My Research

I mainly work in representation theory of (and harmonic analysis on) \(p\)-adic groups. A number of years ago, I did some work in discrete optimization on the side.  You can see a list of of my publications.

What is [group] representation theory?

The appropriate answer depends on how much mathematics you already know. Here are three brief answers, in order of increasing sophistication:

  1. [For everyone:] the study of symmetry;
  2. [For science and engineering types:] glorified linear algebra;
  3. [For mathematicians:] the study of how groups act on vector spaces.

By “symmetry”, I don’t just mean the fact that snowflakes, butterflies, and pine cones exhibit various patterns (though this is a reasonable starting point for the subject). The term also refers to analogous phenomena that are not always directly tied to physical objects. Symmetries can have more than three (and in fact can have infinitely many) dimensions.

Representation theory is useful in physics, physical chemistry, the study of certain differential equations, group theory, number theory, and elsewhere.

What are \(p\)-adic groups?

I strongly believe that if one understands a subject well enough, then one should be able to explain its essence to any intelligent person. Unfortunately, I don’t understand this subject well enough, and am not sure that anyone does yet. Thus, this explanation is aimed at math types only. Sorry about that.

The set \( \mathbb{Q} \) of rational numbers is incomplete, which is a technical way of saying that it contains gaps. But what does it mean to have a gap?

Consider the sequence \( 1, 1.4, 1.414, 1.4142, 1.41421, 1.414213, \ldots \), which appears to converge to  \( \sqrt{2} \). But \(\sqrt{2}\) is not rational, as the Pythagoreans knew, so we have to say that the sequence has no limit in \(\mathbb{Q}\). However, the sequence is a Cauchy sequence, which is a technical way of saying that the terms get closer and closer together fast enough that there really ought to be a limit. This is how we detect a gap in \(\mathbb{Q}\).

The topological process of completion fills in all of the gaps, and we end up with the real line \(\mathbb{R}\).

Notice that our notion of “gap” depended on our notion of Cauchy sequence, which in turn depended on our notion of “distance” between rational numbers. For example, we implicitly assumed that the distance between \( 1.414 \) and \( 1.4142 \) is \( |1.414 – 1.4142|  = .0002\), which is pretty small. This depends on our knowledge of the absolute value function \( | \cdot | \).

However, there is more than one way to define an absolute value function on \( \mathbb{Q} \). For example, here’s another absolute value function (called the \(2\)-adic absolute value), denoted \( | \cdot |_2 \): Given a nonzero rational number \(r\), we can always write it in the form \(2^m (a/b)\), where \(a\) and \(b\) are odd, and \(m\) is an integer. Then \( | r | _2 = 2^{-m} \). We also let \( |0|_2 = 0\).  Thus, the more divisible a number is by \(2\), the smaller its \(2\)-adic absolute value.

If we replace \(2\) above by any prime number \(p\), we can similarly define a \(p\)-adic absolute value \(| \cdot |_p \). Fixing a prime \(p\) and thus an absolute value \( |\cdot |_p \), we obtain a new notion of what it means for a sequence to be Cauchy. Thus, if we complete the space \(\mathbb{Q}\) with respect to this absolute value, we get a new space. This space, like \(\mathbb{R}\), is field. It is called the field of \(p\)-adic numbers, and is denoted \(\mathbb{Q}_p\).

In some ways, \(\mathbb{Q}_p\) resembles \(\mathbb{R}\). For example, each is a domain where calculus makes sense. But in other ways, \(\mathbb{Q}_p\) is nothing like \(\mathbb{R}\). For example, in \(\mathbb{Q}_p\), a series converges if and only if its terms converge to zero. Isn’t that nice?

A \(p\)-adic group is essentially a matrix group, but where the entries of the matrix live in \(\mathbb{Q}_p\) or in one of its finite extensions.

Some of the motivation for studying \(p\)-adic groups comes from number theory. For example, the Langlands program, a far-reaching sequence of conjectures in number theory and arithmetic geometry, depends on an understanding of \(p\)-adic groups (and much else).