Listed within this section are my peer-reviewed publications and student projects I have mentored that resulted in a peer-reviewed publication. I describe my current projects in the "Research" section of this website. In the "Collaborative Work with Students" section of this website, I describe these projects in more detail.
“Implementing Specifications-Based Grading into College Mathematics Courses Using a Points-Based System”
[Began summer 2022]
Contribution of Paper: In this paper, I outline my current implementation of specifications grading. Previously in 2019, I co-authored a paper outlining a somewhat different implementation of specifications grading using a pass/fail rubric, which involved no partial credit. Since then I have adopted the use of a points-based rubric, and I have found this presentation of my grading system to be more easily understood by my students. In this paper, I describe in detail all of the Word, Excel, and LaTex files I use to keep myself and my students up to date on their progress toward each letter grade. The instructions to create these files are provided so that others may create their own. I also display several visuals that can be compiled from the grades to show progress of the class as a whole, or of single students over the course of the semester.
"Using Technology to Aid in Implementing Specifications Grading"
[Began summer 2022]
"Bessmerntyı Realization Theorem for Symmetric Matrices of Multivariate Rational Functions over Arbitrary Fields"
(with Aaron Welters, Anthony Stefan, and Ian Orzel) [Began fall 2022]
"On Kharitonov’s Theorem: The breakthrough in robust stability of polynomials" (with Aaron Welters, Anthony Stefan)
“On the Difference of Two Numbers raised to the same Exponent” (with N. Hallmark)
[Began in 2019]
"Continuity of the Roots of Nonmonic Polynomials" (with Aaron Welters)
[Submitted to the American Mathematical Monthly on January 26, 2024]
“Orbifolds of lattice vertex algebras” (with B. Bakalov, V.G. Kac, and I. Todorov)
Published in Jpn. J. Math, Jun 2023 (102 pages)
[The Jpn. J. Math. has rank A from this independent list]
Brief story of this project: This projected lasted for over 8 years, which extends the work I did for my dissertation. Our goal, once I joined in 2014 after I graduated, was to apply the insights from my dissertation work to tackle some of the technical nuances causing many of the complications. While my work did help to simplify the equations, it was still far too difficult to work out completely. This left us stuck for long periods of time. But fortunately, while I visited Bojko in the spring of 2019, he had an insight which allowed us to push through those difficulties. We were then able to obtain the general results we sought after and also worked out many different examples. Finally, after a lengthy review process by myself and the other authors, we have submitted our 102-page article to the Japanese Journal of Mathematics, which is known for publishing long papers (and the other authors have previously published papers there).
Victor is a Soviet and American mathematician at MIT and he is known for his work in the representation theory of infinite dimensional Lie algebras. Victor arrived at MIT in 1977 and was promoted to full professor in 1981. Victor has several mathematical theorems and quantities named after him. He co-discovered the Kac–Moody algebras and used the Weyl–Kac character formula for them to reprove the Macdonald identities. He classified the finite-dimensional simple Lie superalgebras and found the Kac determinant formula for the Virasoro algebra. He is also known for the Kac–Weisfeiler conjectures with Boris Weisfeiler. Having a paper published with him, especially at a teaching-focused school, is a sizable mathematical achievement.
Contribution of Paper: Many other papers that are published in the area of orbifold representations corresponding to an even lattice only specify the details for one specific lattice and isometry. Our work gives explicit results and formulas that apply more generally, for example to any isometry as long as the order is fixed. We also consider lattices in general, and work out the details for permutation orbifolds which is an important topic in physics. Some of our techniques, however, being from a very old project, are unfortunately outdated. So we will likely not continue this work further.
Published in PRIMUS, Oct 2019 (23 pages)
[Impact Factor and rank C from this independent list]
Contribution of Paper: This paper is a summary of my experiences using standards-based grading (see this for some definitions). In standards-based grading, it is typical to use a pass/fail rubric with no partial credit. We outline how grades are tabulated and combine to produce the A, B, C, D letter grades. We describe several ways that we have offered reassessment opportunities to our students and visuals that depict student progress. We conclude that the system we employed increased quality of student learning and that the reassessment process allowed our students who were struggling the most to become successful by the end. We end the paper with several student testimonials.
Published in Comm. In Alg, Volume 45, Issue 7, Oct 2016 (18 pages)
[Impact Factor and rank B from this independent list]
Contribution of Paper: This paper is a direct continuation from my dissertation work. In my dissertation, I describe the representation theory of an orbifold vertex algebra corresponding to an order-two automorphism. These mathematical objects and their representations have some features that are important in physics (namely the S-matrix, the fusion rules, and the quantum/asymptotic dimensions of irreducible representations). In this work, I compute the fusion rules and quantum dimensions for the representations I computed within my dissertation. This work continues to be distinct from other papers, as it describes explicit constructions using a generic setting; many other papers being published only describe one specific example.
“Orbifolds of Lattice Vertex Algebras Under an Isometry of Order Two” (with B. Bakalov)
Published in J. Algebra, Volume 441, Nov 2015 (27 pages)
[Impact Factor and rank A* from this independent list]
Contribution of Paper: This paper is a summary of my dissertation results. We describe the construction to obtain a vertex algebra from a positive-definite, even lattice, then we construct explicit representations of the orbifold vertex algebra under an order-two automorphism (that is acting on the entire vertex algebra). These representations have some features that are important in physics, namely, the S-matrix, the fusion rules, and the quantum/asymptotic dimensions of irreducible representations. One of the main contributions of this work was to discover a way to make some of the equations more tractable, which resulted in the explicit formulas for the representations.
Published in Pi Mu Epsilon, Volume 12, 2007 (5 pages)
Contribution of Paper: The Simpson's Rule error bound is a well-known result in calculus which gives an upper bound for the error from estimating area under a curve using parabolas (as long as the function has a continuous fourth derivative). Most calculus books omit a proof of this error estimate, as well as for the Trapezoidal Rule error bound. However, my search for a proof led me to a set of notes that describe a way to prove the Trapezoidal Rule error bound. The proof involves only elementary notions. I studied this proof technique and extended it to provide an elementary proof of the Simpson's Rule error bound.
These are the papers I have been asked to referee:
"Using Differential Equations to Model a Cockatoo on a Spinning Wheel as part of Spring 2022 the SCUDEM V Modeling Challenge", Rose-Hulman Undergraduate Mathematics Journal, Spring 2022
“Experimental assessment and grading scheme for reporting student preparedness”, PRIMUS, Spring 2021
“Fusion rules for orbifolds of affine and parafermion vertex operator algebras”, Israel Journal of Mathematics, Spring 2020
“Intertwining operators among twisted modules”, Journal of Algebra, Summer 2017
“2-permutations of lattice vertex operator algebras: higher rank”, Journal of Algebra, Summer 2016
“Characterizations of the vertex operator algebras V_L^T and V_L^O”, Summer 2015
1. “Using Differential Equations to Model Phoretic Parasitism as Part of SCUDEM Challenge”
Students: Nathan Hallmark, Zachary Fralish, Jonathan Marshal
Accepted for publication in the Int. Electronic Journal of Math. Education (Jan 2021)
(This work was presented by the students at the ASPiRE undergraduate conference at FGCU 2022.)
2. “Using Differential Equations to Model Predator-Prey Relations as Part of SCUDEM Modeling Challenge”
Students: Anthony Stefan, Zachary Fralish, Bernard Tyson III
Accepted for publication in Rose-Hulman Undergraduate Math Journal (Aug 2019)
(This work was presented by the students at the Joint Mathematics Meetings in Baltimore 2019.)
1. Effects of specifications grading on student test and math anxiety
In this paper, I will explore the possible effects that the reassessment process embedded within specifications grading may have on a student's test anxiety and math anxiety. I will use validated measures of each (STAI-5 for test anxiety and AMAS for math anxiety) to conduct a paired t-test using a pre-post quasi-experimental design (quasi-experimental design means having a control group but no random assignment was used; the pre scores will serve as the control). I received IRB approval from Florida Southern to collect the data. I will also collect demographic information to investigate possible correlations to the measures of anxiety.
2. Getting student buy-in when using alternative grading schemes
All of the papers I have read that describe an implementation of alternative assessment point out that student buy-in is important. Yet none of them dive into what they actually do help to guarantee that all their students actually understand the grading system and how to benefit from it. In my experience, this is no easy task and took me years to figure out. I now have some guidelines for certain things I do that I believe increase my students' comprehension of my grading system. In this paper, I will describe how I view student buy in and all the things I do to achieve it. This paper will be a sequel to my current paper in preparation.
3. Constructions of Bessmertnyi realizations
This project involves the Bessmertnyi realization theory for rational functions of several complex variables, which is well known, but its general construction is still an open question. In contrast to Bessmertnyi’s approach of solving large systems of linear equations, we use an operator theoretical approach based on the theory of Schur complements. This leads to a simpler and more “natural” construction to solving the realization problem, as we need only apply elementary algebraic operations to Schur complements such as sums, products, inverses, and compositions. A novelty of our approach is the use of Kronecker product as opposed to the matrix product in the realization problem. As such, our synthetic approach leads to a solution to the realization problem that has the potential for further extensions and applications within multidimensional systems theory, especially for those linear models associated with electric circuits, networks, and composites.
An application of these techniques was shown to provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in Helton et al. (2006). These results on realizability were extended from the real field to an arbitrary field different from characteristic 2. The new approach we take is only based on elementary results from the theory of determinants, the theory of Schur complements, and basic properties of polynomials.
Our next steps are to work out which rational matrix functions have a Bessmertnyi realization and conditions for which functions do not. This classification of the Bessmertnyi class of functions (and the related Milton class of functions) would have important consequences for the applicability of the state space methods we use in our synthetic approach.