Let $A$ be the product of an abelian variety and a torus defined over a number field $K$. Fix some prime number $\ell$. If $\unicode[STIX]{x1D6FC}\in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak{p}$ of $K$ such that the reduction $(\unicode[STIX]{x1D6FC}\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{p})$ is well-defined and has order coprime to $\ell$. This set admits a natural density. By refining the method of Jones and Rouse [Galois theory of iterated endomorphisms, Proc. Lond. Math. Soc. (3) 100(3) (2010), 763–794. Appendix A by Jeffrey D. Achter], we can express the density as an $\ell$-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of $\ell$) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.

The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size ℓ a large graph G on n vertices can have. Clearly, this number is $\left( {\matrix{n \cr k}}\right)$ for every n, k and $\ell \in \left\{ {0,\left( {\matrix{k \cr 2}} \right)}\right\}$. We conjecture that for every n, k and $0 \lt \ell \lt \left( {\matrix{k \cr 2}}\right)$ this number is at most $ (1/e + {o_k}(1)) {\left( {\matrix{n \cr k}} \right)}$. If true, this would be tight for ℓ ∈ {1, k − 1}.

In support of our ‘Edge-statistics Conjecture’, we prove that the corresponding density is bounded away from 1 by an absolute constant. Furthermore, for various ranges of the values of ℓ we establish stronger bounds. In particular, we prove that for ‘almost all’ pairs (k, ℓ) only a polynomially small fraction of the k-subsets of V(G) have exactly ℓ edges, and prove an upper bound of $ (1/2 + {o_k}(1)){\left( {\matrix{n \cr k}}\right)}$ for ℓ = 1.

Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun’s sieve, as well as graph-theoretic and combinatorial arguments such as Zykov’s symmetrization, Sperner’s theorem and various counting techniques.

We begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

In a singularly perturbed limit, we analyse the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci., 20, Suppl., (2010)], which models urban crime with police intervention. In our extended RD model, the field variables are the attractiveness field for burglary, the criminal population density and the police population density. Our model includes a scalar parameter that determines the strength of the police drift towards maxima of the attractiveness field. For a special choice of this parameter, we recover the ‘cops-on-the-dots’ policing strategy of Jones et. al., where the police mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocal eigenvalue problems (NLEPs) that characterise the linear stability of these hotspot steady states to ${\cal O}$(1) timescale instabilities. For a cops-on-the-dots policing strategy, we prove that a multi-hotspot steady state is linearly stable to synchronous perturbations of the hotspot amplitudes. Alternatively, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical–numerical method is used to construct linear stability phase diagrams in the police vs. criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model, this provides a parameter range where the effect of a cops-on-the-dots policing strategy is to only displace crime temporally between neighbouring spatial regions. Our hybrid approach to study the NLEPs combines rigorous spectral results with a numerical parameterisation of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.

Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell \neq p$ be a prime number, and $\text{W}_{F}$ the Weil group of $F$. We classify equivalence classes of $\text{W}_{F}$-semisimple Deligne $\ell$-modular representations of $\text{W}_{F}$ in terms of irreducible $\ell$-modular representations of $\text{W}_{F}$, and extend constructions of Artin–Deligne local constants to this setting. Finally, we define a variant of the $\ell$-modular local Langlands correspondence which satisfies a preservation of local constants statement for pairs of generic representations.

We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.

The mathematical theory of Optimal Transport (OT) is now one of the most visible and also most active fields in nonlinear analysis, with two Fieldsmedal winners Cedric Villani and Alessio Figalli among its protagonists. The origins of OT date back to the works of GaspardMonge in the 18th century and those of Leonid Kantorovich in the ’1940s. However, what is considered as the foundations of the modern theory have mostly been established in several ground-breaking works at the end of the 20th century: among the most prominent contributors, there is Yann Brenier, who characterized optimal transport in terms of connections with PDEs and hydrodynamics in the 1980’s; there is RobertMcCann, who introduced the powerful concept of displacement convexity in the 1990’s; there is Felix Otto, who demonstrated the implications of OT to nonlinear evolution equations around the millennium.

We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order $O(n{\rm{/}}\sqrt {{\rm{log}} \ n} )$. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor.