Research Interests

I'm interested in Grothendieck-Witt groups and derived categories of schemes, embedding problems in Galois theory, moduli problems, density problems in algebraic number theory, p-adic analysis, curves in characteristic p, Galois module structure problems, Iwasawa theory, and the Langlands program.

Arithmetic geometry and quadratic forms

My current dissertation research is on the arithmetic theory of quadratic (and line bundle-valued) quadratic forms over schemes.

The main emphasis is on the construction of cohomological invariants (or characteristic classes) for line bundle-valued quadratic forms. When the quadratic form takes values in a line bundle that is not a square in the Picard group of the scheme, then my construction provides truly new invariants. It "interpolates" between the 2nd Hasse-Witt (or Stiefel-Whitney) class and the 1st Chern class.

In my eyes, there are many applications of a general theory of invariants for line bundle-valued quadratic forms:
  • Classifying line bundle-valued forms of low (i.e. < 7) rank. My thesis extends the classical work on this to the line bundle-valued context.
  • Line bundle-valued quadratic forms don't have a usual Clifford algebra, but they have a "generalized Clifford algebra." My invariant helps in describing this generalized Clifford algebra.
  • The Grothendieck-Riemann-Roch formula is very useful for studying Chern classes of vector bundles. The question of whether exists an "Orthogonal Grothendieck-Riemann-Roch" formula for characteristic classes of quadratic forms, has been posed by various people, and answered in some nice cases. It turns out that from the recent advances in the Grothendieck-Witt theory of transfers (i.e. pushforwards), one is forced to consider quadratic forms with values in dualizing sheaves, and hence characteristic classes of such objects. This was the original motivation for my thesis work.
  • A representation theoretic version of my invariant exists for "essentially self-dual representations" i.e. those that are self-dual up to a twist by a 1-dimensional representation. One can use this in the study of Galois representations (and l-adic sheaves) arising from motives. Just as Deligne related the local root number to the 2nd Stiefel-Whitney class of a local orthogonal Galois representation, my invariant should be linked to the local root numbers of essentially self-dual representations of orthogonal type.
  • Just a Serre and Fröhlich relate the Stiefel-Whitney classes of orthogonal representations to Hasse-Witt invariants of trace forms and embedding problems in Galois theory, my new invariant should help solve additional embedding problems. This would apply to groups arising from subgroups of similitude groups and the associated four-fold extensions arising from my construction.
  • The Grothendieck-Witt groups of line bundle-valued quadratic forms are the algebraic analogues of twisted KO-groups. There is a substantial literature in physics applying twisted KO-theory to quantum field theory. In different guises, classes in twisted KO-theory can represent charges in boundary topological field theory and spaces of momenta of states in lattice models in solid state physics. For example, Kane and Mele (currently at Penn) recently used an invariant in the twisted KO-theory of elliptic curves to define a new topological classification of the quantum spin Hall phase. Topological analogues of my invariant could provide a general new invariant in twisted KO-theory, one that may carry a physical significance.
I am also interested in "twisted Lagrangian varieties." These are the moduli spaces of Lagrangian subspaces of a line bundle-valued quadratic form, which is not necessarily metabolic. This is related to the question of the existence of non-vanishing global sections on vector bundles. If a vector bundle of even rank has a non-vanishing global section, or more generally, a line-sub-bundle, then it's middle exterior power will be a metabolic form, either quadratic or alternating depending on the rank modulo 4, with values in the determinant line bundle. The converse is not necessarily true, but knowledge of information about metabolic structures, i.e. of the twisted Lagrangian variety, of a middle exterior power form will give refined information on the existence of non-vanishing global sections. Or at least that's my hope.

Probability questions in algebraic number theory

A natural question to ask is:
What's the probability that a monic polynomial with integer coefficients has all integer roots?
The answer certainly depends on what you take as a probability measure. The analogous question for polynomials with coefficients in the ring of p-adic integers is well-defined because there is a natural measure on this space, coming from the p-adic Haar measure.

My undergraduate thesis answered the above question for p-adic polynomials, giving a nice recursion for the probability and studying the asymptotics. Buhler (my undergraduate thesis advisor), Goldstein, Moews, and Rosenberg later published a paper with the same result and a finer study of the asymptotics.

Of a given fixed degree, there are a finite number of isomorphism classes of étale algebra over a local field, and for each one, why not ask the same question:
What's the probability that a monic polynomial with p-adic integer coefficients generates a given étale algebra?
My undergraduate thesis answered this more general question for polynomials of degree 2 and 3 and for the unique unramified extension of every degree. Already evident in my thesis work is that for a fixed étale algebra A over the p-adics, the probability that a given polynomial will generate that algebra is a rational function in p, and that as p goes to infinity, this probability converges to |DA|/w(A), the p-adic absolute value of the discriminant of a divided by the number of automorphisms of A.

This factor comes up in recent work of Manjul Bhargava on the distribution of number field discriminants. In some sense, it turns out to be the "right" weighting factor when counting local étale algebras.

I also like to draw pictures. You can visualize the splitting behavior of p-adic polynomials. Below is a map of the space of monic quadratic polynomials over the 2-adic integers. They split in the black region and are irreducible in the white region.

Splitting of 2-adic quadratic polynomials.

A more colorful picture shows exactly which of the eight isomorphism classes of étale quadratic algebras over the 2-adic numbers a polynomial generates. If you are interested in these pictures, please contact me.