Graduate STEM Fellow Profile
Thesis: Cameron-Liebler Line Classes and Affine Two-Intersection Sets
College/University: University of Colorado Denver
Research Advisor: Stanley Payne
Degree Sought: Ph.D. Applied Mathematics
Department: Department of Mathematical and Statistical Sciences
Research Focus: Mainly on sets of points and lines in finite projective spaces having combinatorially interesting properties.
Description of Research
My research focuses on what are called finite projective spaces, which are systems of points and lines satisfying some fixed properties (an important one is that any two points determine a line). The most basic of these are the spaces PG(3,q), where q is a power of some prime number; while these share some properties of 3-dimensional space, the points each have 4 coordinates from an algebraic structure (called a field) having q elements. A spread of the finite projective space PG(3,q) is a covering of the points by disjoint lines; spreads are well-studied, and have a variety of applications in different settings ranging from design theory to cryptography. For my thesis, I am studying what are known as Cameron-Liebler line classes of parameter x, which are sets of lines in PG(3,q) that share precisely x lines with every spread of the space. Cameron and Liebler originally conjectured that there were no nontrivial examples of these line classes, but there were some discovered later by Bruen and Drudge in PG(3,q), for odd values of q, having parameter (q2 +1)/2, and later, an example was found by Penttila in PG(3,4). I have discovered new examples of these line classes having parameter (q2 -1)/2 for several odd values of q, and I believe they form part of an infinite family of such examples. Furthermore, when q=9, these line classes are closely tied to a set of points in the affine plane AG(2,9) having the nice property that all lines in the plane have exactly two possible numbers of points in this set they can contain. Point sets with this property are very common in projective planes, but not many are known to exist in affine planes, so this is a result of considerable interest. I hypothesize that I will find this type of point set for affine planes AG(2, 9e), which are not currently known to exist.
Example of how my research is integrated into my GK-12 experience
While my research has been difficult to bring into the classroom directly, I have been able to bring in my knowledge of geometric patterns and counting techniques. Letting the students work with these concepts helps them form a clear idea of why it is important to be able to create and use formulas when working with number patterns. Throughout the course of the year, I hope to be able to bring in some ideas that relate to cryptography and the theory of error-correcting codes, as I feel these are areas where students can get a sense of what math is really used for and how it is involved with the cutting edge of technology.