Papers
Click 'pdf' for the most
recent versions of my papers. The arXiv versions might be outdated.
 Stability in the cohomology of the space of complex
irreducible polynomials in several variables
Abstract: We prove that the space of complex irreducible
polynomials of degree d in n variables satisfies two forms of
homological stability: first, its cohomology stabilizes as d
increases, and second, its compactly supported cohomology stabilizes
as n increases. Our topological results are inspired by counting
results over finite fields due to Carlitz and Hyde.
 Obstructions to choosing distinct points on cubic plane
curves
Abstract: Every smooth cubic plane curve has 9 inflection
points, 27 sextatic points, and 72 "points of type nine". Motivated
by these classical algebrogeometric constructions, we study the
following topological question: Is it possible to continuously
choose n distinct unordered points on each smooth cubic plane curve
for a natural number n? This question is equivalent to asking if
certain fiber bundle admits a continuous section or not. We prove
that the answer is no when n is not a multiple of 9. Our result
resolves a conjecture of Benson Farb.
 Analytic number theory for 0cycles
Abstract: There is a wellknown analogy between integers and
polynomials over F_{q},
and a vast literature on analytic number theory for polynomials.
From a geometric point of view, polynomials are equivalent to
effective 0cycles on the affine line. This leads one to ask: Can
the analogy between integers and polynomials be extended to 0cycles
on more general varieties? In this paper we study prime
factorization of effective 0cycles on a geometrically connected
variety V over F_{q},
emphasizing the analogy between integers and 0cycles. For example,
inspired by the works of Granville and Rhoades, we prove that the
prime factors of 0cycles on V are typically Poisson
distributed.
 Twisted cohomology of configuration spaces and spaces of
maximal tori via pointcounting
Abstract: We consider two families of algebraic varieties Y_{n} indexed
by natural numbers n: the configuration space of unordered
ntuples of distinct points on C, and the space of unordered
ntuples of linearly independent lines in C^{n}.
Let W_{n}
be any sequence of virtual S_{n}representations
given by a character polynomial, we compute H^{i}(Y_{n},
W_{n}) for all i and all n in terms of
double generating functions. One consequence of the computation is a
new recurrence phenomenon: the stable twisted Betti numbers lim_{n→∞}H^{i}(Y_{n},
W_{n}) are linearly recurrent in i. Our
method is to compute twisted pointcounts on the F_{q}points
of certain algebraic varieties, and then pass through the
GrothendieckLefschetz fixed point formula to prove results in
topology. We also generalize a result of ChurchEllenbergFarb about
the configuration spaces of the affine line to those of a general
smooth variety.
 Homology of braid groups, the Burau representation, and
points on superelliptic curves over finite fields
Abstract: The (reduced) Burau representation V_{n}
of the braid group B_{n}
is obtained from the action of B_{n}
on the homology of an infinite cyclic cover of the disc with n
punctures. In this paper, we calculate H_{*}(B_{n},
V_{n}). As an application, we show that the
expected number of points on a random superelliptic curve over F_{q} is equal
to q.
Undergraduate Research

Optimal control with budget constraints and
resets
with Ryo Takei, Zachary
Clawson, Slav Kirov, and Alex
Vladimirsky.
in
SIAM Journal on Control and Optimization 53/2: 712–744 (2015).
Abstract: . We consider both discrete and continuous control
problems constrained by a fixed budget of some resource, which may
be renewed upon entering a preferred subset of the state space. In
the discrete case, we consider deterministic shortest path problems
on graphs with a full budget reset in all preferred nodes. In the
continuous case, we derive augmented PDEs of optimal control, which
are then solved numerically on the extended state space with a
full/instantaneous budget reset on the preferred subset. We
introduce an iterative algorithm for solving these problems
efficiently. The method’s performance is demonstrated on a range of
computational examples, including optimal path planning with
constraints on prolonged visibility by a static enemy observer.
Not for publication
In November 2013 I
passed my topic exam. Here
is my topic proposal.