1,169 research outputs found
The many faces of modern combinatorics
This is a survey of recent developments in combinatorics. The goal is to give
a big picture of its many interactions with other areas of mathematics, such
as: group theory, representation theory, commutative algebra, geometry
(including algebraic geometry), topology, probability theory, and theoretical
computer science
High Dimensional Expanders
Expander graphs have been, during the last five decades, the subject of a
most fruitful interaction between pure mathematics and computer science, with
influence and applications going both ways (cf. [Lub94], [HLW06], [Lub12] and
the references therein). In the last decade, a theory of "high dimensional
expanders" has begun to emerge. The goal of the current paper is to describe
some paths of this new area of study.Comment: Paper to be presented as a Plenary talk at ICM 201
Computing Optimal Morse Matchings
Morse matchings capture the essential structural information of discrete
Morse functions. We show that computing optimal Morse matchings is NP-hard and
give an integer programming formulation for the problem. Then we present
polyhedral results for the corresponding polytope and report on computational
results
Spaces of Hermitian operators with simple spectra and their finite-order cohomology
The topology of spaces of Hermitian operators in with non-simple
spectra was studied by V.Arnold in a relation with the theory of adiabatic
connections and the quantum Hall effect. The natural filtration of these spaces
by the sets of operators with fixed numbers of eigenvalues defines the spectral
sequence, providing interesting combinatorial and homological information on
this stratification.
We construct a different spectral sequence, also counting the homology groups
of these spaces and based on the universal techniques of {\em topological order
complexes} and resolutions of algebraic varieties, generalizing the
combinatorial inclusion-exclusion formula and similar to the construction of
finite degree knot invariants.
This spectral sequence degenerates at the term , is (conjecturally)
multiplicative, and as grows then it converges to a stable spectral
sequence counting the cohomology of the space of infinite Hermitian operators
without multiple eigenvalues, all whose terms are finitely
generated. It allows us to define the finite degree cohomology classes of this
space, and to apply the well-known facts and methods of the topological theory
of flag manifolds to the problems of geometrical combinatorics, especially
concerning the continuous partially ordered sets of subspaces and flags
Topological lower bounds for the chromatic number: A hierarchy
This paper is a study of ``topological'' lower bounds for the chromatic
number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978,
in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology.
This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with
all -element subsets of as vertices and all pairs of
disjoint sets as edges, has chromatic number . Several other proofs
have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz,
Greene, and others), all of them based on some version of the Borsuk--Ulam
theorem, but otherwise quite different. Each can be extended to yield some
lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe
that \emph{every} finite graph may be represented as a generalized Kneser
graph, to which the above bounds apply.)
We show that these bounds are almost linearly ordered by strength, the
strongest one being essentially Lov\'asz' original bound in terms of a
neighborhood complex. We also present and compare various definitions of a
\emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz
and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but
the construction is simpler and functorial, mapping graphs with homomorphisms
to -spaces with -maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea
Some old and new problems in combinatorial geometry I: Around Borsuk's problem
Borsuk asked in 1933 if every set of diameter 1 in can be covered by
sets of smaller diameter. In 1993, a negative solution, based on a
theorem by Frankl and Wilson, was given by Kahn and Kalai. In this paper I will
present questions related to Borsuk's problem.Comment: This is a draft of a chapter for "Surveys in Combinatorics 2015,"
edited by Artur Czumaj, Angelos Georgakopoulos, Daniel Kral, Vadim Lozin, and
Oleg Pikhurko. The final published version shall be available for purchase
from Cambridge University Pres
Face numbers of 4-Polytopes and 3-Spheres
In this paper, we discuss f- and flag-vectors of 4-dimensional convex
polytopes and cellular 3-spheres. We put forward two crucial parameters of
fatness and complexity: Fatness F(P) := (f_1+f_2-20)/(f_0+f_3-10) is large if
there are many more edges and 2-faces than there are vertices and facets, while
complexity C(P) := (f_{03}-20)/(f_0+f_3-10) is large if every facet has many
vertices, and every vertex is in many facets. Recent results suggest that these
parameters might allow one to differentiate between the cones of f- or
flag-vectors of -- connected Eulerian lattices of length 5 (combinatorial
objects), -- strongly regular CW 3-spheres (topological objects), -- convex
4-polytopes (discrete geometric objects), and -- rational convex 4-polytopes
(whose study involves arithmetic aspects). Further progress will depend on the
derivation of tighter f-vector inequalities for convex 4-polytopes. On the
other hand, we will need new construction methods that produce interesting
polytopes which are far from being simplicial or simple -- for example, very
``fat'' or ``complex'' 4-polytopes. In this direction, I will report about
constructions (from joint work with Michael Joswig, David Eppstein and Greg
Kuperberg) that yield -- strongly regular CW 3-spheres of arbitrarily large
fatness, -- convex 4-polytopes of fatness larger than 5.048, and -- rational
convex 4-polytopes of fatness larger than 5-epsilon
Complexes of not -connected graphs
Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not -connected -hypergraphs on vertices. We
show that the complex of not -connected graphs has the homotopy type of a
wedge of spheres of dimension . This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the -action on the homology of the complex is also determined. For
complexes of not -connected -hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not -connected graphs we provide a formula
for the generating function of the Euler characteristic
Computing Convex Hulls in the Affine Building of SL_d
We describe an algorithm for computing the convex hull of a finite collection
of points in the affine building of SL_d(K), for K a field with discrete
valuation. These convex hulls describe the relations among a finite collection
of invertible matrices over K. As a consequence, we bound the dimension of the
tropical projective space needed to realize the convex hull as a tropical
polytope.Comment: 18 pages, 11 figure
Asymptotics of Weil-Petersson geodesics II: bounded geometry and unbounded entropy
We use ending laminations for Weil-Petersson geodesics to establish that
bounded geometry is equivalent to bounded combinatorics for Weil-Petersson
geodesic segments, rays, and lines. Further, a more general notion of
non-annular bounded combinatorics, which allows arbitrarily large
Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson
geodesics. As an application, we show the Weil-Petersson geodesic flow has
compact invariant subsets with arbitrarily large topological entropy.Comment: 39 Pages, 3 figures. Minor revision
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