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Homotopical Algebra and Higher Structures (hybrid meeting)
Homotopical algebra and higher category theory play an increasingly important role in pure mathematics, and higher methods
have seen tremendous development in the last couple of decades. The talks delivered at the workshop described some of the latest progress in this area
and applications to various problems of algebra, geometry, and combinatorics
Peak reduction technique in commutative algebra
The "peak reduction" method is a powerful combinatorial technique with
applications in many different areas of mathematics as well as theoretical
computer science. It was introduced by Whitehead, a famous topologist and group
theorist, who used it to solve an important algorithmic problem concerning
automorphisms of a free group. Since then, this method was used to solve
numerous problems in group theory, topology, combinatorics, and probably in
some other areas as well.
In this paper, we give a survey of what seems to be the first applications of
the peak reduction technique in commutative algebra and affine algebraic
geometry.Comment: survey; 10 page
Bell polynomials in combinatorial Hopf algebras
Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934.
These polynomials have numerous applications in Combinatorics, Analysis,
Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve
combinatorial objects (set partitions, set partitions in lists, permutations,
etc.). So it seems natural to investigate analogous formulae in some
combinatorial Hopf algebras with bases indexed by these objects. The algebra of
symmetric functions is the most famous example of a combinatorial Hopf algebra.
In a first time, we show that most of the results on Bell polynomials can be
written in terms of symmetric functions and transformations of alphabets. Then,
we show that these results are clearer when stated in other Hopf algebras (this
means that the combinatorial objects appear explicitly in the formulae). We
investigate also the connexion with the Fa{\`a} di Bruno Hopf algebra and the
Lagrange-B{\"u}rmann formula
Recent Trends in Combinatorics
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 Verdière 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\u27s 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.
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.https://lib.dr.iastate.edu/math_books/1000/thumbnail.jp
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