# Curriculum Vitae

### Education

• PhD, Mathematics, The University of Wisconsin Milwaukee
• MA, Mathematics, The University of Wisconsin Milwaukee
• BA, Mathematics, The College of New Jersey

### Employment

• Assistant Professor, Mercyhurst University (2013 - Present)
• GAANN Fellow, The University of Wisconsin Milwaukee (2009 - 2013)
• Graduate Teaching Assistant, The University of Wisconsin Milwaukee (2007 - 2009)

### Teaching

##### Mercyhurst University
• Math 110, Math Applications: Art
• Math 110, Math Applications: Nature
• Math 111, College Algebra
• Math 118, Math for the Natural Sciences
• Math 150, Linear Algebra
• Math 170, Calculus I
• Math 265, Transition to Advanced Mathematics
• Math 280, Modern Algebra I
• Math 281, Modern Algebra II
• Math 400, Topics in Mathematics: Combinatorics
• MIS 224, Mobile Application Development
• MIS 370, Client Side Programming
• DATA 562, Data Visualization with JavaScript (Graduate)
##### University of Wisconsin Milwaukee
• Math 095, Beginning Algebra, Instructor
• Math 105, Intermediate Algebra, Instructor
• Math 117, Trigonometry, Instructor
• Math 231, Calculus and Analytic Geometry I, Instructor
• Math 232, Calculus and Analytic Geometry II, Instructor
• Math 431, Numerical Analysis, Computer Lab Instructor

### Research

• The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, with Pamela E Harris and Erik Insko. Journal of Combinatorics, Vol 7, No 1, 2016.

Abstract: Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula is factorial in the rank of the Lie algebra and the value of the partition function is unknown. In this paper we address the difficult question: What are the contributing terms to the multiplicity of the zero weight in the adjoint representation of a finite dimensional Lie algebra? We describe and enumerate the cardinalities of these sets (through linear homogeneous recurrence relations with constant coefficients) for the classical Lie algebras of Type B, C, and D, the Type A case was computed by the first author. In addition, we compute the cardinality of the set of contributing terms for non-zero weight spaces in the adjoint representation. In the Type B case, the cardinality of one such non-zero-weight is enumerated by the Fibonacci numbers. We end with a computational proof of a result of Kostant regarding the exponents of the respective Lie algebra for some low rank examples and provide a section with open problems in this area.
Earlier version available on arXiv

• Invariant polynomial functions on tensors under a product of orthogonal groups. Transactions of the American Mathematical Society, Vol 368, No 2, 2016.

Abstract: Let $$K$$ be the product $$O_{n_1} \times O_{n_2} \times \cdots \times O_{n_r}$$ of orthogonal groups. Let $$V = \bigotimes_{i=1}^r \mathbb{C}^{n_i}$$, the $$r$$-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the $$K$$-invariant algebra of degree $$d$$ homogeneous polynomial functions on $$V$$ . To accomplish this, we compute a formula for the number of matchings which commute with a fixed permutation. Finally, we provide formulas for the invariants and describe a bijection between a basis for the space of invariants and the isomorphism classes of certain $$r$$-regular graphs on $$d$$ vertices, as well as a method of associating each invariant to other combinatorial settings such as phylogenetic trees.
Earlier version available on arXiv

• The measurement of quantum entanglement and enumeration of graph coverings, with Michael W Hero and Jeb F Willenbring. AMS Contemporary Mathematics Series, Vol 557, 2011.

Abstract: We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical state space of a multi-particle system in which each particle has finitely many outcomes upon observation. Moreover, these invariant functions separate the entangled and unentangled states, and are therefore viewed as measurements of quantum entanglement. When the ranks of the unitary groups are large, we provide a graph theoretic interpretation for the dimension of the invariants of a fixed degree. We also exhibit a bijection between isomorphism classes of finite coverings of connected simple graphs and a basis for the space of invariants. The graph coverings are related to branched coverings of surfaces.
Earlier version available on arXiv

### Talks

• MAA Allegheny Mountain Section Meeting, Westminster College, PA, 2014
• Colloquium, The United States Military Academy, West Point, NY, 2013
• Dissertation Defense, UWM, 2013
• Algebra and Combinatorics Seminar, University of Wisconsin Madison, 2013
• Joint Mathematics Meeting Special Session on Lie Algebras, Algebraic Transformation Groups, and Representation Theory, San Diego, CA, 2013
• Colloquium, The United States Military Academy, West Point, NY, 2012
• MAA Mathfest, Madison, WI, 2012
• Applied and Computational Mathematics Seminar, UWM, 2012
• Algebra Seminar, UWM, 2008 - 2013

• Mary Jaskowak, "The Fourth Dimension", MAA Allegheny Mountain Section Meeting, Duquesne University, Spring 2017. Co-advised with Roger Griffiths and Pat Kelly.
• Andrea Flores, "World Income Distribution Dynamics: Analyzing the Dynamics of World Income Distribution Using a Markov Transition Method", MAA Allegheny Mountain Section Meeting, Gannon University, Spring 2016. Co-advised with Roger Griffiths.
• Michael Monaco, "Geometry and the Erlangen Program", MAA Allegheny Mountain Section Meeting, Washington and Jefferson College, Spring 2015.

### Conferences and Workshops

• Allegheny College Mathematics Faculty Lecture, Fall 2018
• MAA Allegheny Mountain Section Meeting, Duquesne University, Spring 2017
• Section NeXT Workshop, Indiana University of Pennsylvania, Fall 2016
• ICERM Illustrating Mathematics, Brown University, 2016
• MAA Allegheny Mountain Section Meeting, Gannon University, Spring 2016
• Section NeXT Workshop, Clarion University, Fall 2015
• MAA Allegheny Mountain Section Meeting, Washington and Jefferson College, Spring 2015
• Section NeXT Workshop, Penn State Behrend, Fall 2014
• MAA Allegheny Mountain Section Meeting, Westminster College, Spring 2014
• Section NeXT Workshop, Slippery Rock University, Fall 2013
• Joint Mathematics Meeting, San Diego CA, 2013
• MAA Mathfest, Madison WI, 2012
• Lie Theory and Its Applications Conference in Honor of Nolan Wallach, San Diego CA, 2011

### Membership

• American Mathematical Society
• Mathematical Association of America