My main goal as an instructor is to help my students become independent learners.
My classes are often based in lecture, though I ask a lot of questions in class in order to lead students through the material. I also occasionally have students solve problems in small groups. By observing students actually working on problems, I can better gauge their understanding. Also, group work forces students to communicate mathematically with their peers, which means that they need to internalize the mathematics to a deeper extent.
I assign projects in many classes, like Linear Algebra and the History of Mathematics. Students are asked to come up with a problem or question and then solve or answer it using techniques learned in the course. Projects are a good way to foster independent learning, because instead of only answering someone else's question, students have to come up with the question themselves, and then answer it. This process can help students begin to comprehend independent mathematical research.
As students progress, I occasionally direct independent studies, lead undergraduate research, and supervise Honors theses. These students work on their own, engaging in research that is original to them, and perhaps even original in the world of mathematics.
From time to time, I teach courses outside the mathematics department, for the General Education and Honors programs. Dr. Bhattacharyya and I developed a course for Pacific Seminar II that relates mathematics to social justice. I thoroughly enjoy these kinds of classes because they take mathematics to other disciplines and bring other disciplines to mathematics. By doing so, students learn how to contextualize mathematics among other great achievements of human thought.
My dissertation was in the field of representation theory, a nice amalgam of abstract algebra and linear algebra. In particular, I studied the fusion algebras of different quasi-Hopf algebras arising from finite groups. I have gone on to investigate fusion algebras arising from other similar Hopf and quasi-Hopf algebras.
Another main area of interest is in mathematics education. In 2015, Lincoln Unified School District here in Stockton was awarded a California Math-Science Partnership (CaMSP), and I am the higher education PI (Principal Investigator). We are providing professional development to approximately 75 K-8 teachers in the district around various STEM topics. The teachers are then creating curricular units following a PBL (Project-Based Learning) pedagogy. Our hypothesis is that these professional development activities, combined with the act of writing, teaching, and then refining their PBL units, will lead to a deeper content knowledge and broader pedagogical knowledge around mathematics and its teaching.
Recently, I have also become interested in the history of mathematics. Having taken several years of Latin in high school, I can now state that I am indeed using it and helping out in the world-wide effort to translate all of 18th-century mathematical giant Leonhard Euler's works from Latin to a modern language (in my case, English). I have translated two papers so far and the third is in progress. I hope to continue this work, and who knows, I might even contribute to discovering something new in the works of this mathematical great!