This volume presents some of the research topics discussed at the 2014-2015 Annual Thematic Program Discrete Structures: Analysis and Applications at the Institute for Mathematics and its Applications during Fall 2014, when combinatorics was the focus. Leading experts have written surveys of research problems, making state of the art results more conveniently and widely available. The three-part structure of the volume reflects the three workshops held during Fall 2014. In the first part, topics on extremal and probabilistic combinatorics are presented; part two focuses on additive and analytic combinatorics; and part three presents topics in geometric and enumerative combinatorics. This book will be of use to those who research combinatorics directly or apply combinatorial methods to other fields.
Section 1: Extremal and Probabilistic Combinatorics.- Problems Related
to Graph Indices in Trees.- The edit distance in graphs: methods, results and
generalizations.- Repetitions in graphs and sequences.- On Some Extremal
Problems for Cycles in Graphs.- A survey of Turan problems for expansions.-
Survey on matching, packing and Hamilton cycle problems on hypergraphs.-
Rainbow Hamilton cycles in random graphs and hypergraphs.- Further
applications of the Container Method.- Independent transversals and
hypergraph matchings - an elementary approach.- Giant components in random
graphs.- Infinite random graphs and properties of metrics.- Nordhaus-Gaddum
Problems for Colin de Verdičre Type Parameters, Variants of Tree-width, and
Related Parameters.- Algebraic aspects of the normalized Laplacian.-
Poset-free Families of Subsets.- Mathematics of causal sets.-
Section 2: Additive and Analytic Combinatorics.- Lectures on Approximate
groups and Hilbert's 5th Problem.- Character sums and arithmetic
combinatorics.- On sum-product problem.- Ajtai-Szemerédi Theorems over
quasirandom groups.- Section 3: Enumerative and Geometric Combinatorics.-
Moments of orthogonal polynomials and combinatorics.- The combinatorics of
knot invariants arising from the study of Macdonald polynomials.- Some
algorithmic applications of partition functions in combinatorics.- Partition
Analysis, Modular Functions, and Computer Algebra.- A survey of consecutive
patterns in permutations.- Unimodality Problems in Ehrhart Theory.- Face
enumeration on simplicial complexes.- Simplicial and Cellular Trees.-
Parametric Polyhedra with at least k Lattice Points: Their Semigroup
Structure and the k-Frobenius Problem.- Dynamical Algebraic Combinatorics and
the Homomesy Phenomenon.
