Jadyn V. Breland

I am a mathematician fascinated by the notion of symmetry, and I study it in two ways.

The first is through linear algebra Symmetry, made precise via the language of finite groups, can be modeled by matrices, or equivalently, by the way a group acts on a vector space. This simple idea grew into an entire field: the representation theory of finite groups. I work over number systems of prime characteristic, where the ordinary theory collapses and progress demands new ideas backed by serious algebraic and categorical machinery. My work is guided by the local-to-global philosophy, which posits that the representation theory of a finite group is secretly controlled by certain smaller (local) subgroups. The field's deepest open problems, such as Broué's abelian defect group conjecture, are precise expressions of this philosophy, and they motivate everything I do.

My second approach trades linear algebra for counting and takes me into the field of algebraic combinatorics. I study Coxeter groups, abstract groups generated by reflections. Coxeter groups provide a broad generalization of the symmetric group (my favorite group) and give rise to rich combinatorics, where even simple questions produce objects worth studying in their own right. For example, every element of a Coxeter group can be written as a product of reflections, and the shortest such spellings are called reduced words. Counting these words is a classic and difficult problem; my research attacks it by organizing the words into a graph, where questions about the group can be translated into questions about the shape of the graph. While the whole graph is still a mystery, certain subgraphs, called braid graphs, turned out to have a hidden geometric structure, and they have become a central object of study in my work.

In the classroom, I strive to create a learning environment where students are engaged in authentic mathematical practice: they explore, experiment, make conjectures, and defend their reasoning. I incorporate elements of active and inquiry-based learning into all of my courses, because students learn mathematics by doing mathematics. I work to make my courses equitable and inclusive, so that every student has the opportunity to experience the unmistakable joy of mathematical discovery.