Abstract: Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose n distinct points on every smooth cubic plane curve, for each given positive integer n? We give an affirmative answer to the question when n=9 and 18 (the smallest open cases), and a negative answer for infinitely many n's.
Stability of the cohomology of the space of complex irreducible polynomials in several variables [click to expand]
in International Mathematics Research Notices, rnz296 (December 2019).
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 [click to expand]
in Advances in Mathematics, 340, 211-220 (December 2018).
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 [click to expand]
in Mathematical Proceedings of the Cambridge Philosophical Society, 1-24 (October 2017).
Abstract: There is a well-known 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 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 an arbitrary connected variety V over F_q, 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 are typically Poisson distributed.
Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting [click to expand]
Abstract: We consider two families of algebraic varieties Y_n indexed by natural numbers n: the configuration space of unordered n-tuples of distinct points on the complex plane C , and the space of unordered n-tuples 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 limn→∞dimH^i(Y_n;W_n) are linearly recurrent in i. Our method is to compute twisted point-counts on the F_q-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 [click to expand]
in Israel Journal of Mathematics,Volume 220, Issue 2, pp 739-762 (June 2017).
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. The group homology H_∗(B_n;V_n) of braid groups with coefficients in the complexified reduced Burau representation is calculated. Our topological calculation has the following arithmetic interpretation (which also has different algebraic proofs): the expected number of points on a random superelliptic curve of a fixed genus over F_q is exactly q.
Optimal control with budget constraints and resets [click to expand]
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.