I mainly work in commutative algebra and algebraic
geometry with a view towards understanding basic
properties of moduli spaces. I have been working
specifically to develop our understanding of the
"geography" of Hilbert schemes, meaning that I search for
regularities in the topology and geometry of collections
of Hilbert schemes.

Abstract.
We study the component structures of some
standard-graded Hilbert schemes closely related to a
Hilbert scheme of curves studied by Gotzmann. In
particular, we encounter examples of singular
lex-segment points lying on two and three
irreducible components. We find further singular
lex-segment points at nearby Hilbert schemes. We
conclude by showing that the analogous example at
the Hilbert scheme of twisted cubics also has a
singular lex-segment point.

Abstract.
We extend the recent classification of Hilbert
schemes with two Borel-fixed points to arbitrary
characteristic. We accomplish this by synthesizing
Reeves' algorithm for generating strongly stable
ideals with the basic properties of Borel-fixed
ideals and the author's previous work classifying
Hilbert schemes with unique Borel-fixed points.

Abstract.
We investigate the geography of Hilbert schemes
parametrizing closed subschemes of projective space
with specified Hilbert polynomials. We classify
Hilbert schemes with unique Borel-fixed points via
combinatorial expressions for their Hilbert
polynomials. These expressions naturally lead to an
arrangement of nonempty Hilbert schemes as the
vertices of an infinite full binary tree. Here we
discover regularities in the geometry of Hilbert
schemes. Specifically, under natural probability
distributions on the tree, we 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.