Isoperimetry in integer lattices

The edge isoperimetric problem for a graph $G$ is to determine, for each $n$, the minimum number of edges leaving any set of $n$ vertices. In general this problem is NP-hard, but exact solutions are known in some special cases, for example when $G$ is the usual integer lattice. We solve the edge isoperimetric problem asymptotically for every Cayley graph on $\mathbb Z^d$. The near-optimal shapes that we exhibit are zonotopes generated by line segments corresponding to the generators of the Cayley graph.Comment: Published in Discrete Analysi

Isoperimetry in integer lattices

Isoperimetry in integer lattices, Discrete Analysis 2018:7, 16 pp. The isoperimetric problem, already known to the ancient Greeks, concerns the minimisation of the size of a boundary of a set under a volume constraint. The problem has been studied in many contexts, with a wide range of applications. The present paper focuses on the discrete setting of graphs, where the boundary of a subset of vertices can be defined with reference to either vertices or edges. Specifically, the so-called edge-isoperimetric problem for a graph $G$ is to determine, for each $n$, the minimum number of edges leaving any set $S$ of $n$ vertices. The vertex isoperimetric problem asks for the minimum number of vertices that can be reached from $S$ by following these edges. For a general graph $G$ this problem is known to be NP-hard, but exact solutions are known for some special classes of graphs. One example is the $d$-dimensional hypercube, which is the graph on vertex set $\{0,1\}^d$ with edges between those binary strings of length $d$ that differ in exactly one coordinate. The edge-isoperimetric problem for this graph was solved by Harper, Lindsey, Bernstein, and Hart, and the extremal sets include $k$-dimensional subcubes obtained by fixing $d-k$ of the coordinates. The edge-isoperimetric problem for the $d$-dimensional integer lattice whose edges connect pairs of vertices at $\ell_1$-distance 1, was solved by Bollobás and Leader in the 1990s, who showed that the optimal shapes consist of $\ell_\infty$-balls. More recently, Radcliffe and Veomett solved the vertex-isoperimetric problem for the $d$-dimensional integer lattice on which edges are defined with respect to the $\ell_\infty$-distance instead. In the present paper the authors solve the edge-isoperimetric problem asymptotically for every Cayley graph on $G=\mathbb{Z}^d$, and determine the near-optimal shapes in terms of the generating set used to construct the Cayley graph. In particular, this solves the edge-isoperimetric problem on $\mathbb{Z}^d$ with the $\ell_\infty$-distance. We now describe the results in more detail. Given any generating set $U$ of $G$ that does not contain the identity, the Cayley graph $\Gamma(G,U)$ has vertex set $G$ and edge set $\{(g,g+u): g\in G, u\in U\}$. This construction includes both the $\ell_1$ and the $\ell_\infty$ graph defined above, by considering the generating sets $U_1=\{(\pm 1, 0,\dots,0),\dots,(0,\dots,0,\pm 1)\}$ and $U_\infty=\{-1,0,1\}^d\setminus\{0,0,0\}$, respectively. The near-optimal shapes obtained by the authors are so-called zonotopes, generated by line segments corresponding to the generators of the Cayley graph. More precisely, if $U=\{u_1,u_2,\dots,u_k\}$ is a set of non-zero generators of $G$, then the near-optimal shapes are the intersections of scaled copies of the convex hull of the sum set $\{0,u_1\}+\{0,u_2\}+\dots+\{0,u_k\}$ with $\mathbb{Z}^d$. For example, when $d=2$, then the zonotope for the $\ell_\infty$ problem is an octagon obtained by cutting the corners off a square through points one third of the way along each side. In contrast to the aforementioned edge-isoperimetry results, which were solved exactly using compression methods, the approach in this paper is an approximate one. It follows an idea of Ruzsa, who solved the vertex-isoperimetry problem in general Cayley graphs on the integer lattice by approximating the discrete problem with a continuous one. While the continuous analogue in the present paper is a natural one, it is not clear that it is indeed a good approximation to the original problem, and it is here that the main combinatorial contribution of the paper lies. It concludes with several open problems and directions for further work

Partial Differential Equations

The workshop dealt with nonlinear partial differential equations and some applications in geometry, touching several different topics such as minimal surfaces and harmonic maps, equations in conformal geometry, geometric flows, extremal eigenvalue problems and optimal transport

50 years of first passage percolation

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2+log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.Comment: 160 pages, 17 figures. This version has updated chapters 3-5, with expanded and additional material. Small typos corrected throughou

Optimization and stability problems for eigenvalues of linear and non linear operators

In this thesis we study eigenvalue problems with different boundary conditions for some operators of linear and non linear type. In particular, we focus our study on Steklov and Robin boundary conditions, obtaining isoperimetric inequalities and the relative stability results with different hypothesis on the class of sets considered. A stability result in terms of the perimeter is also obtained for the first Dirichlet eigenvalues of the Laplacian operator

Stored energies in electric and magnetic current densities for small antennas

Electric and magnetic currents are essential to describe electromagnetic stored energy, as well as the associated quantities of antenna Q and the partial directivity to antenna Q-ratio, D/Q, for general structures. The upper bound of previous D/Q-results for antennas modeled by electric currents is accurate enough to be predictive, this motivates us here to extend the analysis to include magnetic currents. In the present paper we investigate antenna Q bounds and D/Q-bounds for the combination of electric- and magnetic-currents, in the limit of electrically small antennas. This investigation is both analytical and numerical, and we illustrate how the bounds depend on the shape of the antenna. We show that the antenna Q can be associated with the largest eigenvalue of certain combinations of the electric and magnetic polarizability tensors. The results are a fully compatible extension of the electric only currents, which come as a special case. The here proposed method for antenna Q provides the minimum Q-value, and it also yields families of minimizers for optimal electric and magnetic currents that can lend insight into the antenna design.Comment: 27 pages 7 figure