About us

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.

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

  10:00 am, Friday, 6th Mar 2020
  (Location to be decided)

  Schedule:
  
  10.00-11.00: Waltraud Lederle
  11.00-11.30: Morning Tea
  11.30-12.30: Mark Pengitore
  12.30-14.00: Lunch
  14.00-15.00: Jeroen Schillewaert
  15.00-15.30: Afternoon Tea
  15.30-16.00: François Thilmany

  "Fixed points for group actions on $2$-dimensional affine buildings"
     — Senior Lecturer Jeroen Schillewaert
  
  Abstract:
  We prove a local-to-global result for fixed points of groups acting on $2$-dimensional affine buildings (possibly non-discrete, and not of type $\tilde{G}_{2}$). In the discrete case, our theorem establishes two conjectures by Marquis. (joint work with Koen Struyve and Anne Thomas)
  "Lattices of minimal covolume in $\mathrm{SL}_n$"
     — Dr François Thilmany
  
  Abstract:
  A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\mathrm{SL}_2(\mathbb{R})$. In general, given a semisimple Lie group $G$ over some local field $F$, one may ask which lattices in $G$ attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel's result, Lubotzky determined that a lattice of minimal covolume in $\mathrm{SL}_2(F)$ with $F=\mathbb{F}_q((t))$ is given by the so-called characteristic $p$ modular group $\mathrm{SL}_2(\mathbb{F}_q[1/t])$. He noted that, in contrast with Siegel’s lattice, the quotient by $\mathrm{SL}_2(\mathbb{F}_q[1/t])$ was not compact, and asked what the typical situation should be: "for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice?".
  In the talk, we will review some of the known results, and then discuss the case of $\mathrm{SL}_n(\mathbb{R})$ for $n > 2$. It turns out that, up to automorphism, the unique lattice of minimal covolume in $\mathrm{SL}_n(\mathbb{R})$ ($n > 2$) is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case.
  "Conjugacy and dynamics in the almost automorphism group of a tree"
     — Dr Waltraud Lederle
  
  Abstract:
  We define the almost automorphism group of a regular tree, also known as Neretin's group, and determine when two elements are conjugate. (joint work with Gil Goffer)
  "Translation-like actions on nilpotent groups"
     — Assistant Prof Mark Pengitore
  
  Abstract:
  Whyte introduced translation-like actions of groups as a geometric generalization of subgroup containment. He then proved a geometric reformulation of the von Neumann conjecture by demonstrating that a finitely generated group is non amenable if and only it admits a translation-like action by a non-abelian free group. This provides motivation for the study of what groups can act translation-like on other groups. As a consequence of Gromov’s polynomial growth theorem, virtually nilpotent groups can act translation-like on other nilpotent groups. We demonstrate that if two nilpotent groups have the same growth, but non-isomorphic Carnot completions, then they can't act translation-like on each other. (joint work with David Cohen)

• Symmetry in Newcastle

  12:00 pm, Friday, 3rd Apr 2020
  (Location to be decided)

Publications

Coming soon...