Number theory studies the algebra of the integers, from factorisation theory to finding integer solutions to polynomial equations. It turns out that the algebra of multiplication arises naturally in a plethora of different contexts, such as symmetries and differential operators. The abstract study of these has led to the notions of groups and rings which are the foundation of modern algebra.

The Algebra and Number Theory group has an extremely broad range of research interests. Within Algebra, their research spans a wide spectrum including homological algebra, group theory, quantum groups, representation theory and noncommutative algebra. Members of the group are interested in classical, analytical and geometric number theory.

#### Group Member

- David Angell
- Arnaud Brothier
- Peter Brown
- Daniel Chan
- Changhao Chen
- Peter Donovan
- Jie Du
- James Franklin
- Pinhas Grossman
- David Harvey
- Randell Heyman
- Mike Hirschhorn
- Bryce Kerr
- Chi Mak
- Alina Ostafe
- John Roberts
- Igor Shparlinski
- Tim Trudgian (UNSW Canberra)
- Mircea Voineagu
- Norman Wildberger
- Liangyi Zhao

#### Research Interests

**David Angell **is interested in number theory and combinatorics, particularly continued fractions, irrationality and transcendence. A selection of extension articles for secondary students can be found at his personal homepage. The image at the top of the page illustrates an exponential sum involving the function f(n)=n/dd+n^2/mm+n^3/yy, where dd.mm.19yy is David's birthday.

**Arnaud Brothier** is interested primarily on operator algebras, a cross-field of many areas of mathematics and physics, in particular von Neumann algebras, Jones' Subfactor Theory and their interactions with group theory, ergodic theory, representation theory and conformal field theory.

**Peter Brown **has been working in the area of number theory, specifically on elliptic curves and has recently been looking at some problems in analytic number theory.

**Daniel Chan **is interested in various noncommutative algebras arising from noncommutative algebraic geometry. These include orders, Sklyanin algebras, Clifford algebras and twisted co-ordinate rings. He has studied noncommutative Grothendieck duality theory and the McKay correspondence.

**Changhao Chen **is interested in additive combinatorics, exponential sums and character sums, Fourier analysis, random graphs, and random fractals.

**Peter Donovan**, now semi-retired, has publications in algebraic geometry (localisation at fixed points). algebraic topology (related to geometrical physics), representation theory (including the first non-trivial progress towards a key finiteness conjecture in the modular representation theory of finite groups), homological algebra and the insecurity of Japanese naval ciphers in WW2. Currently his interests are returning to modular representation theory and geometrical physics.

**Jie Du**'s interests lie in the representation theories on algebraic and quantum groups, finite groups of Lie type, finite dimensional algebras, and related topics. His recent work has concentrated mainly on the Ringel-Hall approach to quantum groups and q-Schur and generalised q-Schur algebras and their associated monomial and canonical basis theory. He is also interested in combinatorics arising from generalised symmetric groups, Kazhdan-Lusztig cells and representations of finite algebras.

**James Franklin**'s mathematical interests lie in the evaluation of extreme risks. He is also working on structuralist philosphy of mathematics. His most recent book was *What Science Knows*, on knowledge in science and mathematics, while previous ones were on Australian philosophy and the history of probability and evidence evaluation.

**Pinhas Grossman** is interested in fusion categories and planar algebras. He is particularly interested in the representation theory of fusion categories coming from von Neumann algebras.

**David Harvey**'s research interests lie in the areas of computational number theory, polynomial and integer arithmetic, and arithmetic geometry.

**Bryce Kerr **is interested in additive combinatorics, exponential sums and character sums.

**Mike Hirschhorn **studies applications of q-series to problems in additive number theory. A greater part of his work is bound up in elucidating results due to Ramanujan.

**Chi Mak** is interested in Coxeter groups, complex reflection groups and their Hecke algebras.

**Alina Ostafe** is interested in algebra and number theory, particularly in algebraic dynamical systems, polynomials and rational functions over finite fields and their applications to pseudorandom number generators and cryptography. In her research, she uses various tools of analytic number theory (exponential and character sums, additive combinatorics) and commutative algebra (discriminants, resultants, Hilbert’s Nullstellensatz).

**John Roberts**' research is in dynamical systems (sometimes popularly termed "Chaos Theory"), which seeks to understand how systems change with time and how this evolution can be understood, classified and predicted. This area of research is an exciting interdisciplinary field which relates to, and uses ideas from pure and applied mathematics, physics and computer science. His current work focuses on two broad areas: the study of integrable systems (ordered dynamics based on rotations) and the study of arithmetic dynamics (a hybrid of dynamical systems with number theory).

**Min Sha **is interested in number theory and its applications. Currently, his main interest lies in elliptic curves and linear recurrence sequences, and their intersections with related fields.

**Igor Shparlinski **has a broad range of research interests, from number theory (such as exponential and character sums, finite fields, smooth numbers, linear recurrence sequences), cryptography (especially elliptic curve cryptography and pseudorandom number generators) and computational aspects including algorithm design, computational complexity and quantum cryptography.

**Mircea Voineagu**'s research lie at the intersection of algebraic geometry, algebraic topology and homological algebra. In particular, he is interested in motivic cohomology, a new cohomological theory for algebraic varieties introduced by V. Voevodsky, and in the K-theory of algebraic varieties.

**Norman Wildberger**'s interest in the theory of hypergroups has led him into description of various finite hypergroups. These have rather interesting algebraic properties and may be applied in the study of diophantine equations.