spaces; Mathematics of Quantum theory; Mathematics of String theory and T-duality. |

Raymond Vozzo | Differential geometry, gerbes, loop groups. |

Hang Wang | Operator algebras and their applications to topology, geometry and representation theory. In particular, non-commutative geometry, index theory of elliptic operators and KK-theory. |

## Research Areas

- Differential geometry - Possible supervisors: Dr Thomas Leistner, Professor Michael Eastwood, Dr David Baraglia
- Finite geometry - Possible supervisors: Dr Sue Barwick
- Mathematical physics - Possible supervisors: Elder Professor Mathai Varghese, Dr David Baraglia, Professor Michael Murray
- Index theory, K-theory and noncommutative geometry - Possible supervisors: Elder Professor Mathai Varghese, Dr David Baraglia, Dr Peter Hochs.
- Complex geometry and complex analysis - Possible supervisors: Professor Finnur Larusson, Associate Professor Nicholas Buchdahl, Professor Michael Eastwood, Dr David Baraglia
- Lie groups, representations and geometry - Possible supervisors: Professor Michael Eastwood, Dr Thomas Leistner, Dr David Baraglia, Professor Michael Murray, Elder Professor Mathai Varghese, Dr Peter Hochs
- Higher geometry and category theory - Possible supervisors: Dr Danny Stevenson, Professor Michael Murray

## Possible PhD Projects

- Dr Thomas Leistner
I am happy to supervise topics in differential geometry, Lie groups and Lie algebras, and mathematical physics. My publications and the slides of talks about my research can be found on my web-page.

The subject of differential geometry are curved spaces (and space-times), known as manifolds. Manifolds are topological spaces for which the standard procedures of analysis (differentiation and integration) can be performed.

They usually come equipped with additional geometric structures that are motivated by mathematical problems or physical applications. A key example are semi-Riemannian manifolds, which are equipped with a metric tensor that defines geometric quantities such as lengths, angles, volume, shortest curves (geodesics) and parallel translation.

The main subject of my research are holonomy groups. The holonomy group of a semi-Riemannian manifold (or another geometric structure) is defined as the group of parallel translations along closed curves. It captures a lot of information about how curved a manifold is and whether it admits additional geometric structures such as a complex structure or a quaternionic structure. Since holonomy groups are algebraic objects, they allow to attack difficult geometric and analytic problems by powerful algebraic tools such as group theory or representation theory. A brief introduction on holonomy groups can be found here.

Possible topics, that are all related to holonomy groups, are the following:

### Classification of holonomy groups

The mentioned algebraic tools can be used in many situation to classify holonomy groups, i.e., to obtain a list of groups that can occur as holonomy group of a certain geometric structure. A famous result of this kind is Berger’s classification of irreducible holonomy groups of Riemannian manifolds. In many situations however, a classification is not yet known but very desirable. Possible projects here are the extension of know classification results holonomy groups of pseudo-Riemannian manifolds but also for other geometric structures for example in conformal geometry. A overview on some classification results can be found here

### Geodesic completeness of compact space-times

The aim of this project is to study the behavior of geodesics (shortest curves) in compact space-times. Whereas geodesics in compact Riemannian manifolds are defined for all times, geodesics in compact Lorentzian manifolds, or space-times, may have singularities in finite time. The aim of the project is to study geodesic completeness for compact Lorentzian manifolds with special holonomy and to extend results in

### Symmetries and holonomy

Symmetries are transformations of a manifold that leave the geometric structure, for example a semi-Riemannian metric, invariant. In many cases there is a close relation between the symmetry group and the holonomy group of manifolds. Projects in this area could explore this relation for specific examples of Lorentzian manifolds, for example one and two.

### Cauchy problems for Lorentzian manifolds

To solve a geometric Cauchy problem means to construct (by solving a PDE) a spacetime with a certain geometry from a spacelike slice that satisfies certain initial conditions. This is a well-known approach for the Einstein equations but can also be performed for space-times with special holonomy. There are many open problems, that could be subject of a project, about how the specific geometry of the slice evolves under a flow that arises as solution of a Cauchy problem. Some of them are described here:

### Conformal geometry

In conformal geometry the metric tensor is only defined up to scaling, and hence not lengths but angles are the fundamental geometric quantities. Invariant descriptions of conformal structures are more involved than for metric structures: there is the description in terms of conformal tractor calculus and that of the conformal ambient metric construction. The conformal ambient metric generalises the realisation of the conformal sphere as section of the light-cone. To construct the ambient metric for specific conformal structures and study its relation to conformal tractor calculus as well as to certain cone constructions in Riemannian geometry could be part of a project in conformal geometry. An example of such a relation is given here.

- Dr Danny Stevenson
My current research interests lie at the intersection of the fields of homotopy theory and category theory; in particular I have an interest in the subject of ∞-categories.

A category is an algebraic structure consisting of objects, together with morphisms which connect various objects. There is an operation of composition of morphisms which shares similar properties to the operation of composition of functions between sets. In fact, one of the archetypal examples of a category is the category Set whose objects are sets, and whose morphisms are the various functions between sets.

An ∞-category is a generalization of the concept of category in which there are objects and morphisms, but there are also 2-morphisms between morphisms, and 3-morphisms between 2-morphisms and so on – in other words there is a notion of i-morphism for every i>1. There are lots of different models of ∞-categories; these models are typically encoded in terms of combinatorial structures known in algebraic topology as simplicial sets.

There is a lot of current interest in ∞-categories; they have applications in homotopy theory, the topology of manifolds, algebraic K-theory, and various areas of mathematical physics amongst other places.

Possible PhD topics in this area include the development of the theory of (∞,2)-categories and developing various theories of fibrations (for instance cartesian and co-cartesian fibrations) in the context of bisimplicial sets.

- Dr David Baraglia
I am happy to discuss potential projects for supervision in any of the following areas: Differential geometry, complex analysis, algebraic geometry, topology, mathematical physics.

### Higgs bundles and ramification

A Riemann surface is a 2-dimensional surface which is covered by coordinate charts, such that the maps between coordinate charts are complex analytic (they satisfy the Cauchy-Riemann equations). An example is the Riemann sphere S, which is the complex plane C together with a point at infinity. One can do complex analysis on Riemann surfaces and many results familiar from complex analysis in the plane have counterparts for Riemann surfaces.

This project is concerned with the study of certain complex analytic objects on Riemann surfaces called Higgs bundles. Higgs bundles arise as special solutions of the Yang-Mills equations in physics and are important in many branches of mathematics, due in part to their substantial role in the Geometric Langlands program, a major active area of current research in geometry. The aim of this project is to extend various aspects of the study of Higgs bundles to the case where the Higgs bundle admits certain singularities, known as ramification. For instance, associated to a Higgs bundle is an auxiliary Riemann surface, called a spectral curve. When the Higgs bundle is ramified, the spectral curve acquires certain singularities. In this project, you will investigate the relationship between ramification of Higgs bundles and singularities of their spectral curves and develop techniques for dealing with such singularities.

### Geometry and arithmetic of character varieties

An algebraic variety is the set of solutions to a system of polynomial equations. For example, the set of solutions to the equation:

y^2 - x^3 + 5x = 0.

Algebraic varieties can be considered over any field, such as the field of complex numbers or a finite field such as the integers modulo a prime number. The Weil conjectures, a set of famous conjectures proposed by André Weil and settled by Dwork, Grothendieck and Deligne, establishes a deep relationship between the topology (or "shape") of algebraic varieties over the complex numbers and the number of points of the corresponding variety considered over finite fields. So counting the number of solutions to a system of polynomial equations over a finite field tells us about the topology of the same system of equations over the complex numbers!

The aim of this project is to apply the Weil conjectures to an important class of varieties known as character varieties. The character variety X(n,G,k) is the space of n-dimensional representations of a group G over a field k. In this project you will use tools from representation theory to count the number of points of X(n,G,F_q), where F_q is a finite field of order q and use this to understand the topology of X(n,G,C), where C is the complex numbers.

- Professor Michael Murray
Currently my research centres around the area of pure mathematics known as higher geometry. You can find my publications.

Higher geometry is a combination of differential geometry, topology and category theory. Within that broad area I am mostly interested in the theory of bundle gerbes which, along with my students and colleagues, I have been developing since the middle of the 1990s. Bundle gerbes are particular geometric objects related to three-dimensional topology. They have applications within mathematics and also to string theory in mathematical physics. An explanation of what a bundle gerbe is which should be accessible to a second or third year undergraduate can be found.

With Danny Stevenson I currently have two MPhil students working in this general area: Parsa Kavkani and Kimberley Becker and other past students in this area include Vincent Schlegel (MPhil), Raymond Vozzo (PhD), David Roberts (PhD), Stuart Johnson (PhD) and Danny Stevenson (PhD).

What specific projects I would suggest for an MPhil or PhD depend on what I am currently working on. Please contact me if you would like to discuss them.

- Professor Finnur Larusson
I do research in complex geometry and complex analysis. Recent work of mine has also involved topology, algebraic geometry, group actions, and the theory of minimal surfaces. Here is a brief description of one PhD project, among other possible projects, that I would be happy to offer to a suitably prepared student.

There is a strong connection between complex analysis and homotopy theory in certain geometric settings. The prototypical theorem of this kind is due to Gromov and says that if X is a Stein manifold and Y is an elliptic manifold — these are two important classes of complex manifolds — then every continuous map from X to Y can be deformed to a holomorphic map. More is true. The inclusion of the space of holomorphic maps from X to Y into the space of continuous maps is what is called a weak homotopy equivalence, meaning that the two spaces have, in a sense, the same “rough shape”. Over the past 15 years or so, the interaction between complex analysis and homotopy theory has evolved into a subfield in its own right, called Oka theory. It has turned out to have fruitful connections with several other areas of mathematics. For a brief survey of Oka theory, see my article in the Notices of the American Mathematical Society.

Equivariant Oka theory takes into account symmetries of X and Y. If we have a complex Lie group acting on X and Y, we might like to restrict our attention to equivariant maps from X to Y, that is, maps that respect the action. In 2016, my collaborators Frank Kutzschebauch and Gerald Schwarz and I proved an equivariant version of Gromov’s theorem.

- Elder Professor Mathai Varghese
Aspects of Geometric Group Theory, braids and knots Chiral de Rham complex and vertex operator algebras, Geometry and topology of loop spaces of manifolds, Symplectic geometry and geometric quantization, K-theory and twisted analogues, Atiyah-Singer index theory of elliptic operators and generalizations, Spectral theory of elliptic operators, Noncommutative geometry, Mathematics of the fractional quantum Hall effect, Mathematics of String Theory and T-duality. Positive scalar curvature and topological obstructions

- Dr Peter Hochs
My favourite mathematical results are ones that link different areas in maths to each other. I mainly work in index theory, which is the study of relations between geometry, topology and analysis via differential equations on geometric spaces. In many cases, index theory can also be used to related these fields to group theory and noncommutative geometry. Below are two possible Ph.D. projects I am happy to supervise, but I am open to discussing others in this area.

### The Kasparov product in index theory

Kasparov developed KK-theory in the 1980s. It is a simultaneous generalisation of K-homology and K-theory, two tools used to study topological spaces and their "noncommutative analogues", C*-algebras. The most powerful ingredient of KK-theory is the Kasparov product. This generalises many earlier constructions in geometry, topology and analysis. For example, it can be used to describe how differential operators on a space decompose into equations on the parts the space is made up of. Another important result involving the Kasparov product is a vast generalisation of the Atiyah-Singer index theorem, which started the field of index theory in the 1960s.

In this project, we focus on applications of the Kasparov product to index theory. The initial goals are to

• state and prove a generalisation of the index theorem for families of differential operators

• formulate and generalise an index of deformed Dirac operators that has been used successfully in representation theory

• study indices on noncompact orbifolds, which are spaces with singularites, and their relations to indices of transversally elliptic operators.

### Geometric decompositions of representations

A representation of a group describes how the group acts on a vector space. Representations for example describe symmetries in quantum mechanics, but they are used in many other areas as well. For a particularly relevant class of groups, semisimple Lie groups, representation theory was developed by Harish Chandra and many others since the 1950s. There are representations of such groups of different types, such as discrete series, tempered, unitary and admissible representations, where each of these types is more general the preceding one.

Index theory can be used to relate representation theory to geometry. One application of this idea is the study of restrictions of irreducible representations to subgroups. If G is a group and H is a subgroup, then a representation of G is irreducible if it does not break up into smaller ones. We can restrict an irreducible representation of G to H, but then it becomes reducible in general. It is an important question how that restriction breaks up into smaller representations. For discrete series representations, and H a large enough compact subgroup of G, this problem was solved geometrically by Paradan. He used index theory to do this, and the idea from physics that "quantisation commutes with reduction". That idea is a relation between the roles of symmetry in classical and quantum mechanics. Paradan's work was generalised to tempered representations by Song, Yu and myself. In this project, we will look into generalising such results and finding new applications.