Spectral graph theory is a fast developing field in modern discrete mathematics with important applications in computer science, chemistry and operational research. By merging combinatorial techniques with algebraic and analytical methods it creates new approaches to hard discrete problems and gives new insights in classical Linear Algebra. The proposed minisymposium will bring together leading...

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of c...

The Shortest lattice Vector Problem is central in lattice-based cryptography, as well as in many areas of computational mathematics and computer science, such as computational number theory and combinatorial optimisation. We present an algorithm for solving it in time 2 and space 2, where n is the lattice dimension. This improves the best previously known algorithm, by Micciancio and Voulgaris ...

A subgroup G ≤ Z exhibits the Specker phenomenon if every homomorphism G → Z maps almost all unit vectors to 0. We give several combinatorial characterizations of the cardinal se, the size of the smallest G ≤ Z exhibiting the Specker phenomenon. We also prove the consistency of b < e, where b is the unbounding number and e the evasion number. Our results answer several questions addressed by Bl...

* During my seven years at Caltech, I had the pleasure of knowing Herb Ryser as a teacher and a colleague. It was difficult not to be inspired by Herb. He held such a deep understanding of combinatorics but was still honestly fascinated by the subject. Through his research, his books, and his students he added immeasurably to the wealth of combinatorial mathematics. + Work partially supported b...

Multiple ergodic averages, such as the average of expressions like f1(T x) f2(T x) . . . fk(T x), were first studied in the ergodic theoretic proof of Szemerédi’s Theorem on arithmetic progressions. It turns out that all constraints on such averages (in a sense that we describe) have an algebraic character, arising from identities in nilpotent groups. We discuss these averages, several generali...

A generalization of the Seidel-Entringer-Arnold method for calculating the alternating permutation numbers (or secant-tangent numbers) leads to a new operation on sequences, the boustrophedon transform. This paper was published (in a somewhat different form) in J. Combinatorial Theory, Series A, 76 (1996), pp. 44–54. Present address: Mathematics Department, MIT, Cambridge, MA Present address: A...

In this lecture we be begin our tour of discrete mathematics and algorithms by looking at one of the most fundamental and classic problems in combinatorial optimization and graph theory, computing the edge connectivity of an undirected, unweighted graph. In the next few lectures we will take a more principled and fundamental approach to graph theory and connectivity. The goal of today is to get...

Our paper has two goals: i) We propose the combinatorial approach to facilitate the calculation of the number of spanning trees for five new classes of graphs. ii) We use a new powerful operation (subdivision) to get larger graphs from a given graph. In particular, we derive the explicit formulas for the subdivision of ladder, fan, triangular snake, double triangular snake and the total graph o...

In this paper, we investigate the number of spanning trees in planar graphs with two cut vertices. We propose a combinatorial approach based on the contraction method, in order to derive spanning trees recursions for this class of graphs. Applications on different graphs showed the efficiency and the simplicity of our recursive approach in comparison with the classic matrix tree theorem. Mathem...