Isabella Novik

Syzygies of oriented matroids

We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids, and oriented matroids. These are StanleyReisner ideals of complexes of independent sets and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of R. Stanley’s formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by D. Bayer, S. Popescu, and B. Sturmfels []. We resolve the combinatorial problems posed in [3] by computing Mobius invariants of graphic and cographic arrangements in terms of Hermite polynomials.

How Neighborly Can a Centrally Symmetric Polytope Be

AbstractWe show that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n + d) vertices, where

Upper bound theorems for homology manifolds

k(d,n)=\Theta\left(\frac{d}{1+\log ((d+n)/d)}\right).

A Centrally Symmetric Version of the Cyclic Polytope

We also show that this bound is tight.

Gorenstein rings through face rings of manifolds

AbstractIn this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that βk(Δ)⩽Σ{βi(Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where βi(Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.)We prove an analog of the UBC for all other even-dimensional homology manifolds.Kuhnel conjectured that for every 2k-dimensional combinatorial manifold withn vertices,

A Note on Geometric Embeddings of Simplicial Complexes in a Euclidean Space

\left( { - 1} \right)^k \left( {\mathcal{X}\left( \Delta \right) - 2} \right) \leqslant \left( {\begin{array}{*{20}c} {n - k - 2} \\ {k + 1} \\ \end{array} } \right)/\left( {\begin{array}{*{20}c} {2k + 1} \\ k \\ \end{array} } \right)

Lower Bound Theorems and a Generalized Lower Bound Conjecture for balanced simplicial complexes

. We prove this conjecture for all 2k-dimensional homology manifolds withn vertices, wheren≥4k+3 orn≤3k+3. We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.

Remarks on the upper bound theorem

Abstract We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 1≤j<k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least