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 The arXiv versions might be outdated.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
Abstract: Every smooth cubic plane curve has 9 inflection points, 27 sextatic points, and 72 "points of type nine". Motivated by these classical algebro-geometric 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 0-cycles
Abstract: There is a well-known analogy between integers and polynomials over Fq, and a vast literature on analytic number theory for polynomials. From a geometric point of view, polynomials are equivalent to effective 0-cycles on the affine line. This leads one to ask: Can the analogy between integers and polynomials be extended to 0-cycles on more general varieties? In this paper we study prime factorization of effective 0-cycles on a geometrically connected variety V over Fq, emphasizing the analogy between integers and 0-cycles. For example, inspired by the works of Granville and Rhoades, we prove that the prime factors of 0-cycles on V are typically Poisson distributed.
- Twisted cohomology of configuration spaces and spaces of
maximal tori via point-counting
Abstract: We consider two families of algebraic varieties Yn indexed by natural numbers n: the configuration space of unordered n-tuples of distinct points on C, and the space of unordered n-tuples of linearly independent lines in Cn. Let Wn be any sequence of virtual Sn-representations given by a character polynomial, we compute Hi(Yn, Wn) 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 limn→∞Hi(Yn, Wn) are linearly recurrent in i. Our method is to compute twisted point-counts on the Fq-points of certain algebraic varieties, and then pass through the Grothendieck-Lefschetz fixed point formula to prove results in topology. We also generalize a result of Church-Ellenberg-Farb 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 Vn of the braid group Bn is obtained from the action of Bn on the homology of an infinite cyclic cover of the disc with n punctures. In this paper, we calculate H*(Bn, Vn). As an application, we show that the expected number of points on a random superelliptic curve over Fq is equal to q.
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.