Survey and Review

Abstract

This issue of SIAM Review contains three Survey and Review papers. When taken together, they show the enormous variety of mathematical ideas and techniques currently being used by applied mathematicians and also the wide spectrum of application fields now being tackled, ranging from very classic areas like electromagnetism to the investigation of the functioning of the brain or to the analysis of human mobility. The first paper is “Randomized Reference Models for Temporal Networks,” by Laetitia Gauvin, Mathieu Génois, Márton Karsai, Mikko Kivelä, Taro Takaguchi, Eugenio Valdano, and Christian L. Vestergaard. Random graphs are used to gain understanding of complex networks. For instance, one may estimate the size of the largest connected component of a graph with $N$ nodes and $E$ edges by generating random samples from the family of all such graphs. Techniques to generate those samples are referred to as randomized reference models. The present paper does not deal with (static) graphs, but with temporal networks, such as the network of all mobile phone users in a given city, where an edge from a node $i$ to a node $j$ appears during a time interval $(t,\tau)$ if users $i$ and $j$ hold a conversation in such interval. The paper surveys many recent developments and provides tools that systematize and unify the approaches currently being used in different application areas. The second paper is “Love--Lieb Integral Equations: Applications, Theory, Approximations, and Computations,” by Leandro Farina, Guillaume Lang, and P. A. Martin. The Love--Lieb equation $$ \hskip2cm u(x) \pm \frac{1}{\pi}\int_{-1}^{1} \frac{\alpha u(y)}{\alpha^2+(x-y)^2}\, {\rm d} y =1,\qquad -1\leq x\leq 1, $$ where $\alpha >0$ is a real parameter and $u$ the unknown function, arises, e.g., in the study of circular plate capacitors and in certain quantum integrable models. The authors present the origin of the equation in a number of applications, a discussion of the existence and uniqueness of the solution, and a number of approaches to approximate $u$ numerically or analytically. Hamza Fawzi, Joao Gouveia, Pablo A. Parrilo, James Saunderson, and Rekha R. Thomas are the authors of the third paper, “Lifting for Simplicity: Concise Description of Convex Sets.” Convex sets $P$ in $\mathbb{R}^n$ appear frequently in optimization and in many other branches of applied mathematics. The paper studies how to represent a given convex set $P$ as the projection onto $\mathbb{R}^n$ of a convex subset $Q$ of a larger space $\mathbb{R}^m$, $m >n$. Such a representation may be very useful in cases where the description of $Q$ is much easier than the description of $P$. For instance, in linear programming, where we seek to maximize $\langle c, x\rangle$ as $x$ ranges in $P$, if $P=\pi(Q)$ with $\pi$ a linear map, one may equivalently maximize the linear function $\langle c,\pi(y)\rangle$ as $y$ ranges in $Q$, an optimization problem over $Q$ that can potentially be much easier to solve. A simple illustration is afforded for the case where $P$ is the unit $\ell_1$-norm ball in $\mathbb{R}^n$, a polytope with $2^n$ facets that nevertheless may be written as a projection of a polytope $Q\subset \mathbb{R}^n$ with only $2n$ facets, thus removing the exponential dependence on $n$ of the number of constraints.