CARMA Research Group

Number Theory, Algorithms and Discrete Mathematics

About us

Leader: Ljiljana Brankovic and Richard Brent

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.

People

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

Activities

  • Symmetry in Newcastle

    6:30 pm, Monday, 8th Mar 2021
    , Online

    Schedule (Zoom):

    18.30-19.30: Charlotte Hoffmann
    19.30-20.00: Break
    20.00-21.00: David Kielak

    "Short words of high imprimitivity rank yield hyperbolic one-relator groups"
       — Ms Charlotte Hoffmann

    Abstract:
    It is a long standing question whether a group of type $F$ that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type $F$ and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than $2$, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most $17$. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I’ll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.
    "Recognising surface groups"
       — A/Prof David Kielak

    Abstract:
    I will address two problems about recognising surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension two. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognising surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan.