A Geometric Theory For Hypergraph Matching

The authors develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: `space barriers' from convex geometry, and `divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, they introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. They determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, their main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, the authors apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in -graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemerédi Theorem. Here they prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical they defer it to a subsequent paper.

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.

This book constitutes the refereed proceedings of the 14th Annual Conference on Theory and Applications of Models of Computation, TAMC 2017, held in Bern, Switzerland, in April 2017. The 45 revised full papers presented together with 4 invited papers were carefully reviewed and selected from 103 submissions. The main themes of TAMC 2017 have been computability, computer science logic, complexity, algorithms, and models of computation and systems theory.

The Kronecker coefficient is the multiplicity of the -irreducible in the restriction of the -irreducible via the natural map , where are -vector spaces and . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. The authors construct two quantum objects for this problem, which they call the nonstandard quantum group and nonstandard Hecke algebra. They show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality.

The authors study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. They solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. They also develop a product formula for solving these asymptotic problems in general. The central tools of their approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale.

Graph Theory has proved to be an extremely useful tool for solving combinatorial problems in such diverse areas as Geometry, Algebra, Number Theory, Topology, Operations Research and Optimization. It is natural to attempt to generalise the concept of a graph, in order to attack additional combinatorial problems. The idea of looking at a family of sets from this standpoint took shape around 1960. In regarding each set as a ``generalised edge'' and in calling the family itself a ``hypergraph'', the initial idea was to try to extend certain classical results of Graph Theory such as the theorems of Turán and König. It was noticed that this generalisation often led to simplification; moreover, one single statement, sometimes remarkably simple, could unify several theorems on graphs. This book presents what seems to be the most significant work on hypergraphs.

Let be a simple classical algebraic group over an algebraically closed field of characteristic with natural module . Let be a closed subgroup of and let be a non-trivial irreducible tensor-indecomposable -restricted rational -module such that the restriction of to is irreducible. In this paper the authors classify the triples of this form, where is a disconnected maximal positive-dimensional closed subgroup of preserving a natural geometric structure on .

The deformation theory of automorphic representations is used to study local properties of Galois representations associated to automorphic representations of general linear groups and symplectic groups. In some cases this allows to identify the local Galois representations with representations predicted by a local Langlands correspondence.