Algebraic geometry has a long history. Its origins are the geometry of the set of solutions to systems of polynomial equations in several variables. While such down-to-earth problems remain the focus of algebraic geometry today, the language and techniques have changed enormously.

One of the advantages of algebraic geometry over other types of geometries is its ability to handle singularities and its computability. This is because its roots lie in algebra, as opposed to differential equations.

The Algebraic Geometry group at the University of Melbourne is broad, with interests in number theory, representation theory, K-theory, moduli theory and elsewhere.

### Algebraic geometry is part of the Pure Mathematics Research Group

## Academic Staff

### Prof Christian Haesemeyer

##### Professor

Motives, algebraic cycles, K-theory

### A/Prof Jack Hall

##### Associate Professor and ARC Future Fellow

Algebraic stacks, moduli spaces, formal and analytic geometry, deformation theory

### Dr Johanna Knapp

##### Senior Lecturer and ARC Future Fellow

String theory, algebraic geometry, Gauge theory

### Dr Peter McNamara

##### Senior Lecturer

Representation theory

### Dr Daniel Murfet

##### Lecturer

Mathematical logic, algebraic geometry, topological field theories

### Prof Paul Norbury

##### Professor

Gauge theory, mathematical physics, algebraic geometry, moduli spaces

### Prof Arun Ram

##### Professor

Combinatorics, Representation Theory, algebraic geometry, number theory,

algebraic topology and mathematical physics

### Prof Kari Vilonen

##### Professor and ARC Laureate Fellow

Representation theory, algebraic geometry, algebraic analysis

### Dr Chenyan Wu

##### Senior Lecturer

Representation theory, algebraic geometry, algebraic number theory

## Postdocs

### Dr Scott Mullane

##### ARC DECRA Fellow (2023 – present)

Algebraic geometry and the geometry of moduli spaces

### Dr Dougal Davis

##### Research Fellow (2022 – present)

Algebraic geometry and representation theory

### Dr Lance Gurney

##### Research Fellow (2021 – present)

Number theory, arithmetic geometry, cohomology

### Dr June Park

##### Research Fellow (2022 – present)

Arithmetic theory of algebraic varieties and moduli spaces

## Graduate Researchers

Oliver Li (Hall)

Fei Peng (Hall)

## Affiliates

### A/Prof Alex Ghitza

##### Associate Professor

computational number theory, Galois representations, Shimura varieties, Automorphic forms

### Dr Yaping Yang

##### Senior Lecturer

Geometric representation theory, Quantum groups

### Dr Gufang Zhao

##### Senior Lecturer

Algebra, Geometry & Topology

## Previous Members

#### Dr Mehdi Tavakol

##### Research Fellow (2018-2021)

## Moduli and Stacks Learning Seminar, 2023

### Semester 2, 2023: Alper–Halpern-Leinster–Heinloth

#### Organizer: Fei Peng

**Time:** Wednesdays 2-3pm, Peter Hall 162

The topic of the seminar is to work through Existence of moduli spaces for algebraic stacks.

### Semester 1, 2023: The Picard Stack

#### Organizer: Jack Hall and Fei Peng

**Time:** Tuesdays, 3:15-4:15pm, Peter Hall 162

The topic of this seminar is to prove that the Picard stack and functor are algebraic.

## Arithmetic Geometry Seminar, 2021

If you would like to attend or be added to the mailing list please email lance.gurney@unimelb.edu.au.

### Semester 2, 2021: oo-categories

#### Organiser: Lance Gurney

**Time:** Fridays 5-6pm.

The topic of the seminar is to work through the foundations of oo-categories.

### Semester 1, 2021: Prismatic Cohomology

#### Organisers: Lance Gurney, Jack Hall, Christian Haesemeyer

**Time: **Fridays 3:15pm-5:00pm

The topic of the seminar is the recent prismatic cohomology of Bhatt-Scholze and its interpretation via stacks due to Drinfeld here and here. [At some point this paper by Kisin will probably pop-up].

The talks will hopefully have notes posted afterwards. A more detailed reference for some things will be found in the evolving notes on prisms (draft) which also covers the case of DVRs other than **Z*** _{p}*.

21 May **Jack Hall** *Lefschetz theorems via localization*

I will discuss a new approach to Lefschetz style hyperplane theorems using localizations of triangulated categories. The approach is also applicable to the Fargues–Fontaine curve, an object of recent interest in arithmetic geometry.

14 May **Lance Gurney** *Prismatization* Notes

The aim is to define prismatization and (time permitting) prismatic cohomology.

7 May **Lance Gurney** *Some computations* Notes

I gave an explicit description of the divisor in Sigma and of fibre products over Sigma.

30 April **Lance Gurney ***What is a prism? (continued)* Notes

24 April **Lance Gurney ***What is a prism?* Notes

I will begin with some motivational remarks about prismatic cohomology. After this I’ll explain the definition of a prism and of Drinfeld’s stack $\Sigma$. Time permitting some associated analogies from the theory of curves over finite fields (and shtukas).

**Previous Seminars**

Please contact a member of our group if you are interested in an MSc project in Algebraic Geometry.

## Relevant Coursework

### Semester 1

MAST90136 Algebraic Number Theory (even years)

MAST90023 Algebraic Topology (even years)

### Semester 2

MAST90068 Groups, Categories & Homological Algebra (even years)

MAST90097 Algebraic Geometry (even years)

MAST90056 Riemann Surfaces and Complex Analysis (odd years)

MAST90017 Representation theory (odd years)

## Current MSc Students

Adam Monteleone (Hall-Davis)

Matthew Bowkett (Hall)

## Previous MSc Students

Bowen Hafey (Hall, 2022)

Fei Peng (Hall, 2022)

### FT210100405

Integral transforms and moduli theory (Hall)

### DP210103397

Moduli, invariants, and algebraisation (Gepner, Haesemeyer, Hall, Neeman)

### DP180103891

Frobenius Manifolds From A Geometrical And Categorical Viewpoint (Murfet, Norbury)

### DP180101445

Real Groups, Hodge Theory, And The Langlands Program (Vilonen)

### DP170102328

Algebraic Invariants Of Singularities (Haesemeyer)