Random graphs were introduced as a convenient example for demonstrating the impossibility of ‘complete disorder’ by Erdos, who also thought that these objects will never become useful in the applied areas outside of pure mathematics. In this talk, I will view random graphs as objects in the field of applied mathematics and discus how the application-driven objectives have set new directions for studying random graphs. I will focus on characterising the sizes of connected components in graphs with a given degree distribution, on the percolation-like processes on such structures, and on generalisations to the coloured graphs. These theoretical questions have interesting implications for studying resilience of networks with nontrivial structures, and for materials science where they explain kinetics-driven phase transitions. Even more surprisingly, the results reveal intricate connections between random graphs and non-linear partial differential equations indicating new possibilities for their analysis.

Spatial networks are often the most natural way to represent spatial information of different kinds. One of the outstanding problems in current spatial network research is to effectively quantify the heterogeneity of the discrete-valued spatial distributions underlying a spatial graph. In this talk we will presentsome recent alternative approaches to estimate heterogeneity in spatial networks based on simple dynamical processes running on them.

 In Classification Theory, Shelah defined several cardinal invariants of a complete theory which detect the presence of certain trees among the definable sets, which in turn quantify the complexity of forking.  In later model-theoretic developments, local versions of these invariants were recognized as marking important dividing lines - e.g. simplicity and NTP2.  Around these dividing lines, a dichotomy theorem of Shelah states that a theory has the tree property if and only if it is witnessed in one of two extremal forms--the tree property of the first or second kind--and it was asked if there is a 'quantitative' analogue of this dichotomy in the form of a certain equation among these invariants.  We will describe these model-theoretic invariants and explain why the quantitative version of the dichotomy fails, via a construction that relies upon some unexpected tools from combinatorial set theory. 

 

The principal ideal theorem (1930) guaranteed that any number field K would embed into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will finish the proof of their cohomological result and thus fully justify how it settled the class field tower problem.

The principal ideal theorem (1930) ascertained that any number field K embeds into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will sketch the proof of their cohomological result and explain how it settled the class field tower problem.

A particularly active area of research in modern algebraic number theory is the study of a class of numbers, called periods. In their simplest form, periods are integrals of rational functions over domains defined by rational in equations. They form a ring, which encompasses all algebraic numbers, logarithms thereof and \pi. They arise in the study of modular forms, cohomology and quantum field theory, and conjecturally have a sort of Galois theory.

We will take a whirlwind tour of these numbers, before discussing non-periods. In particular, we will sketch the construction of an explicit non-period, often forgotten about.

L-functions have become a vital part of modern number theory over the past century, allowing comparisons between arithmetic objects with seemingly very different properties. In the first part of this talk, I will give an overview of where they arise, their properties, and the mathematics that has developed in order to understand them. In the second part, I will give a sketch of the beautiful result of Herbrand-Ribet concerning the arithmetic interpretations of certain special values of the Riemann zeta function, the prototypical example of an L-function.

Let $[N] = \{1,..., N\}$ and let $A$ be a subset of $[N]$. A result of Sárközy in 1978 showed that if the difference set $A-A = \{ a - a’: a, a’ \in A\}$ does not contain any number which is one less than a prime, then $A = o(N)$. The quantitative upper bound on $A$ obtained from Sárközy’s proof has be improved subsequently by Lucier, and by Ruzsa and Sanders. In this talk, I will discuss my work on this problem. I will give a brief introduction of the iteration scheme and the Hardy-Littlewood method used in the known proofs, and our major arc estimate which leads to an improved bound.

A subset of the integers larger than 1 is called $\textit{primitive}$ if no member divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ over $n$ in a primitive set $A$ is universally bounded for any choice of $A$. In 1988, he famously asked if this universal bound is attained by the set of prime numbers. In this talk we shall discuss some recent progress towards this conjecture and related results, drawing on ideas from analysis, probability, & combinatorics.