We have reworked our website, with a new focus on our research strengths. If something is broken or missing, please let us know!

# Number Theory, Algorithms and Discrete Mathematics

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

12:00 pm, Friday, 1st Nov 2019
V109, Mathematics Building

Schedule:

12-1: Anthony Dooley
1-2: Lunch
2-3: Colin Reid
3-3.30: Tea
3.30-4.30: Michael Barnsley

"Classification of non-singular systems and critical dimension"
— Prof Anthony Dooley

Abstract:
A non-singular measurable dynamical system is a measure space $X$ whose measure $\mu$ has the property that $\mu$ and $\mu \circ T$ are equivalent measures (in the sense that they have the same sets of measure zero). Here $T$ is a bimeasurable invertible transformation of $X$. The basic building blocks are the \emph{ergodic} measures. Von Neumann proposed a classification of non-singular ergodic dynamical systems, and this has been elaborated subsequently by Krieger, Connes and others. This work has deep connections with C*-algebras. I will describe some work of myself, collaborators and students which explore the classification of dynamical systems from the point of view of measure theory. In particular, we have recently been exploring the notion of critical dimension, a study of the rate of growth of sums of Radon-Nikodym derivatives $\Sigma_{k=1}^n \frac{d\mu \circ T^k}{d\mu}$. Recently, we have been replacing the single transformation $T$ with a group acting on the space $X$.

"Piecewise powers of a minimal homeomorphism of the Cantor set"
— Dr Colin Reid

Abstract:
Let $X$ be the Cantor set and let $g$ be a minimal homeomorphism of $X$ (that is, every orbit is dense). Then the topological full group $\tau[g]$ of $g$ consists of all homeomorphisms $h$ of $X$ that act 'piecewise' as powers of $g$, in other words, $X$ can be partitioned into finitely many clopen pieces $X_1,...,X_n$ such that for each $i$, $h$ acts on $X_i$ as a constant power of $g$. Such groups have attracted considerable interest in dynamical systems and group theory, for instance they characterize the homeomorphism up to flip conjugacy (Giordano--Putnam--Skau) and they provided the first known examples of infinite finitely generated simple amenable groups (Juschenko--Monod). My talk is motivated by the following question: given $h\in\tau[g]$ for some minimal homeomorphism $g$, what can the closures of orbits of $h$ look like? Certainly $h\in\tau[g]$ is not minimal in general, but it turns out to be quite close to being minimal, in the following sense: there is a decomposition of $X$ into finitely many clopen invariant pieces, such that on each piece $h$ acts a homeomorphism that is either minimal or of finite order. Moreover, on each of the minimal parts of $h$, then either $h$ or $h^{-1}$ has a 'positive drift' with respect to the orbits of $g$; in fact, it can be written in a canonical way as a conjugate of a product of induced transformations (aka first return maps) of $g$. No background knowledge of topological full groups is required; I will introduce all the necessary concepts in the talk.

"Dynamics on Fractals"
— Prof Michael Barnsley

Abstract:
I will outline a new theory of fractal tilings. The approach uses graph iterated function systems (IFS) and centers on underlying symbolic shift spaces. These provide a zero dimensional representation of the intricate relationship between shift dynamics on fractals and renormalization dynamics on spaces of tilings. The ideas I will describe unify, simplify, and substantially extend key concepts in foundational papers by Solomyak, Anderson and Putnam, and others. In effect, IFS theory on the one hand, and self-similar tiling theory on the other, are unified.
The work presented is largely new and has not yet been submitted for publication. It is joint work with Andrew Vince (UFL) and Louisa Barnsley. The presentation will include links to detailed notes. The figures illustrate 2d fractal tilings.
By way of recommended background reading I mention the following awardwinning paper: M. F. Barnsley, A. Vince, Self-similar polygonal tilings, Amer. Math. Monthly 124 (1017) 905-921.
• #### Mahler Lecture

4:00 pm, Monday, 25th Nov 2019
LSTH, Life Sciences Lecture Theatre
"Transcendence and dynamics"
Dr Holly Krieger
Abstract:
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.