Ed Swartz - Papers

g-elements of matroid complexes, Journal of Comb. Theory Ser. B, 88 (2003), 369-375. pdf file

Lower bounds for h-vectors of k-CM, independence and broken circuit complexes, SIAM J. Disc. Math., 18 (2004/05), 647-661. pdf file

Topological representations of matroids, J. Amer. Math. Soc., 16 (2003), 427-442.  pdf file

(with Tamas Hausel)  Intersection forms of toric hyperkaehler varieties, Proc. Amer. Math. Soc., 134 (2006), 2403-2409   math.AG/0306369

(with Kathryn Nyman)  Inequalities for the h- and flag h-vecotrs of geometric lattices, Disc. and Comp. Geom., 32 (2004), 533-548.  pdf file

g-elements, finite buildings and higher Cohen-Macaulay connectivity, J. Combin. Theory Ser. A, 113 (2006), 1305-1320. math.CO/0512086

Face enumeration: from spheres to manifolds. J. Europ. Math. Soc., 11 (2009), 449-485. math/0709.3998

(with I. Novik)  Face ring multiplicity via CM-connectivity sequences,  Canadian J. of Mathematics, 61 (2009), 888-903, arXive:math.AC/0606.5246

(with J.Chestnut and J. Sapir)  Enumerative properties of triangulations of spherical bundles over S^1, European J. Comb., 29 (2008), 662-671. arXive:math.CO/0611.5039

(with I. Novik)  Socles of Buchsbaum modules, complexes and posets  Adv. in Math., 222 (2009), 2059-2084. arXive: mathCO/0711.0783

(with I. Novik)  Applications of Dehn-Sommerville relations,  Disc. and Comp. Geom., 42 (2009), 261-276. pdf file

Topological finiteness for edge-vertex enumeration, Adv. in Math., 219 (2008), 1722-1728. pdf file

(with I. Novik)) Gorenstein rings through face rings of manifolds, Composit. Math., 145 (2009), 993-1000. arXiv:0806.1017

(with F. Lutz and T. Sulanke) f-vectors of 3-manifolds, Elec. J. of Comb. 16 (2)(2009), R13. pdf file

(with E. Miller and I. Novik) Face rings of simplicial complexes with singularities, Math. Ann., 351 (2011), pg. 857-875. arXive:1001.2812   (replaces arXive 0908.1433)
The face ring of a pure simplicial complex modulo m generic linear forms is a ring with finite local cohomology if and only if the link of every face of dimension m or more is nonsingular.  There is a slightly weaker generalization to squarefree modules.  

(with C. Klivans) Projection volumes of hyperplane arrangements.  Discrete and Computational Geometry, 46 (2011), 417-426. arXive:1001.5095
The average projection volumes of the maximal cones of a finite real hyperplane arrangement are given by the chraracterisitic polynomial of the arrangement. The angle sums of a zonotope are given by the characteristic polynomial of the dual of the intersection lattice of the arrangement.

(with I. Novik) Face numbers of pseudomanifolds with isolated singularities, Math. Scan. 110 (2012), 198-222,  arXive:1004.5100
We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the $f$-vector can be completely characterized are described. 

(with M. Hughes) Quotients of spheres by linear actions of tori, arXive:1205.6387
To every linear quotient of a sphere by a real torus there is an associated rationally represented matroid whose Tutte polynomial determines the integral homology of both the full quotient space and the subspace of rationally singular points.   We also determine when the quotient space is a manifold and, more specifically, when it is a sphere.

The average dual surface of a cohomology class and  minimal simplicial decompositions of infinitely many lens spaces,  arXive:1310.1991
  Discrete normal surfaces are normal surfaces whose intersection with each tetrahedron of a triangulation has at most one component.  They are natural Poincar\'e duals to $1$-cocycles with $\ZZ/2\ZZ$-coefficients.  We show that for a simplicial poset and  a fixed cohomology class the average Euler characteristic of the associated discrete normal surfaces only depends on the $f$-vector of the triangulation.  As an application we determine the minimum simplicial poset representations, also known as crystallizations, of lens spaces $L(2k,q),$ where $2k=qr+1.$