A classical result of Dirac says that every $n$-vertex graph with minimum degree at least $\frac{n}{2}$ contains a Hamilton cycle. A `discrepancy' version of Dirac's theorem was shown by Balogh--Csaba--Jing--Pluhár, Freschi--Hyde--Lada--Treglown, and Gishboliner--Krivelevich--Michaeli as follows. Every $r$-colouring of the edge set of every $n$-vertex graph with minimum degree at least $(\frac{1}{2} + \frac{1}{2r} + o(1))n$ contains a Hamilton cycle where one of the colours appears at least $(1+o(1))\frac{n}{r}$ times. In this paper, we generalize this result by asymptotically determining the maximum possible value $f_{r,\alpha}(n)$ for every $\alpha \in [\frac{1}{2}, 1]$ such that every $r$-colouring of the edge set of every $n$-vertex graph with minimum degree at least $\alpha n$ contains a Hamilton cycle where one of the colours appears at least $f_{r,\alpha}(n)$ times. In particular, we show that $f_{r,\alpha}(n) = (1-o(1)) \min\{(2\alpha - 1)n, \frac{2\alpha n}{r}, \frac{2n}{r+1}\}$ for every $\alpha\in [\frac{1}{2} + \frac{1}{2r}, 1]$. A graph $H$ is called an $\alpha$-residual subgraph of a graph $G$ if $d_H(v)\ge \alpha d_{G[V(H)]}(v)$ for every $v\in V(H)$. Extending Dirac's theorem in the setting of random graphs, Lee and Sudakov showed the following. The Erdős--Rényi random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, typically has the property that every $(\frac{1}{2} +o(1))$-residual spanning subgraph contains a Hamilton cycle. Motivated by this, we prove the following random version of our `discrepancy' result. The random graph $G \sim G(n,p)$, with $p$ above the Hamiltonicity threshold, typically satisfies that every $r$-colouring of the edge set of every $\alpha$-residual spanning subgraph of $G$ contains a Hamilton cycle where one of the colours appears at least $f_{r,\alpha}(n)$ times.

In this paper we give an algorithm to find the 3-torsion subgroup of the Jacobian of a smooth plane quartic curve. We describe $3-$torsion points in terms of cubics which triply intersect the curve, and use this to define a system of equations whose solution set corresponds to the coefficients of these cubics. We compute the points of this zero-dimensional, degree $728$ scheme first by approximation, using homotopy continuation and Newton-Raphson, and then using lattice reduction or continued fractions to obtain accurate expressions for these points. We describe how the Galois structure of the field of definition of the $3$-torsion subgroup can be used to compute local conductor exponents, including at $p=2$.

Failures of cooperation cause many of society's gravest problems. It is well known that cooperation among many players faced with a social dilemma can be maintained thanks to the possibility of punishment, but achieving the initial state of widespread cooperation is often much more difficult. We show here that there exist strategies of `targeted punishment' whereby a small number of punishers can shift a population of defectors into a state of global cooperation. The heterogeneity of players, often regarded as an obstacle, can in fact boost the mechanism's effectivity. We conclude by outlining how the international community could use a strategy of this kind to combat climate change.

Motivated by recent developments on random polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. This process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. It was shown by Harrison and Williams (1987) that RBM in a polyhe- dral domain has an invariant measure in product form if a certain skew-symmetry condition holds. We show that (modulo technical assumptions) the generalised RBM has an invariant measure in product form if (and essentially only if) the same skew-symmetry condition holds, independent of the choice of potential. The invari- ant measure of course does depend on the potential. Examples include TASEP-like particle systems, generalisations of Brownian motion with rank-dependent drift and diffusions connected to the generalised Pitman transform.

We construct a set of $2^n$ points in $\mathbb{R}^n$ such that all pairwise Manhattan distances are odd integers, which improves the recent linear lower bound of Golovanov, Kupavskii and Sagdeev. In contrast to the Euclidean and maximum metrics, this shows that the odd-distance set problem behaves very differently to the equilateral set problem under the Manhattan metric. Moreover, all coordinates of the points in our construction are integers or half-integers, and we show that our construction is optimal under this additional restriction.

We develop an effective algorithm to compute the derivative of a Bianchi modular form with respect to weight space as it varies in a $p$-adic family. This method is entirely local at the modular form, and does not compute the family anywhere outside an infinitesimal neighbourhood. We numerically verify some conjectures surrouding smoothness of the eigenvariety (equivalently, uniqueness of families) and the ``direction over weight space'' of the family. The methods are also applied to study elliptic modular forms and their $\mathcal{L}$-invariants.

For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.