I am interested in algebraic geometry and commutative algebra. I
especially enjoy studying computational and combinatorial aspects of
Hilbert schemes and deformation theory, and have been specifically
working to develop our understanding of the geography of Hilbert
schemes. Please see my research statement for an
overview.

Abstract. We
investigate the geography of Hilbert schemes that parametrize closed
subschemes of projective space with a specified Hilbert
polynomial. We classify Hilbert schemes with unique Borel-fixed
points via combinatorial expressions for their Hilbert
polynomials. We realize the set of all nonempty Hilbert schemes as a
probability space and prove that Hilbert schemes are irreducible and
nonsingular with probability greater than $0.5$.

Abstract. We
study logarithmic jet schemes of a log scheme and generalize a
theorem of M. Mustata from the case of ordinary jet schemes to the
logarithmic case. If X is a normal local complete intersection log
variety, then X has canonical singularities if and only if the log
jet schemes of X are irreducible.

Abstract. We prove that a random
Hilbert scheme that parametrizes the closed subschemes with a fixed
Hilbert polynomial in some projective space is irreducible and
nonsingular with probability greater than 0.5. To consider the set
of nonempty Hilbert schemes as a probability space, we transform
this set into a disjoint union of infinite binary trees,
reinterpreting Macaulay’s classification of admissible Hilbert
polynomials. Choosing discrete probability distributions with
infinite support on the trees establishes our notion of random
Hilbert schemes. To bound the probability that random Hilbert
schemes are irreducible and nonsingular, we show that at least half
of the vertices in the binary trees correspond to Hilbert schemes
with unique Borel-fixed points.

Abstract. A Hilbert scheme is a
parameter space for all subschemes of projective space with a fixed
Hilbert polynomial. Hilbert schemes are fundamental moduli spaces,
whose local geometry is studied via the deformation theory of
projective schemes. We give a concise introduction to deformation
theory, computing specific examples of the cotangent cohomology of
projective schemes. We then give a detailed account of the power
series ansatz, a procedure for computing versal pairs of local
moduli functors, and studying the local geometry of Hilbert
schemes. This is explained through two concrete examples. We end
with some open questions about Hilbert schemes and research
goals.

Abstract. A section of the total
tangent space of a scheme $X$ of finite type over a field $k$,
i.e. a vector field on $X$, corresponds to an $X$-valued $1$-jet on
$X$. In the language of jets the notion of a vector field becomes
functorial, and the total tangent space constitutes one of an
infinite family of jet schemes $J_m(X)$ for $m \geq 0$. We prove
that there exist families of ``logarithmic'' jet schemes
$J_m^{\mathcal D}(X)$ for $m \geq 0$, in the category of $k$-schemes
of finite type, associated to any given $X$ and its family of
divisors $\mathcal D = (D_1, \dotsc, D_r)$. The sections of
$J_1^{\mathcal D}(X)$ correspond to so-called vector fields on $X$
with logarithmic poles along the family of divisors $\mathcal D =
(D_1, \dotsc, D_r)$. To prove this, we first introduce the
categories of pairs $(X, \mathcal D)$ where $\mathcal D$ is as
mentioned, an $r$-tuple of (effective Cartier) divisors on the
scheme $X$. The categories of pairs provide a convenient framework
for working with only those jets that pull back families of
divisors.