Caroline J. Klivans

The Bergman complex of a matroid and phylogenetic trees

We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of the proper part of its lattice of flats. In addition, we show that the Bergman fan B˜(Kn) of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn × R. This leads to a proof that the link of the origin in Tn is homeomorphic to the order complex of the proper part of the partition lattice Πn.

Simplicial matrix-tree theorems

First published in Transactions of the American Mathematical Society in volume 361 (2009), no. 11, 6073--6114, published by the American Mathematical Society

A Non-Partitionable Cohen-Macaulay Simplicial Complex

Abstract A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.

A Cheeger-type inequality on simplicial complexes

In this paper, we consider a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterer and Kahle. A Cheeger-type inequality is proved, which is similar to a result on graphs due to Fan Chung. This inequality is then used to study the relationship between coboundary expanders on simplicial complexes and their corresponding eigenvalues, complementing and extending results found by Gundert and Wagner. In particular, we find these coboundary expanders do not satisfy natural Buser or Cheeger inequalities.

Threshold graphs, shifted complexes, and graphical complexes

We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We present a number of structural results and relations among them including new characterizations of the class of threshold graphs.

Scheduling problems

We introduce the notion of a scheduling problem which is a boolean function S over atomic formulas of the form x i ? x j . Considering the x i as jobs to be performed, an integer assignment satisfying S schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to S. We prove that this counting function is a polynomial in the number of time slots allowed. Scheduling polynomials include the chromatic polynomial of a graph, the zeta polynomial of a lattice, and the Billera-Jia-Reiner polynomial of a matroid.To any scheduling problem, we associate not only a counting function for solutions, but also a quasisymmetric function and a quasisymmetric function in non-commuting variables. These scheduling functions include the chromatic symmetric functions of Sagan, Gebhard, and Stanley, and a close variant of Ehrenborgs quasisymmetric function for posets.Geometrically, we consider the space of all solutions to a given scheduling problem. We extend a result of Steingrimsson by proving that the h-vector of the space of solutions is given by a shift of the scheduling polynomial. Furthermore, under certain conditions on the defining boolean function, we prove partitionability of the space of solutions and positivity of fundamental expansions of the scheduling quasisymmetric functions and of the h-vector of the scheduling polynomial.

Chip-firing and energy minimization on M-matrices

We consider chip-firing dynamics defined by arbitrary M-matrices. M-matrices generalize graph Laplacians and were shown by Gabrielov to yield avalanche finite systems. Building on the work of Baker and Shokrieh, we extend the concept of energy minimizing chip configurations. Given an M-matrix, we show that there exists a unique energy minimizing configuration in each equivalence class defined by the matrix.We consider the class of z-superstable configurations. We prove that for any M-matrix, the z-superstable configurations coincide with the energy minimizing configurations. Moreover, we prove that the z-superstable configurations are in simple duality with critical configurations. Thus for all avalanche-finite systems (including all directed graphs with a global sink) there exist unique critical, energy minimizing and z-superstable configurations. The critical configurations are in simple duality with energy minimizers which coincide with z-superstable configurations.

Visibility constraints on features of 3D objects

To recognize three-dimensional objects it is important to model how their appearances can change due to changes in viewpoint. A key aspect of this involves understanding which object features can be simultaneously visible under different viewpoints. We address this problem in an image-based framework, in which we use a limited number of images of an object taken from unknown viewpoints to determine which subsets of features might be simultaneously visible in other views. This leads to the problem of determining whether a set of images, each containing a set of features, is consistent with a single 3D object. We assume that each feature is visible from a disk of viewpoints on the viewing sphere. In this case we show the problem is NP-hard in general, but can be solved efficiently when all views come from a circle on the viewing sphere. We also give iterative algorithms that can handle noisy data and converge to locally optimal solutions in the general case. Our techniques can also be used to recover viewpoint information from the set of features that are visible in different images. We show that these algorithms perform well both on synthetic data and images from the COIL dataset.

Obstructions to Shiftedness

Abstract In this paper we show results on the combinatorial properties of shifted simplicial complexes. We prove two intrinsic characterization theorems for this class. The first theorem is in terms of a generalized vicinal preorder. It is shown that a complex is shifted if and only if the preorder is total. Building on this we characterize obstructions to shiftedness and prove there are finitely many in each dimension. In addition, we give results on the enumeration of shifted complexes and a connection to totally symmetric plane partitions.

Projection Volumes of Hyperplane Arrangements

We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones are given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans that this held for all finite real reflection arrangements. The methods used are geometric and combinatorial. As a consequence, we determine that the angle sums of a zonotope are given by the characteristic polynomial of the order dual of the intersection lattice of the arrangement.