The notion of trace is ubiquitous in mathematics. A recurring theme is that traces turn complicated objects into simpler ones, discarding much of the information in the original object but retaining enough to say something useful about it. Many familiar invariants arise as traces of identity maps; for example, the trace of the identity map on a real or complex vector space is the vector space’s dimension. While I am primarily interested in traces arising in algebraic settings, the framework I use to study traces is so general that it also has topological applications, such as the Euler characteristic (i.e. vertices minus edges plus faces). One reason traces are so useful is that they are compatible with many kinds of structure. For example, the following identities hold for matrices A and B:
(1) tr(A + B) = tr(A) + tr(B) and (2) tr(A ⊗ B) = tr(A) · tr(B).
These properties are not specific to linear algebra; they hold for vastly more general notions of trace. I study the interaction of traces with other kinds of structure. However, traces alone are not enough to describe these interactions; a dual notion of "cotrace" is also needed. Building on Ponto’s work, I described the structure necessary to support a notion of cotrace, and established properties of the cotrace analogous to those of the trace. Cotraces provide similar information to traces in some settings; for example, in the case of representations, a certain cotrace captures precisely the same information as the group character, which is also an example of a trace. Other examples, however, are not adequately explained by traces, so the cotrace is a distinct and necessary component of this framework.
Ponto’s traces and my cotraces are both formulated in the language of category theory, which provides a way of describing common structure across different areas of mathematics. This abstract perspective is valuable because it often allows us to make mathematical constructs and theorems more accessible by extracting the core ideas from the technical details of their original presentation. Moreover, we are often able to prove vastly more general versions of these results and port them over to other mathematical contexts. For example, the tools that I have built to understand Joseph Lipman’s work create a bridge to the theory of group representations and 2-representations.
You can find more a more extensive description of my research program in my research statement.