My research program includes topics from pure and and applied mathematics, mathematics education, and data science policy in higher education. Overall, my research interests are interdisciplinary and collaborative, and are well-suited for student projects.
In applied mathematics, I currently study topics in finite elements. I am interested in finite element exterior calculus, the Periodic Table of Finite Elements, and the implementation of new finite elements.
As a graduate student, I explored topics in homological algebra and combinatorial topology. My PhD thesis described connections between the Koszul and Cohen-Macaulay properties.
In the context of undergraduate mathematics education research, I investigate students' proof validation skills.
Finally, I have recently contributed to data science education and policy documents (including workshop reports and white papers).
Papers
Data Science Leadership Summit; Summary Report (with Lucy C. Erickson, Vandana P. Janeja, and Jeannette M. Wing)
Toolkit: Preparing a Poster (with Azadeh Rafizadeh), MAA Focus, Vol. 38, No. 6, December 2018/January 2019
Computational Serendipity and Tensor Product Finite Element Differential Forms (with Andrew Gillette and Victoria Sanders), accepted
Many conforming finite elements on squares and cubes are elegantly classified into families by the language of finite element exterior calculus and presented in the Periodic Table of the Finite Elements. Use of these elements varies, based principally on the ease or difficulty in finding a "computational basis" of shape functions for element families. The tensor product family, Q-rΛk, is most commonly used because computational basis functions are easy to state and implement. The trimmed and non-trimmed serendipity families, S-rΛk and SrΛk respectively, are used less frequently because they are newer to the community and, until now, lacked a straightforward technique for computational basis construction. This represents a missed opportunity for computational efficiency as the serendipity elements in general have fewer degrees of freedom than elements of equivalent accuracy from the tensor product family. Accordingly, in pursuit of easy adoption of the serendipity families, we present complete lists of computational bases for both serendipity families, for any order r ≥ 1 and for any differential form order 0≤k≤n, for problems in dimension n=2 or 3. The bases are defined via shared subspace structures, allowing easy comparison of elements across families. We use and include code in SageMath to find, list, and verify these computational basis functions.
Trimmed Serendipity Finite Element Differential Forms (with Andrew Gillette), Mathematics of Computation, accepted and in press
We introduce the family of trimmed serendipity finite element differential form spaces, defined on cubical meshes in any number of dimensions, for any polynomial degree, and for any form order. The relation between the trimmed serendipity family and the (non-trimmed) serendipity family developed by Arnold and Awanou [Math. Comp. 83(288) 2014] is analogous to the relation between the trimmed and (non-trimmed) polynomial finite element differential form families on simplicial meshes from finite element exterior calculus. We provide degrees of freedom in the general setting and prove that they are unisolvent for the trimmed serendipity spaces in all cases of immediate relevance to application: spatial dimension n up to 4, any differential form order k, and polynomial order r up to 10. In these cases, the sequence of trimmed serendipity spaces with a fixed polynomial order r provides an explicit example of a system described by Christiansen and Gillette [ESAIM:M2AN 50(3) 2016], namely, a minimal compatible finite element system on n-dimensional cubes containing order r-1 polynomial differential forms.
Weakly Cohen-Macaulay posets and a class of finite-dimensional graded quadratic algebras, Journal of Algebra 487 (2017), pp. 138-160
To a finite ranked poset Γ we associate a finite-dimensional graded quadratic algebra RΓ. Assuming Γ satisfies a combinatorial condition known as uniform, RΓ is related to a well-known algebra, the {\it splitting algebra} AΓ. First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset Γ, we ask: Is RΓ Koszul? The Koszulity of RΓ is related to a combinatorial topology property of Γ called Cohen-Macaulay. Kloefkorn and Shelton proved that if Γ is a finite ranked cyclic poset, then Γ is Cohen-Macaulay if and only if Γ is uniform and RΓ is Koszul. We define a new generalization of Cohen-Macaulay, weakly Cohen-Macaulay, and we note that this new class includes posets with disconnected open subintervals. We prove: if Γ is a finite ranked cyclic poset, then Γ is weakly Cohen-Macaulay if and only if RΓ is Koszul.
Splitting algebras: Koszul, Cohen-Macaulay and numerically Koszul (with Brad Shelton), Journal of Algebra 422C (2015), pp. 660-685
We study a finite dimensional quadratic graded algebra RΓ defined from a finite ranked poset Γ. This algebra has been central to the study of the splitting algebras AΓ introduced by Gelfand, Retakh, Serconek and Wilson. Those algebras are known to be quadratic when Γ satisfies a combinatorial condition known as uniform. A central question in this theory has been: when are the algebras Koszul? We prove that RΓ is Koszul and Γ is cyclic and uniform if and only if the poset Γ is Cohen-Macaulay. We also show that the cohomology of the order complex of Γ can be identified with certain cohomology groups defined internally to the ring RΓ, HΓ(n, 0) whenever Γ is Cohen-Macaulay. Finally, we settle in the negative the long-standing question: Does numerically Koszul imply Koszul for algebras of the form RΓ.