Lucas Sabalka's Homepage
I have moved: I am now an Assistant Professor at Saint Louis University.
I am a Riley Assistant Professor at Binghamton University, to work with Ross Geoghegan. I have been at Binghamton since January 2009. Before that, I was a Krener Assistant Professor at the University of California, Davis to work with 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.(more)
Curriculum Vitae, etc.
- Curriculum Vitae (updated 19 Nov 2011)
- Research Statement (updated 19 Nov 2011)
- Teaching Statement (updated 19 Nov 2011)
- Publication List (updated 19 Nov 2011)
I am a geometric group theorist. Geometric group theory (also known as combinatorial group theory) is a highly interdisciplinary field focusing on the study of groups via their actions on geometric spaces. Geometric group theory uses the tools and approaches of algebraic topology, combinatorics, commutative algebra, semigroup theory, hyperbolic geometry, geometric analysis, computational group theory, computational complexity theory, logic, dynamical systems, probability theory, and other areas. It is a young and fast-growing field, with much of the work in the area accomplished within the past 30 years.
My work has embraced the interdisciplinary nature of my field -- I have published theorems which could be classified in each of: group theory, commutative algebra, algebraic topology, combinatorics, coding theory, mathematical robotics, computational complexity theory, and differential geometry. For example, I have used tools as diverse as: exterior face algebras and Stanley-Reisner rings, differential forms, Fox calculus, cohomology rings, discrete Morse theory, face polynomials of simplicial complexes, linear codes, configuration spaces, fundamental groups, and coarse curvature conditions. My work has appeared or been accepted in top journals in a number of fields, including: the International Journal of Algebra and Computation; the Journal of Combinatorial Theory Series A; the Journal of Pure and Applied Algebra; and Algebraic and Geometric Topology.
Papers below are linked to their journal of publication when possible, and all available appear on the ArXiv.
- A Classifying Space for Pure Braided Thompson's Group
With Keith Jones. In Progress. (project description)
- Generalized expanders and Lipschitz
With Jerry Kaminker. In Progress. (project description)
- Submanifold Projection for Out(F_n)
With Matt Clay and Dmytro Savchuk. In Progress. (project description)
On restricting subsets of bases in relatively free groups
With Dmytro Savchuk. To appear, International Journal of Algebra and Computation. (abstract)
On the geometry of a proposed curve complex analogue for
With Dmytro Savchuk. Submitted. (abstract)
Face vectors of subdivided simplicial complexes
With Emanuele Delucchi and Aaron Pixton. Discrete Mathematics (appeared online 1 Oct 2011, to appear in print). (abstract)
Projection-forcing multisets of weight changes
With Josh Brown Kramer. Journal of Combinatorial Theory, Series A, 117(8): 1136-1142, 2010. (abstract)
Multidimensional online motion planning for a spherical robot
With Josh Brown Kramer. International Journal of Computational Geometry and Applications, 20(6):653-684, 2010. (abstract)
Presentations of graph braid groups
With Daniel Farley. To appear, Forum Mathematicum. (abstract)
rigidity and the isomorphism problem for tree braid groups
Groups, Geometry, and Dynamics,3(3):469-523, 2009. (abstract)
the cohomology rings of tree braid groups
With Daniel Farley. Journal of Pure and Applied Algebra, 212(1):53-71, 2007. (abstract)
of right-angled Artin groups into graph braid groups
Geometriae Dedicata, 124:191-198, 2007. (abstract)
Discrete Morse theory and graph braid groups
With Daniel Farley. Algebraic and Geometric Topology, 5:1075-1109, 2005. (abstract)
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.
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 Calendar.
Math 601A: Topics in Topology: Mapping Class Groups and Out(F_n)
MW3:30pm-5:00pm in LN-2201
- Math 601A: Topics in Topology: Mapping Class Groups and Out(F_n)
- I have led three Research Experiences for undergraduates (Summer 2007, Summer 2008). The Summer 2008 project was with students Paul Prue and Travis Scrimshaw, both now graduate students at UC Davis, on braid groups on graphs. They have a preprint improving Aaron Abrams's `sufficient subdivision' theorem. Scrimshaw also wrote a second paper based on my REU, on which graph braid groups are classical braid groups.
Conferences(see slides below)
- Special Session on Geometric, Combinatorial, and Computational Group Theory*, University of Utah, October 2011.
- Special Session on Algorithmic and Geometric Properties of Groups and Semigroups*, University of Nebraska, Lincoln, October 2011.
- Special Session on Geometry of Arithmetic Groups*, Cornell, September 2011.
- Special Session on Computational Algebra, Groups, and Applications*, Las Vegas, April 2011.
- Spring Topology and Dynamics Conference, University of Texas at Tyler, March 2011.
- Wasatch Topology Conference*, University of Utah, December 2010.
- Workshop on the Geometry of the Outer Automorphism Group of a Free Group*, American Institute of Mathematics, October 2010.
- Approaches to Group Theory (Ken Brown Conference), Cornell, October 2010.
(* = invited address)
- Oct 2011: On restricting subsets of bases in relatively free groups, at Utah.
- Oct 2011: On restricting subsets of bases in relatively free groups, at UNL.
- Sep 2011: On restricting subsets of bases in relatively free groups, at Cornell.
- Apr 2011: Geometry of Curve Complex Analogues for Out(F_n), at Las Vegas.
- Mar 2011: Geometry of Curve Complex Analogues for Out(F_n), at STDC.
- Nov 2010: Geometry of Curve Complex Analogues for Out(F_n), II, at Binghamton. This is part II, where part I was given by Dima Savchuk, but it is self-contained.
- Oct 2010: On the Geometry of the Free Factorization Graph, at AIM.
- Oct 2010: On the Geometry of the Free Factorization Graph, at Ken Brown Fest, Cornell.
- Sep 2010: Geometry of the Free Splitting Graph (now called the free factorization graph), at UNL.
- Apr 2010: Geometry of Out(F_n), at Binghamton.
- Mar 2010: Geometry of Out(F_n), at STDC.
- I write Ask-A-Scientist columns for the local paper, the Binghamton Press and Sun Bulletin (also appearing in the Ithaca Journal and the Elmira Gazette), answering questions from local school children. Here are my articles:
- A friend has started an interesting new website, MinutesPlease.com . Check it out; if you like it, Digg it.
- We Can Solve It
- Help save the world.
- Interesting mathematics
- Since this blog of interesting coarse math cites me, I'm citing it!