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Statistical physics models with hard constraints, such as the discrete hard-core gas model (random independent sets in a graph), are inherently combinatorial and present the discrete mathematician with a relatively comfortable setting for…
Statistical models that possess symmetry arise in diverse settings such as random fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the…
We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modeled by a closed polygon drawn on the square diagonal lattice, with possible crossings describing pairs of strands of…
A variant of the classical optimal transportation problem is: among all joint measures with fixed marginals and which are dominated by a given density, find the optimal one. Existence and uniqueness of solutions to this variant were…
In this article we study the expanding properties of random perturbations of contracting Lorenz maps satisfying the summability condition of exponent 1. Under general conditions on the maps and perturbation types, we prove stochastic…
This article discusses two recent works by the author, one with Brown and Hurtado on Zimmer's conjecture and one with Bader, Miller and Stover on totally geodesic submanifolds of real and complex hyperbolic manifolds. The main purpose of…
We present an introductory overview of several challenging problems in the statistical characterisation of turbulence. We provide examples from fluid turbulence in three and two dimensions, from the turbulent advection of passive scalars,…
A meander can be seen as a pair of transversally intersecting simple closed curves on a 2-sphere. We consider pairs of transversally intersecting simple closed curves on a closed oriented surface of arbitrary genus g. The number of such…
We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$, $d\ge 1$, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…
Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson…
We study the statistical properties of orientation and rotation dynamics of elliptical tracer particles in two-dimensional, homogeneous and isotropic turbulence by direct numerical simulations. We consider both the cases in which the…
We study optimization problems for partially hinged rectangular plates, modeling bridge roadways, in the presence of real and artificial obstacles. Real obstacles represent structural constraints to avoid, while artificial ones are…
A classical problem of statistical inference is the valid specification of a model that can account for the statistical dependencies between observations when the true structure is dense, intractable, or unknown. To address this problem, a…
We produce new examples of Riemannian manifolds with scalar curvature lower bounds and collapsing behavior along codimension 2 submanifolds. Applications of this construction are given, primarily on questions concerning the stability of…
Multi-lane totally asymmetric simple exclusion processes with interactions between the lanes have recently been investigated actively. This paper proposes a two-lane model with extended Langmuir kinetics on a periodic lattice. Both…
The ground state properties of spin-1/2 ladders are studied, emphasizing the role of frustration and ring exchange coupling. We present a unified field theory for ladders with general coupling constants and geometry. Rich phase diagrams can…
We present an extended version of an invited talk given on the International Conference "Turbulent Mixing and Beyond". The dynamical and statistical description of stably stratified turbulent boundary layers with the important example of…