Lucas Sabalka's Homepage

(Photo of Lucas Sabalka)
Address: Department of Mathematics and Computer Science
Saint Louis University, St Louis, MO 63103
Office: RH 214
Office Phone: (314) 977-2435
Fax: (314) 977-1452
Email: lsabalka@slu.edu
Research Interests: geometric group theory, computational geometry, algebraic topology, combinatorics

About Me

Since the Fall of 2012 I have been an Assistant Professor at Saint Louis University. I am looking forward to working with the many interesting people with research related to my own here.

In August 2012, I was officially trained as a Leader in the Climate Reality Leadership Corps by the Climate Reality Project. I am excited to give talks on both mathematics and climate reality. Take a look at my Climate Reality profile.

Before joining SLU, I was a Riley Assistant Professor at Binghamton University, to work with Ross Geoghegan. I was there from January 2009 to May 2012. 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 Braided Thompson's Group
    With Matthew Zaremsky. In Progress. (project description)

  2. LYM Inequalities for Graded Posets
    With Joshua Brown Kramer. In Progress. (project description)

  3. Submanifold projection for Out(F_n)
    With Dmytro Savchuk. Preprint. (project description)

  4. On restricting subsets of bases in relatively free groups (Erratum)
    With Dmytro Savchuk. International Journal of Algebra and Computation, 22(4): 12250030 [8 pages], 2012. (abstract)

  5. On the geometry of the edge splitting complex
    With Dmytro Savchuk. To appear, Groups, Geometry, and Dynamics. (abstract)

  6. Face vectors of subdivided simplicial complexes
    With Emanuele Delucchi and Aaron Pixton. Discrete Mathematics 312(2): 248-257, 2012. (abstract)

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

  8. Multidimensional online motion planning for a spherical robot
    With Joshua Brown Kramer. International Journal of Computational Geometry and Applications, 20(6):653-684, 2010. (abstract)

  9. Presentations of graph braid groups
    With Daniel Farley. Forum Mathematicum, 24(4): 827-859, 2012. (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



Teaching

Present
Fall 2013
  • Math 142: Calculus I (webages on SLU Global)
    MTWF10:00am-10:50am in RH-223
    MTWF1:10pm-2:00pm in RH-223
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.

Book Projects


Conferences

(see slides below)

(* = invited address)

Slides


Links