Lucas Sabalka

Discrete Morse theory and graph braid groups

If is any finite graph, then the unlabelled configuration space of n points on , denoted U C n , is the space of n-element subsets of . The braid group of on n strands is the fundamental group of U C n . We apply a discrete version of Morse theory to these U C n , for any n and any , and provide a clear description of the critical cells i n every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space U C n strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in of degree at least 3 (and k is thus independent of n). AMS Classification 20F65, 20F36; 57M15, 57Q05, 55R80

On the cohomology rings of tree braid groups

Abstract Let Γ be a finite connected graph. The (unlabelled) configuration space U C n Γ of n points on Γ is the space of n -element subsets of Γ . The n -strand braid group of Γ , denoted B n Γ , is the fundamental group of U C n Γ . We use the methods and results of [Daniel Farley, Lucas Sabalka, Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005) 1075–1109. Electronic] to get a partial description of the cohomology rings H ∗ ( B n T ) , where T is a tree. Our results are then used to prove that B n T is a right-angled Artin group if and only if T is linear or n 4 . This gives a large number of counterexamples to Ghrist’s conjecture that braid groups of planar graphs are right-angled Artin groups.

Face vectors of subdivided simplicial complexes

Brenti and Welker have shown that, for any (d-1)-dimensional simplicial complex X, the f-vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number -2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric motivation to generalize the combinatorial statements to arbitrary subdivision methods: any subdivision method will exhibit similar limit behavior and symmetry. Our techniques allow us to compute explicit formulas for the values of the limit roots in the case of barycentric subdivision.

On rigidity and the isomorphism problem for tree braid groups

We solve the isomorphism problem for braid groups on trees with n D 4 or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations using the cohomology of tree braid groups, couching our discussion in the language of differential forms. Second, we show that, given a tree braid group BnT on n D 4 or 5 strands, H � .Bn T/ is an exterior face algebra. Finally, we prove that one may reconstruct the tree T from a tree braid group BnT for n D 4 or 5. Among other corollaries, this third step shows that, when n D 4 or 5, tree braid groups BnT and trees T (up to homeomorphism) are in bijective correspondence. That such a bijection exists is not true for higher dimensional spaces, and is an artifact of the 1-dimensionality of trees. We end by stating the results for right-angled Artin groups corresponding to the main theorems, some of which do not yet appear in the literature.

MULTIDIMENSIONAL ONLINE MOTION PLANNING FOR A SPHERICAL ROBOT

We consider three related problems of robot movement in arbitrary dimensions: coverage, search, and navigation. For each problem, a spherical robot is asked to accomplish a motion-related task in an unknown environment whose geometry is learned by the robot during navigation. The robot is assumed to have tactile and global positioning sensors. We view these problems from the perspective of (non-linear) competitiveness as defined by Gabriely and Rimon. We first show that in 3 dimensions and higher, there is no upper bound on competitiveness: every online algorithm can do arbitrarily badly compared to the optimal. We then modify the problems by assuming a fixed clearance parameter. We are able to give optimally competitive algorithms under this assumption. We show that these modified problems have polynomial competitiveness in the optimal path length, of degree equal to the dimension.

On the geometry of the edge splitting complex

The group Out of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which Out acts, in analogy with the curve complex for the mapping class group. Here, we focus on one of these proposed analogues: the edge splitting complex E n, equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton E 1n algebraically, and use our characterization to find lower bounds on distances between points in this graph. Our distance calculations allow us to find quasiflats of arbitrary dimension in E n. This shows that E n: is not hyperbolic, has infinite asymptotic dimension, and is such that every asymptotic cone is infinite dimensional. These quasiflats contain an unbounded orbit of a reducible element of Out. As a consequence, there is no coarsely Out-equivariant quasiisometry between E n and other proposed curve complex analogues, including the regular free splitting complex F n, the (nontrivial intersection) free factorization complex FF n, and the free factor complex Fn. Mathematics Subject Classification (2010). 20F65, 20E36.

Projection-forcing multisets of weight changes

Let F be a finite field. A multiset S of integers is projection-forcing if for every linear function @f:F^n->F^m whose multiset of weight changes is S, @f is a coordinate projection up to permutation and scaling of entries. The MacWilliams Extension Theorem from coding theory says that S={0,0,...,0} is projection-forcing. We give a (super-polynomial) algorithm to determine whether or not a given S is projection-forcing. We also give a condition that can be checked in polynomial time that implies that S is projection-forcing. This result is a generalization of the MacWilliams Extension Theorem and work by the first author.

Modeling bed exit likelihood in a camera-based automated video monitoring application

Hospital inpatients often fall during an exit from the bed or in the ensuing seconds and minutes. Existing fall prevention technologies fail to provide adequate lead time to a patient bed exit or exhibit high rates of false alarms. To address these limitations and reduce risk of falls for patients and hospitals, we have developed a 3D camera-based system, named Ocuvera, for monitoring patients at risk of falling, without requiring a human monitor. The developed automated system looks for cues that predict a likely bed exit. If the system determines that the risk to patient safety is high, the system alerts nursing staff, often with enough lead time to prevent the exit. In this paper we discuss the algorithmic pipeline of the developed system, starting with the raw camera feed and ending with alarms. Emphasis will be placed on computer vision models of behaviors and objects, as well as a machine-learned bed exit risk model.

ON RESTRICTING SUBSETS OF BASES IN RELATIVELY FREE GROUPS

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety AmAn, and let A = {a1,…, ar} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {al+1,…, ar} for some l < r, then S is a subset of a basis for the relatively free group on {a1,…, al}.