# Closure of the Exponential Embedding of the Nilpotent Cone

### Summary

Algebra; representation theory; algebraic groups; sheaves; geometry

### Supervisor(s)

### Research Location

School of Mathematics and Statistics

### Program Type

PHD

### Synopsis

Given a reductive algebraic group over the complex numbers, two of the most important varieties associated to it are its nilpotent cone (the set of nilpotent elements in its Lie algebra) and its affine Grassmannian. In the case of the general linear group, the nilpotent cone just consists of all nilpotent n by n matrices, while the affine Grassmannian consists of all full-rank lattices in an n-dimensional vector space over the field of Laurent series. There is an embedding of the nilpotent cone in the affine Grassmannian using the exponential map. The aim of this project is to describe the closure of the image of this embedding, which is a projective variety on which the group acts: in particular, to find defining equations for this variety, parametrize the orbits, and investigate the singularities of the orbit closures.

### Additional Information

A good Honours degree (or equivalent) majoring in some algebraic area of pure mathematics is essential. Prior knowledge of group representation theory and algebraic geometry is highly desirable.

The School of Mathematics and Statistics has a large and active Pure Mathematics research group including (as of 2017) more than twenty ongoing academic and research staff members, more than ten fixed-term research staff members, and more than twenty HDR students. Seminar series in Algebra, Computational Algebra, Geometry and Topology, and Partial Differential Equations showcase the research of the group and its many visitors.

In addition to the academic requirements set out in the Science Postgraduate Handbook, you may be required to satisfy a number of inherent requirements to complete this degree. Example of inherent requirement may include:

- Confidential disclosure and registration of a disability that may hinder your performance in your degree;
- Confidential disclosure of a pre-existing or current medical condition that may hinder your performance in your degree (e.g. heart disease, pace-maker, significant immune suppression, diabetes, vertigo, etc.);
- Ability to perform independently and/or with minimal supervision;
- Ability to undertake certain physical tasks (e.g. heavy lifting);
- Ability to undertake observatory, sensory and communication tasks;
- Ability to spend time at remote sites (e.g. One Tree Island, Narrabri and Camden);
- Ability to work in confined spaces or at heights;
- Ability to operate heavy machinery (e.g. farming equipment);
- Hold or acquire an Australian driver’s licence;
- Hold a current scuba diving license;
- Hold a current Working with Children Check;
- Meet initial and ongoing immunisation requirements (e.g. Q-Fever, Vaccinia virus, Hepatitis, etc.)

## Want to find out more?

Contact us to find out whatâ€™s involved in applying for a PhD. Domestic students and International students

Contact Research Expert to find out more about **participating in this opportunity**.

**Browse for other opportunities** within the School of Mathematics and Statistics .

### Keywords

Algebra; representation theory; algebraic groups; sheaves; geometry

### Opportunity ID

The opportunity ID for this research opportunity is: **2271**

### Other opportunities with Professor Anthony Henderson