Lucas Sabalka's Homepage

Address:	Department of Mathematical Sciences Binghamton University, SUNY Binghamton, NY 13902-6000
Office:	LN 2220
Office Phone:	(607) 777-4246
Fax:	(607) 777-2450
Email:	sabalka@math.binghamton.edu
Research Interests:	geometric group theory, algebraic topology, computational geometry

About Me

I am a Riley Assistant Professor at Binghamton University. I have been at Binghamton since January 2009. Before that, I was a Krener Assistant Professor at the University of California, Davis under the guidance of Misha Kapovich. I received my Ph.D. from the University of Illinois at Urbana-Champaign in May 2006 for my dissertation Braid Groups on Graphs. My Ph.D. advisor was Ilya Kapovich.

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In 2002, I was fortunate enough to be awarded both a UIUC mathematics department VIGRE fellowship and a National Science Foundation Graduate Research Fellowship.

Also in 2002, I received my undergraduate Bachelor of Science degree With Highest Distinction and With Honors from the University of Nebraska-Lincoln with majors in mathematics, history, and computer science and minors in psychology and physics. My undergraduate advisers were Susan Hermiller and John Meakin.

To see some of my personal pictures, click here; to see some of my mathematical pictures, click here.

Curriculum Vitae, etc.

Curriculum Vitae (updated 30 Oct 2009)

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Research

I am currently working on three main topics.

In my dissertation, I studied the notion of braid groups on graphs, frequently in joint work with Dan Farley. Dan Farley and I just finished a new paper on presentations of these groups. I am also leading an extended REU (Summer 2008-present) with students Paul Prue and Travis Scrimshaw, both of UC Davis. They have a preprint improving Aaron Abrams's `sufficient subdivision' theorem.
Conversations about braid groups on graphs have led me to interesting problems in mathematical robotics (from the computational geometry point of view). I am currently working on certain online navigation problems for mobile robots in unknown arbitrary-dimensional environments with Josh Brown Kramer. Simulations for this project may be found on our Robot Motion Homepage. I am also working with Elon Rimon on other applications in this area.
I have lately become intrigued with Out(F_n), the outer automorphism group of the free group. In particular, because of conversations with Ilya Kapovich and with Moon Duchin and various people at UC Davis, I have been thinking about curve complex analogues for Out(F_n).

I also have latent projects with Jerry Kaminker on generalized expanding graphs; and with Moon Duchin on horofunction boundaries, for instance of the mapping class group.

Papers below are linked to their journal of publication. Many also appear in the ArXiv.

F-vectors of subdivided simplicial complexes.
With Emanuele Delucchi. In Progress. (abstract)

In a recent paper of Brenti and Welker, it was shown that for any simplicial $n$-dimensional 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 give an alternate and intuitive proof of this fact here, and compute explicit formulas for the values of these roots. In the process, we observe an interesting symmetry of these roots about the real number $-2$.
Generalized expanders and Lipschitz cohomology.
With Jerry Kaminker. In Progress. (abstract)

Romain Tessera has introduced a generalized notion of expander. In this paper, we study a moduli space of Tessera some expanders. This is used to study the obstruction to being able to uniformly embed a metric space into a Hadamard manifold. We also study how these expanders obstruct the property that all the cohomology of a discrete group is Lipschitz (in the sense of \cite{ConnesGromovMoscovici}). In general this work is aimed at understanding how classes in the cohomology of a group might satisfy the Novikov conjecture, yet not come, in some sense, from a boundary.
Projection-forcing multisets of weight changes.
with Josh Brown Kramer. Submitted. (abstract)

Let $\F$ be a finite field. A multiset $S$ of integers is \emph{projection-forcing} if for every linear map $\phi: \F^n \to \F^m$ whose multiset of weight changes is $S$, $\phi$ is a coordinate projection up to permutation of entries. The MacWilliams Extension Theorem from coding theory says that $S = \{0, 0, \ldots, 0\}$ 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 The MacWilliams Extension Theorem and work by the first author.
Multidimensional online mobile robot planning.
With Josh Brown Kramer. To appear, International Journal of Computational Geometry and Applications. (abstract)

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.
Presentations of graph braid groups
With Daniel Farley. Submitted. (abstract)

Let $\Gamma$ be a graph. The (unlabeled) configuration space $\ucng$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $\bng$, is the fundamental group of $\ucng$. This paper extends the methods and results of \cite{FarleySabalka1}. For any graph $\Gamma$, we describe the relators of a corresponding graph braid group computed using a version of Forman's discrete Morse theory. In particular, we describe presentations for graph braid groups with $2$ strands.
On rigidity and the isomorphism problem for tree braid groups.
Groups, Geometry, and Dynamics,3(3):469-523, 2009.\\ (abstract)

We solve the isomorphism problem for braid groups on trees with $n = 4$ 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 $B_nT$ on $n = 4$ or $5$ strands, $H^*(B_nT)$ is an exterior face algebra. Finally, we prove that one may reconstruct the tree $T$ from a tree braid group $B_nT$ for $n = 4$ or $5$. Among other corollaries, this third step shows that, when $n = 4$ or $5$, tree braid groups $B_nT$ 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.
On the cohomology rings of tree braid groups .
With Daniel Farley. Journal of Pure and Applied Algebra, 212(1):53-71, 2007. (abstract)

Let $\Gamma$ be a finite connected graph. The (unlabelled) configuration space $\ucng$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $\bng$, is the fundamental group of $\ucng$. We use the methods and results of \cite{FS1} to get a partial description of the cohomology rings $H^{\ast}(B_n T)$, 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.
Embeddings of right-angled Artin groups into graph braid groups.
Geometriae Dedicata, 124:191-198, 2007. (abstract)

We construct an embedding of any right-angled Artin group $G(\Delta)$ defined by a graph $\Delta$ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of $\Delta$. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.
Discrete Morse theory and graph braid groups.
With Daniel Farley. Algebraic and Geometric Topology, 5:1075-1109, 2005. (abstract)

If $\Gamma$ is any finite graph, then the \emph{unlabelled configuration space of $n$ points on $\Gamma$}, denoted $\ucng$, is the space of $n$-element subsets of $\Gamma$. The \emph{braid group of $\Gamma$ on $n$ strands} is the fundamental group of $\ucng$. We apply a discrete version of Morse theory to these $\ucng$, for any $n$ and any $\Gamma$, and provide a clear description of the critical cells in 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 proofof a theorem due to Ghrist: the space $\ucng$ strong deformation retracts onto a CW complex of dimension at most $k$, where $k$ is the number of vertices in $\Gamma$ of degree at least $3$ (and $k$ is thus independent of $n$).
Geodesics in the braid group on three strands.
In Group theory, statistics, and cryptography, volume 360 of Contemporary Mathematics,
pages 133-150. Amer. Math. Soc., Providence, RI, 2004. (abstract)

This is a version of my undergraduate thesis, prepared under advisors Susan Hermiller and John Meakin.

We study the geodesic growth series of the braid group on three strands, $B_3 := \langle a,b|aba = bab \rangle$. We show that the set of geodesics of $B_3$ with respect to the generating set $S := \{a,b\}^{\pm 1}$ is a regular language, and we provide an explicit computation of the geodesic growth series with respect to this set of generators. In the process, we give a necessary and sufficient condition for a freely reduced word $w \in S^*$ to be geodesic in $B_3$ with respect to $S$. Also, we show that the translation length with respect to $S$ of any element in $B_3$ is an integer.

Dissertation

Braid Groups on Graphs.
PhD thesis, U. of Illinois at Urbana-Champaign, 2006.
This is my doctoral dissertation. It includes all results from papers 2-4 and some results from 5 and 6. It is meant to be a more readable, compact, and polished compilation of the earlier work.

Geometry and Topology Seminar

Teaching

Present

Fall 2009 (Office Hours: MW1:00pm-2:15pm)

Math 130: Mathematics in Action
MWF 12:00pm-1:00pm in LH-02

Past

Binghamton U, Spring 2009-present

Fall 2009

Math 222: Calculus II, Section 4
MWF 12:00pm-1:00pm in LH-12 and T 10:05am-11:30am in EB-Q23
Math 222: Calculus II, Section 6
MWF 2:20pm-3:20pm and R 11:40am-1:05pm in LH-4

Spring 2009

Math 222: Calculus II, Section 2
MWF 12:00pm-1:00pm in FA-209 and T 8:30am-9:55am in LH-003
Math 375: Complex Variables
MWF 8:30am-9:30am in SW-331 and R 8:30am-9:55am in S2-143

UC Davis, Fall 2006-Fall 2008
(most webpages hosted on my.ucdavis.edu)

Fall 2008

Math 16A: Short Calculus I, Section 2

Fall 2007-Spring 2008

2007-2008 Research Focus Group

Spring 2008

Math 147: Topology

2008 Winter

Math 16A: Short Calculus A

2007 Spring

Math 141: Euclidean Geometry

2007 Spring

Math 147: Topology

2007 Winter

Math 16B: Short Calculus B

2006 Fall

Math 210A: Graduate Topics in Geometry

U of Illinois at Urbana-Champaign, Fall 2004-Spring 2006

2006 Spring

Math 225: Introductory Matrix Theory, section T1

2005 Fall

Math 225: Introductory Matrix Theory, section S2

Conferences

(see slides below)

Geometric and Combinatorial Methods in Group Theory and Semigroup Theory, Lincoln, Nebraska, May 2009.
Special Session on Geometric Group Theory, Champaign, IL, March 2009.
Spring Topology and Dynamics Conference*, Milwaukee, Wisconsin, March 2008.
Special Session on Geometric Group Theory*, Louisiana State University, Baton Rouge, Louisiana, March 2008.
Postdoctoral seminar, Program in Geometric Group Theory, MSRI, Fall 2007.
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(* = invited address)

Slides

May 2009: Generalized Expanders, at UNL.
Mar 2009: Generalized Expanders, at UIUC.
Mar 2009: Generalized Expanders, seminar at Lafayette.
Oct 2008: Navigating Blindly (robot navigation), colloquium at SJSU.
Aug 2008: Navigating Blindly (robot navigation), seminars at Binghamton and Cornell.
May 2008: Navigating Blindly (robot navigation), seminar at UC Davis.
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