Research
My research interests include Lie algebras and representation theory, linear algebra and matrix theory, and related topics. Specifically, I’ve done work in derivations of Lie algebras and zero product determined algebras. This page gives a highlevel overview of these topics. Please see my curriculum vitae for details.
 Derivations of Parabolic Lie Algebras
 Zero Product Determined Algebras
 List of Publications
 List of Presentations
Derivations of Parabolic Lie Algebras
A derivation on a Lie algebra \( L \) is a map \( f: L \to L \) satisfying
\[ f\big([x,y]\big) = \big[f(x), y\big] + \big[x, f(y)\big] \]
for all \( x, y \in L \).
I proved that the derivations of a parabolic Lie algebra (a Lie algebra that is realized as parabolic subalgebras of some reductive Lie algebra) are the sums of inner derivations and linear maps into the center that kill the derived algebras.
Explicitly, given a parabolic Lie algebra \( L \), the derivations algebra \( \mathrm{Der} (L) \) decomposes as the direct sum of ideals
\[ \mathrm{Der} (L) = \mathrm{ad} (L) \oplus \mathcal L \]
where
\[ \mathcal L = \left\{ f: L \to L ; f(L) \subseteq Z(L), f([L,L]) = 0 \right\}. \]
These results will appear in the Journal of Lie Theory in 2017 (article).
Zero Product Determined Algebras
An algebra \( A \) is zero product determined if for each bilinear map \( \varphi: A \times A \to V \) satisfying
\[ \varphi(x, y) = 0 \text{ whenever } xy=0 \]
there is a linear map \( f: A^2 \to A \) such that
\[ \varphi(x, y) = f(xy) \]
for all \( x, y \in A \).
Huajun Huang and I wrote a paper on general zero product determined algebras that appeared in Linear and Multilinear Algebra in 2015, and I’m currently working on a second paper concerning specifically zero product determined Lie algebras. I’ll go into more details some time in the future (still working as of Jan 2017).
List of Publications
A selection of my peerreviewed and submitted research articles.

On derivations of parabolic Lie algebras
· Feb, 2017
Published in "Journal of Lie Theory" 
The Matrix Lie Algebra on a onestep ladder is zero product determined
· Dec, 2015
Published in "Alabama Journal of Mathematics" 
On zero product determined algebras
· Feb, 2015
With Huajun Huang. Published in "Linear and Multilinear Algebra"
List of Presentations
Selected conference and seminar presentations.

Applications of Thompson Sampling to Machine Learning
· Feb, 2017
Mathematics and Physics Seminar, California State University, Channel Islands. 
Automatic Differentiation in Haskell
· Aug, 2016
Santa Monica Haskell Users Group 
Linear Algebra and Data Analysis
· Feb, 2016
Mathematics and Physics Seminar, California State University, Channel Islands. 
Linear Lie Algebras, Block Matrices, and Ladder Matrices
· Nov, 2015
MAA Golden Section/SoCalNevada Section Joint Meeting, California Polytechnic State University. 
Upper Triangular Ladder Matrix Algebras, A Preliminary Report
· Oct, 2015
AMS Fall 2015 Western Sectional Meeting, California State University, Fullerton. 
Parabolic Lie Algebras are Zero Product Determined
· Mar, 2015
AMS Spring 2015 Southeastern Sectional Meeting, University of Alabama, Huntsville. 
Derivations of Parabolic Lie Algebras with Applications to Zero Product Determined Algebras
· Nov, 2014
AMS Fall 2014 Southeastern Sectional Meeting, University of North Carolina at Greensboro. 
Symmetry Groups, an introduction to group theory through geometry and graph theory
· Jul, 2011
Auburn University Summer 2011 REU on Algebra and Combinatorics. 
Continuous Symmetry Groups
· Jul, 2010
Auburn University Summer 2010 REU on Algebra and Combinatorics.