Research Interests
My research is mostly in the area of finite geometry which, in a very general sense, constitutes the study of incidence structures containing a finite number of points and lines. This includes the study of graphs, combinatorial designs, polar spaces, and other geometric structures. The nature of these objects puts their study at the intersection of geometry, combinatorics, and algebra. In particular, techniques from permutation group theory, matrix theory, and the theory of finite fields have been used to construct and analyze important examples of finite incidence structures. A major focus is on substructures of a projective space having some specified combinatorial structure; often these objects can be used to define error-correcting codes that are "good" in some sense. Finite polar spaces are an area of particular interest for me; these geometries are defined by quadratic or semilinear forms on finite projective spaces, and are naturally associated with the finite classical groups.
In my dissertation, I applied algebraic and computational methods to find new examples of Cameron-Liebler line classes in PG(3,q); under the Klein correspondence, these are equivalent to tight sets in the hyperbolic polar spaces Q+(5,q). You can download a copy here.
Publications
- Regular sets of lines in rank 3 polar spaces (with Ferdinand Ihringer). (submitted)
- Strong ovoids and Cameron-Liebler sets of generators in polar spaces (with Maarten De Boeck and Jozefien D'haeseleer). (submitted)
- Cameron-Liebler line classes with parameter x=(q+1)2/3 (with Tao Feng, Koji Momihara, Qing Xiang, and Hanlin Zou). Advances in Mathematics, Volume 385 (2021).
- Cameron-Liebler sets of generators in finite classical polar spaces (with Maarten De Boeck, Leo Storme, and Andrea Svob). Journal of Combinatorial Theory, Series A, Volume 167 (2019), pp 340-388. (pdf)
- Classification of 8-dimensional rank two commutative semifields (with Michel Lavrauw). Advances in Geometry, Volume 19, Issue 1 (2019), pp 57-64. (pdf)
- Cameron-Liebler k-classes in PG(2k+1,q) (with Leo Storme and Andries Vansweevelt). Combinatorica, Volume 38, Issue 3 (2018), pp 737-757. (pdf)
- A new family of tight sets in Q+(5,q) (with Jan De Beule, Jeroen Demeyer, and Klaus Metsch). Designs, Codes and Cryptography, Volume 78, Issue 3 (2016), pp 655-678. (pdf)
- Double k-sets in symplectic generalized quadrangles (with Stanley Payne) Designs, Codes and Cryptography, Volume 72, Issue 2 (2014), pp 265-271. (pdf)
- Cameron-Liebler line classes, Designs, Codes and Cryptography, Volume 68, Issue 1-3 (2013), pp 33-37. (pdf)
Selected Research Presentations
- Cameron-Liebler sets in projective and polar spaces, Mathematics Department Colloquium, Michigan Technological University, November 1, 2019.
- Sets of type (m,n) in PG(3,9), International Conference on Finite Geometry and Extremal Combinatorics, University of Delaware, August 21-24 2019.
- Intriguing sets in distance regular graphs, Discrete Mathematics Seminar, University of Delaware, November 26, 2018.
- Eigenvalue techniques in graph theory and combinatorics, Math Seminar Series, California State University Fresno, August 31, 2018.
- Permutation groups and symmetric block designs, Undergraduate Mathematics Seminar, California State University Fresno, October 27, 2017.
- Finite commutative semifields with small rank, Finite Geometries, Fifth Irsee Conference, September 10-16, 2017.
- Classification of 8-dimensional rank two commutative semifields, Combinatorics 2016, May 29-June 4, 2016.
- Cameron-Liebler type sets and completely regular codes in Grassmann graphs, Network Coding and Designs, CAAS Dubrovnik, April 4-8, 2016.
- An infinite family of tight sets in Q+(5,q), Fq12, The 12th International Conference on Finite Fields and Their Applications, July 13-15, 2015.
- A new family of Cameron-Liebler line classes, Finite Geometries, Fourth Irsee Conference, September 14-19, 2014.
- Cameron-Liebler type sets of k-spaces, Seminars on Incidence Geometry, Ghent University, December 6, 2013.
- Tight sets in Q+(5,q), Seminari di Geometria Combinatoria a Napoli, Università degli Studi di Napoli Federico II, November 14, 2013.
- Cameron-Liebler line classes, Eindhoven Seminar Combinatorial Theory, Eindhoven University of Technology, October 2, 2013.
- Cameron-Liebler line classes from cyclic groups, Algebraic Combinatorics: In Memory of Bob Liebler, Colorado State University, November 6, 2011.
- Some new examples of Cameron-Liebler line classes, Finite Geometry Seminar, Ghent University, October 13, 2011.
- Cameron-Liebler line classes and affine two-intersection sets, Finite Geometries, Third Irsee Conference, June 19-25, 2011.
