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Positivity Problems and Conjectures in Algebraic Combinatorics
 in Mathematics: Frontiers and Perspectives
, 1999
"... Introduction. Algebraic combinatorics is concerned with the interaction between combinatorics and such other branches of mathematics as commutative algebra, algebraic geometry, algebraic topology, and representation theory. Many of the major open problems of algebraic combinatorics are related to p ..."
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Introduction. Algebraic combinatorics is concerned with the interaction between combinatorics and such other branches of mathematics as commutative algebra, algebraic geometry, algebraic topology, and representation theory. Many of the major open problems of algebraic combinatorics are related to positivity questions, i.e., showing that certain integers are nonnegative. The significance of positivity to algebraic combinatorics stems from the fact that a nonnegative integer can have both a combinatorial and an algebraic interpretation. The archetypal algebraic interpretation of a nonnegative integer is as the dimension of a vector space. Thus to show that a certain integer m is nonnegative, it suces to nd a vector space Vm of dimension m. Similarly to show that m n, it suces to nd an injective map Vm ! V n or surjective map V n ! Vm . Of course the inequality m n is equivalent to the positivity statement n m 0, while the injectivity of the map ' : Vm ! V n is equivalent to the
Neighborly cubical polytopes
 Discrete & Computational Geometry
, 2000
"... Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubic ..."
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Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubical polytope Cn d maximizes the fvector among all cubical (d − 1)spheres with 2n vertices. While we show that this is true for polytopal spheres if n ≤ d+1, we also give a counterexample for d = 4 and n = 6. Further, the existence of neighborly cubical polytopes shows that the graph of the ndimensional cube, where n ≥ 5, is “dimensionally ambiguous ” in the sense of Grünbaum. We also show that the graph of the 5cube is “strongly 4ambiguous”. In the special case d = 4, neighborly cubical polytopes have f3 = f0 4 log2 f0 4 vertices, so the facetvertex ratio f3/f0 is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch.
Combinatorics with a geometric flavor: some examples
 in Visions in Mathematics Toward 2000 (Geometric and Functional Analysis, Special Volume
, 2000
"... In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound t ..."
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In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete ndimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.
Polytope Skeletons And Paths
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY (SECOND EDITION ), CHAPTER 20
"... ..."
THE SIGNATURE OF A TORIC VARIETY
 DUKE MATHEMATICAL JOURNAL VOL. 111, NO. 2
, 2002
"... We identify a combinatorial quantity (the alternating sum of the hvector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by R. Charney and M. Davis in their work, which in particular showed that its nonnegativity is closely related to a conjecture ..."
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Cited by 6 (3 self)
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We identify a combinatorial quantity (the alternating sum of the hvector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by R. Charney and M. Davis in their work, which in particular showed that its nonnegativity is closely related to a conjecture of H. Hopf on the Euler characteristic of a nonpositively curved manifold. We prove positive (or nonnegative) lower bounds for this quantity under geometric hypotheses on the polytope and, in particular, resolve a special case of their conjecture. These hypotheses lead to ampleness (or weaker conditions) for certain line bundles on toric divisors, and then the lower bounds follow from calculations using the Hirzebruch signature formula. Moreover, we show that under these hypotheses on the polytope, the ith Lclass of the corresponding toric variety is (−1) i times an effective class for any i.
NEIGHBORLY CUBICAL POLYTOPES AND SPHERES
, 2005
"... Abstract. We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a nonpolytop ..."
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Abstract. We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a nonpolytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz, and Wills [16] can be embedded into 3. 1.
The Ring Structure On The Cohomology Of Coordinate Subspace Arrangements
, 1999
"... Every simplicial complex 2 [n] on the vertex set [n] = f1; : : : ; ng denes a real resp. complex arrangement of coordinate subspaces in R n resp. C n via the correspondence 3 7! spanfe i : i 2 g: The linear structure of the cohomology of the complement of such an arrangement is explicitl ..."
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Cited by 4 (0 self)
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Every simplicial complex 2 [n] on the vertex set [n] = f1; : : : ; ng denes a real resp. complex arrangement of coordinate subspaces in R n resp. C n via the correspondence 3 7! spanfe i : i 2 g: The linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of and its links by the Goresky{MacPherson formula. Here we derive, by combinatorial means, the ring structure on the integral cohomology in terms of data of . We provide a nontrivial example of dierent cohomology rings in the real and complex case. Furthermore, we give an example of a coordinate arrangement that yields nontrivial multiplication of torsion elements.
Flag Numbers and FLAGTOOL
 POLYTOPES  COMBINATORICS AND COMPUTATION, VOLUME 29 OF DMV SEMINAR
, 1999
"... FLAGTOOL is a computer program for proving automatically theorems about the combinatorial structure of polytopes of dimensions at most 10. Its starting point is the known linear relations (equalities and inequalities) for ag number of polytopes. After describing the state of the art concerning su ..."
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FLAGTOOL is a computer program for proving automatically theorems about the combinatorial structure of polytopes of dimensions at most 10. Its starting point is the known linear relations (equalities and inequalities) for ag number of polytopes. After describing the state of the art concerning such linear relations we describe various applications of FLAGTOOLS and we conclude by indicating several direction for future research and automation. As an appendix we describe FLAGTOOL's main tools and demonstrate one working session with the program.