Lucas Sabalka's Homepage

(Photo of Lucas Sabalka)
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, combinatorics

About Me


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.

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Curriculum Vitae, etc.

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Research

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.

  1. A Classifying Space for Pure Braided Thompson's Group
    With Keith Jones. In Progress. (project description)

  2. Generalized expanders and Lipschitz cohomology
    With Jerry Kaminker. In Progress. (project description)

  3. Submanifold Projection for Out(F_n)
    With Matt Clay and Dmytro Savchuk. In Progress. (project description)

  4. On restricting subsets of bases in relatively free groups
    With Dmytro Savchuk. To appear, International Journal of Algebra and Computation. (abstract)

  5. On the geometry of a proposed curve complex analogue for Out(F_n)
    With Dmytro Savchuk. Submitted. (abstract)

  6. Face vectors of subdivided simplicial complexes
    With Emanuele Delucchi and Aaron Pixton. Discrete Mathematics (appeared online 1 Oct 2011, to appear in print). (abstract)

  7. Projection-forcing multisets of weight changes
    With Josh Brown Kramer. Journal of Combinatorial Theory, Series A, 117(8): 1136-1142, 2010. (abstract)

  8. 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)

  9. Presentations of graph braid groups
    With Daniel Farley. To appear, Forum Mathematicum. (abstract)

  10. On rigidity and the isomorphism problem for tree braid groups
    Groups, Geometry, and Dynamics,3(3):469-523, 2009. (abstract)

  11. On the cohomology rings of tree braid groups
    With Daniel Farley. Journal of Pure and Applied Algebra, 212(1):53-71, 2007. (abstract)

  12. Embeddings of right-angled Artin groups into graph braid groups
    Geometriae Dedicata, 124:191-198, 2007. (abstract)

  13. Discrete Morse theory and graph braid groups
    With Daniel Farley. Algebraic and Geometric Topology, 5:1075-1109, 2005. (abstract)

  14. 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.

Dissertation



Geometry and Topology Seminar Calendar.


Teaching

Present
Spring 2012
Past

Mentorship
  • 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)

(* = invited address)

Slides


Links

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!