CARMA Research Group
Number Theory, Algorithms and Discrete Mathematics
Leader: Florian Breuer
This group covers a wide range of research interests from number theory, combinatorics and theoretical computer science, and in particular the interplay between these fields as well as links to analysis and algebraic geometry.
Topics of interest include: Diophantine analysis and Mahler functions; the arithmetic of global fields including elliptic curves, Drinfeld modules and associated modular forms; special integer sequences and special values of analytic functions; Hadamard matrices; combinatorics, enumeration and the probabilistic method; graph theory, optimal networks and other discrete structures.
There is a strong focus on computational aspects of such topics, including experimental mathematics, visualisation, computational number theory and the analysis of algorithms.
Potential applications of our work range from coding theory and cryptography through group theory, counting points on algebraic varieties to computer networks and even theoretical physics.
Members of this research group:
- Brian Alspach
- David Bailey
- Ljiljana Brankovic
- Richard Brent
- Florian Breuer
- Stephan Chalup
- Tony Guttman
- Yuqing Lin
- Jim MacDougall
- Andrew Mattingly
- Judy-anne Osborn
- Joe Ryan
- Matt Skerrit
SR202, SR Building
"Generalised Polygons and their Symmetries"
Generalised polygons were first introduced by Jacque Tits in 1959, in the context of studying geometric realisations of the finite simple groups of Lie type. Thus, the study of their symmetry groups and symmetry properties is a rich area of research. My work has focused on studying the point-primitive quadrangles. In my talk I will describe a computer program for testing whether a particular group can act point-primitively on a generalised quadrangle and its application to analysing the almost simple sporadic groups. My work on this program motivated the discovery of a new result dubbed the Line Orbit Lemma, which in turn inspired the conjecturing of the Hemisystem Conjecture, both of which could prove very useful in the analysis of point-primitive quadrangles.
LSTH, Life Sciences Lecture Theatre
"Transcendence and dynamics"
Dr Holly Krieger
Many interesting objects in the study of the dynamics of complex algebraic varieties are known or conjectured to be transcendental, such as the uniformizing map describing the (complement of a) Julia set, or the Feigenbaum constant. We will discuss various connections between transcendence theory and complex dynamics, focusing on recent developments using transcendence theory to describe the intersection of orbits in algebraic varieties, and the realization of transcendental numbers as measures of dynamical complexity for certain families of maps.