Towards a better approximation for sparsest cut?

We give a new $(1+\epsilon)$-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size $n/r$ expand by a factor $\sqrt{\log n\log r}$ bigger, for some small $r$; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-$r$ Lasserre relaxation. The other is combinatorial and involves a new notion called {\em Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which we show exists in the input graph. Both algorithms run in time $2^{O(r)} \mathrm{poly}(n)$. We also show similar approximation algorithms in graphs with genus $g$ with an analogous local expansion condition. This is the first algorithm we know of that achieves $(1+\epsilon)$-approximation on such general family of graphs

Random Tensors and Quantum Gravity

We provide an informal introduction to tensor field theories and to their associated renormalization group. We focus more on the general motivations coming from quantum gravity than on the technical details. In particular we discuss how asymptotic freedom of such tensor field theories gives a concrete example of a natural "quantum relativity" postulate: physics in the deep ultraviolet regime becomes asymptotically more and more independent of any particular choice of Hilbert basis in the space of states of the universe.Comment: Section 6 is essentially reproduced from author's arXiv:1507.04190 for self-contained purpose of the revie

First Passage Percolation on Inhomogeneous Random Graphs

We investigate first passage percolation on inhomogeneous random graphs. The random graph model G(n,kappa) we study is the model introduced by Bollob\'as, Janson and Riordan, where each vertex has a type from a type space S and edge probabilities are independent, but depending on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal weight path, properly normalized follows a central limit theorem. We handle the cases where lambda(n)->lambda is finite or infinite, under the assumption that the average number of neighbors lambda(n) of a vertex is independent of the type. The paper is a generalization the paper by Bhamidi, van der Hofstad and Hooghiemstra, where FPP is explored on the Erdos-Renyi graphs

The Number of Nowhere-Zero Flows on Graphs and Signed Graphs

A nowhere-zero $k$-flow on a graph $\Gamma$ is a mapping from the edges of $\Gamma$ to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in any fixed orientation of $\Gamma$, at each node the sum of the labels over the edges pointing towards the node equals the sum over the edges pointing away from the node. We show that the existence of an \emph{integral flow polynomial} that counts nowhere-zero $k$-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the flow counting functions, and reciprocity laws that interpret the evaluations of the flow polynomials at negative integers in terms of the combinatorics of the graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.