Workshop

All finite graphs satisfy the two properties mentioned in the title. I will explain what I mean by this, and speculate on generalizations and interconnections. This talk will be non-technical: Nothing will be assumed beyond basic linear algebra.

The extension space conjecture, proposed by Sturmfels and Ziegler in 1993, is a conjecture about the topology of a realizable oriented matroid's "extension space", which is a topological model for the set of all extensions of the oriented matroid by a single element. Equivalently, it is a conjecture about the poset of proper zonotopal tilings of a zonotope, namely that this poset is homotopy equivalent to a sphere. In this talk we describe a counterexample to this conjecture in three dimensions.

The chromatic quasisymmetric function of a labeled graph was introduced by Shareshian and myself as a refinement of Stanley’s chromatic symmetric function. We conjectured a refinement of the long standing Stanley- Stembridge e-positivity conjecture, and formulated an algebro-geometric approach to proving this refined conjecture involving Hessenberg varieties. Significant progress in this direction has recently been made by Brosnan and Chow and by Guay-Paquet. In this talk, I will discuss the connection with Hessenberg varieties, and also present some new directions, including results on generalizations to directed graphs obtained by my student Brittney Ellzey

Given a collection of finite sets, Kneser-type problems aim to partition the collection into parts with well-understood intersection pattern, such as in each part any two sets intersect. Since Lovász' solution of Kneser's conjecture, concerning intersections of all k-subsets of an n-set, topological methods have been a central tool in understanding intersection patterns of finite sets. We will develop a method that in addition to using topological machinery takes the topology of the collection of finite sets into account via a translation to a problem in Euclidean geometry. This leads to simple proofs of old and new results.

Starting from Barnette's classical lower bound theorem for simplicial polytopes (LBT), and its proof via framework rigidity by Kalai, I'll then discuss corresponding theorems and conjectures, as well as proof techniques, for sub-classes and super-classes of interest. For example, we'll consider centrally-symmetric simplicial polytopes, flag polytopes, cubical polytopes, polytopes with one non-simplex facet, general polytopes, and analogues for polyhedral spheres and manifolds.

A central problem of algebraic topology is to understand the homotopy groups π_d(X) of a topological space X. 
For the computational version of the problem, it is well known that there is no algorithm to decide whether the 
fundamental group π₁(X) of a given finite simplicial complex X is trivial. On the other hand, there are several 
algorithms that, given a finite simplicial complex X that is simply connected (i.e., with π₁(X) trivial), compute 
the higher homotopy group π_d(X) for any given d ≥ 2.

However, these algorithms come with a caveat: They compute the isomorphism type of π_d(X), d ≥ 2, as an abstract 

finitely generated abelian group given by generators and relations, but they work with very implicit representations 

of the elements of π_d(X). 

We present an algorithm that, given a simply connected simplicial complex X, computes π_d(X) and represents its elements 

as simplicial maps from a suitable triangulation of the d-sphere to X. For fixed d, the algorithm runs in time singly exponential 

in size(X), the number of simplices of X. 

Moreover, we prove that this is optimal: For every fixed d ≥ 2, we construct a family of simply connected simplicial complexes X 

such that for any simplicial map representing a generator of π_d(X), the size of the triangulation of S^d on which the map is defined is 

exponential in size(X). 

Apart from the intrinsic importance of homotopy groups, we view this as a first step 

towards the more general goal of computing explicit maps with specific topological properties, 

e.g., computing explicit embeddings of simplicial complexes or counterexamples to Tverberg-type

problems, and towards quantitative bounds on the complexity of such maps.

Joint work with Marek Filakovský, Peter Franek, and Stephan Zhechev

I will survey some recent progress on complexity of counting integer points in convex polytopes of fixed dimension. A few our results will have a seductively easy yet counterintuitive statements. The talk will assume no background and will be very accessible. Joint work with Danny Nguyen.

Schubert varieties parametrize families of linear spaces intersecting certain hyperplanes of C^n in a predetermined way. In the 1970’s Hansen and Demazure independently constructed resolutions of singularities for Schubert varieties: the Bott-Samelson varieties. In this talk I will describe their relation with associahedra. I will also discuss joint work with Pechenick-Tenner-Yong linking Magyar’s construction of these varieties as configuration spaces with Elnitsky’s rhombic tilings. Finally, based on joint work with Wyser-Yong, I will give a parallel for the Barbasch-Evens desingularizations of certain families of linear spaces which are constructed using symmetric subgroups of the general linear group.

We study the motion of a robotic arm inside a rectangular tunnel. We prove that the configuration space of all possible positions of the robot is a CAT(0) cubical complex. To do this we use a bijection between rooted CAT(0) cubical complexes and a family of combinatorial objects that we call “posets with inconsistent pairs”. This bijection allows us to use techniques from geometric group theory and poset theory to find the optimal way of moving the arm from one position to another. We also compute the diameter of the configuration space, that is, the longest distance between two positions of the robot. This talk will include joint work with Tia Baker, Hanner Bastidas, Cesar Ceballos, John Guo, Megan Owen, Seth Sullivant, and Rika Yatchak, and will assume no previous knowledge of the subject.