My research interests involve studying singularity types in prime characterstic by various means. In most settings, this would mean that I am interested in studying how far some geometric object is from being smooth. However, in the setting of prime characteristic commutative algebra, our notion of smoothness is the concept of a regular ring (e.g. a polynomial ring in finitely many variables over a field). Thus, I like to think about rings which deviate from being regular and studying what type of 'regular-like' properties remain.
F-invariants of simple algebroid plane branches (2024), in preparation
Abstract: In this paper, we investigate the singularities of (a,b)-simple power series, which are described as a power series whose initial form (the sum of its lowest degree terms) is a K*-linear combination of x^a, y^b. Using the integer programs recently defined by Hernández and Witt, we describe an algorithm to compute the F-Threshold of such a power series with respect to a diagonal ideal <x^s, y^t>. We note that this is the first example in literature of computing the F-Thresholds for a class of inhomogeneous equations. We also compute the associated test ideal at the F-pure Threshold.
Basics of Test Ideals (written for a series of 3 talks given in KU's student algebra seminar)
Fedder's Criterion for Frobenius Splittings
Geometric Intuition of Regularity (written for a seminar talk)
F-pure Threshold of various classes of functions:
Given a function, the most basic mathematics looks at understanding the zeros of this function. Even in early grade school classes, we are asked to find the zeros of a function. How do we examine and understand the zeros of a function when there are too many variables to visualize or is of too high degree? Historically, one way mathematicians have looked at investigating this is to think of the zeros of a function as a curve/surface in space and attach a measurement to every point representing how "smooth" the set is at that point. Within the past twenty-some odd years, one such measurement involving passage to positive characteristic, called the F-pure Threshold, has become increasingly important. In this line of research, we are motivated by some of the following goals:
Understanding/Investigating the F-pure threshold of certain functions (currently thinking about semi-quasi homogeneous singularities)
Developing elementary algorithms for computing the F-pure Threshold of homogeneous polynomials with isolated singularities
This research direction has largely been motivated by papers of Hernández - Núñez-Betancourt - Witt - Zhang, Pagi, and Smith - Vraciu.
F-jumping Numbers/Test Ideals of various functions:
These invariants form a finer class of measurement from the F-pure Threshold and give even more understanding of the behavior of a function at a point on its zero set. In a way, these encode higher order behavior than the F-pure Threshold. Currently, based off of work by Hernández - Witt, I have been thinking about computing these invariants for simple algebroid plane branches.
"F-invariants of simple algebroid plane branches" - University of Utah Commutative Algebra Seminar
"F-invariants of simple algebroid plane branches" - Joint Mathematics Meetings
"F-invariants of simple algebroid plane branches" - Kansas Math Graduate Student Conference
"F-invariants of simple algebroid plane branches" - Algebra Day at KU
"F-Thresholds of Semi-quasi Homogeneous Singularities" - URiCA Conference at UNL
"F-Pure Threshold of Semi-quasi Homogeneous Singularities" - University of Utah Graduate Student Commutative Algebra Seminar (BIKES)
"Singularity Theory in Positive Characteristic" - Kansas Math Graduate Student Conference
"Some Facts on F-pure Thresholds of Homogeneous Polynomials" - KU Student Algebra Seminar
"F-Regularity: A quick way to show Cohen-Macaulayness" - CARES
"Geometric Intuition of Regularity" - KU Student Algebra Seminar
"Basics of Test Ideals" - KU Student Algebra Seminar (Series of 3 talks)
"Measuring Singularities'' - KU Graduate Student Seminar
"Regularity and Differential Operators'' - CARES
"F-invariants of simple algebroid plane branches" (Poster here)