Non-technical Research Statement:

My research interests are in algebraic topology, specifically in homotopy theory. Broadly speaking, topology is a generalization of the study of shapes. These shapes can be abstract and hard to visualize, so we study them though algebra. In particular, in homotopy theory, we use algebraic tools called invariants to describe whether two shapes are the same or different.

Homotopy theory has its origins in algebraic topology, but its core tools and ideas have spread and found use in other areas of mathematics. In my research, I apply the computational methods of homotopy theory to answer questions about the modular representation theory of finite groups.

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Below is a brief, undergraduate-level introduction to the kinds of things I think about, namely stable homotopy theory. For further details on my research, check out my Research page.

Homotopy Theory

The idea of homotopy theory is the following: given objects and a notion of equivalence, we would like to determine whether or not two objects are the same or differnt. For example, in algebraic topology, the objects we study are topological spaces, and one notion of equivalence is homotopy equivalence. Examples of topological spaces include circles, spheres, donuts, and anything you can form out of rubber. We say two topological spaces are homotopy equivalent if you can stretch, shrink, or deform one into the other through continuous maps. This means that you are not allowed to cut, pierce, or attach things to the object you start with.

In the eyes of homotopy theorists, it is useful and convenient to replace equality with the notion of homotopy equivalence instead. Making this replacement means we are now considering the homotopy category. This leads to the usual joke that algebraic topologists cannot tell coffee cups apart from donuts, since our notion of equivalence is now too coarse to tell the two apart.

Click to play GIF.
Source: Wikipedia.
However, we can distinguish topological spaces up to homotopy by identifying certain traits (called invariants) such as homotopy groups and (co)homology groups. We have various powerful tools at our disposal to compute these invariants.

Stable Homotopy Theory

We enter the world of stable homotopy theory when we begin to investigate the structure of the homotopy category. For example, if we consider the most fundamental topological spaces, the n-dimensional spheres, we notice some interesting patterns in a table of their homotopy groups:

Colors!
The homotopy groups of spheres.
This is no coincidence: this stabilization of homotopy groups is the content of the Freudenthal suspension theorem. One can form a similar table for any other other topological space (Exercise: If we put a space X in the first row, what should we replace the following rows?). This leads one to look for new objects that can capture this stable information. In other words, we would like to collapse our tables along the anti-diagonal (roughly). The objects that capture this information are called spectra, and studying the homotopy theory of spectra is known as stable homotopy theory.

Colors!
The stable homotopy groups of spheres are the homotopy groups of the sphere spectrum.

Axiomatic Stable Homotopy Theory

It turns out that we can generalize this story to other settings, which brings us to axiomatic stable homotopy theory. We can formalize the story above with categorical axioms, and many of the important theorems and tools of algebraic topology become formal consequences. These axiomatic structures are called stable model categories (or stable infinity categories, depending on your formalism). Some instances of these structures occur in modular representation theory, derived categories of modules over rings, the derived category of quasi-coherent sheaves over nice schemes, etc.

One can then try to prove theorems in these other settings by mimicking or taking inspiration from classical theorems in algebraic topology. On the other hand, one can also try to understand open questions in algebraic topology by understanding the analogous questions in other axiomatic stable frameworks.

Further Reading

For further survey reading, I recommend the following resources:

  • For a quick undergrad-level introduction to the ideas of homotopy theory, see this slide talk that I gave for the UT Austin Math Club.
  • The standard introductory textbooks for algebraic topology are Allen Hatcher's "Algebraic Topology" and Peter May's "A Concise Course in Algebraic Topology".
  • For a quick survey of stable homotopy theory, see Neil Strickland's survey "An Introduction to the Category of Spectra" (An introduction to the stable category).
  • There's a lot of scattered references regarding spectra, as the historical development of spectra was complicated. It turns out that there is no category of spectra that has all the properties that one might want, and there are various flavors of spectra with different properties. However, the standard references are Elmendorf-Kriz-Mandell-May's "Rings, Modules, and Algebras in Stable Homotopy Theory" (#83) and Mandell-May-Schwede-Shipley's "Model Categories of Diagram Spectra" (#96).
  • For a quick survey of axiomatic homotopy theory (model categories), I like W. Dwyer and J. Spalinskies' "Homotopy Theories and Model Categories" (#75).
  • A perhaps more modern approach is to use the language of infinity categories. Jacob Lurie wrote a column in the Notices of the AMS, "What is an ∞-category?".
  • For a quick survey of axiomatic stable homotopy theory, check out this survey by Neil Strickland.
  • For a more in-depth exposition of my research interests, click here for a survey that I wrote about the structure of stable homotopy theories.