Alex Dugas
PhD, UC Berkeley, 2006
BS, Stanford University, 2000
I strive to teach mathematics not as a static body of formulas, theorems and proofs, but rather as an evolving process of inquiry and discovery. This mathematical process often starts with a problem or question that can motivate a mathematical model, while suggesting which theoretical definitions we should make and which theoretical questions are important. By working through and thinking about examples, we search for patterns, make hypotheses and try to generalize our observations. The conclusions we reach are guided by intuition and justified by logic.
In all my courses, from introductory to advanced, I seek to engage my students in this process of mathematical inquiry and problem solving, and ultimately to teach them how to independently approach new problems with the ability to think about them critically and identify how mathematics and logic can be applied to study them.
In my classes, I achieve this through a mix of lecture and individual or group work. I carefully organize my lectures to guide students through their own process of discovery, asking many questions and encouraging students to come up with suggestions or make the next step. Sometimes we make mistakes or we try the wrong approach: this is part of mathematics and part of learning, and it is a crucial skill to be able to recognize when we are on the wrong track or our answer doesn't make sense.
Likewise, through group work activities, I give students a chance to actively practice and master the course material. Mathematics is best learned through practice and can only be fully absorbed through actively thinking about and solving problems. I always urge students to discuss problems in small groups and write out their work on the boards, where I can see it and provide instant feedback. While working in groups students have ample opportunities to ask me questions in class. When possible I answer with more questions to stimulate group discussion, guide the students in the right direction, and help them identify the root of their misunderstanding. At the same time, I hope to teach students to follow this process on their own to become more successful and independent problem solvers.
Much of modern mathematics benefits from the incorporation of abstract algebraic structures. Representation theory is a field that has emerged from the study of concrete representations, often in terms of matrices, of these abstract structures, including groups, rings and algebras. For example, the "imaginary" number i, defined as the square root of -1, can be represented algebraically by a 2x2 matrix
| 0 | -1 |
| 1 | 0 |
or geometrically by a 90 degree rotation of the plane. It follows that the algebra of complex numbers, as well as that of more abstract systems, can be defined and studied concretely using matrices.
My research interests mostly fall within the scope of the Representation Theory of Algebras, a field that encompasses the study of representations of many different algebraic structures, from groups to rings to categories. Many of the questions that motivate my research have their origins in group representation theory, and I am interested in extending aspects of this theory to larger classes of algebras. I use tools from category theory and homological algebra, which together offer a unified perspective for many diverse areas of modern mathematics and can be used to describe deep connections between Representation Theory, Algebraic Geometry, Commutative Algebra, Algebraic Topology and Theoretical Physics. There are ample opportunities for these fields to influence one another, and these connections are an important source of motivation for my own research. Additionally, Graph Theory and Combinatorics play a prominent role in the representation theory of quivers, and offer many possible elementary topics for student research projects.
For more details about my research and publications see my personal webpage.
