Lucas Sabalka's Homepage

Department of Mathematics
University of California, Davis
One Shields Avenue
Davis, California 95616

Office: 2228 Mathematical Sciences Building
Office Phone: (530) 752-0878
Phone: (402) 770-0588

e-mail: sabalka (at) math.ucdavis.edu

Personal Information

My name is Lucas Sabalka, and I am a Krener Assistant Professor at the University of California, Davis. I received my Ph.D. from the University of Illinois at Urbana-Champaign in May 2006 for my dissertation Braid Groups on Graphs.

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 Ph.D. advisor was Ilya Kapovich. My main mathematical area of interest is geometric group theory.

I am currently working on three main topics. Firstly, in my dissertation, I studied the notion of braid groups on graphs, sometimes with my coauthor Dan Farley. Secondly, conversations about braid groups on graphs have led me to interesting problems in robotics. I am currently working on certain online navigation problems for mobile robots in unknown arbitrary-dimensional environments. Thirdly, I am working with Moon Duchin and occasionally Matt Rathbun on horofunction boundaries of various spaces, hopefully including the Cayley graph of the mapping class group. In the process, we are considering the interrelations between a few different curvature conditions. As a side project, a good friend of mine, Josh Brown Kramer, and I are considering a problem in extremal set theory.

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

Since this blog of interesting mathematics cites me, I am citing it!

Papers

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

9. Projection-forcing multisets of weight changes, with Josh Brown Kramer. In progress. Please email me for a current version.
   
   
8. Multidimensional online mobile robot planning, with Josh Brown Kramer. In progress. Please email me for a current version.
   
   
7. Horofunction boundaries, with Matt Rathbun and Moon Duchin, is in the preliminary stages right now.
   
   
6. On presentations of graph braid groups, with Daniel Farley. In progress. Please email me for a current version.
   
   
5. On rigidity and the isomorphism problem for tree braid groups. Submitted, 2007.
   
   
4. On the cohomology rings of tree braid groups , with Daniel Farley. Journal of Pure and Applied Algebra, 212(1):53-71, 2007.
   
   
3. Embeddings of right-angled Artin groups into graph braid groups. Geometriae Dedicata, 124:191-198, 2007.
   
   
2. Discrete Morse theory and graph braid groups, with Daniel Farley. Algebraic and Geometric Topology, 5:1075-1109, 2005.
   
   
1. 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.
   This is a version of my undergraduate thesis, prepared under advisors Susan Hermiller and John Meakin.
   
   

Dissertation

Conference Presentations

• 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.
• Special Session on Combinatorial and Geometric Group Theory*, Miami University, Oxford, Ohio, March 2007.
• Conference on Topology and Robotics*, ETH Zurich, July 2006.
• Conference on Combinatorial and Geometric Group Theory, in honor of A. Yu. Olshanskii, Vanderbilt University, May 2006.
• Spring Topology and Dynamics Conference*, University of North Carolina at Greensboro, March 2006.
• Geometric Groups on the Gulf Coast Conference, University of Southern Alabama, March 2006.
• Conference on Geometric and Probabilistic Methods in Group Theory and Dynamical Systems, Texas A&M University, November 2005.
• Special Session on Geometric Methods in Group Theory and Semigroup Theory, University of Nebraska-Lincoln, October 2005.
• Conference on Asymptotic and Probabilistic Methods in Geometric Group Theory, University of Geneva, Switzerland, June 2005.
• Special Session on Curvature in Group Theory and Combinatorics, 1007th AMS Meeting, University of California-Santa Barbara, April 2005.
• Special Session on Braids and Knots*, 1000th AMS Meeting, University of New Mexico-Albuquerque, October 2004.
• Albany Group Theory Conference, Albany, New York, October 2004.
• Special Session on Combinatorial and Statistical Group Theory, 986th AMS Meeting, Courant Institute, New York, October 2003.
• Hudson River Undergraduate Mathematics Conference, Hamilton College, New York, April 2002.
• Spring Topology and Dynamics Conference, University of Texas at Austin, March 2002.
• Seminar talks given at University of California, Davis; Miami University of Ohio; University of Illinois at Urbana-Champaign; University of Nebraska, Lincoln.