# Papers

Topological complexity of finding flex points on cubic plane curves

with Zheyan Wan.

Preprint (June 2023).

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Abstract: We prove a lower bound for the topological complexity, in the sense of Smale, of the problem of finding a flex point on a cubic plane curve. The key is to bound the Schwarz genus of a cover associated to this problem. We also show that our lower bound for the complexity is close to be the best possible.

Choosing points on cubic plane curves: rigidity and flexibility

with Ishan Banerjee.

in Advances in Mathematics, 418 (April 2023).

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arXiv: arxiv.org/pdf/2101.03824.pdf

Journal: doi.org/10.1016/j.aim.2023.108959

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

in International Mathematics Research Notices, rnz296 (December 2019).

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arXiv: arxiv.org/pdf/1902.01882.pdf

Journal: doi.org/10.1093/imrn/rnz296

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

in Advances in Mathematics, 340, 211-220 (December 2018).

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Journal: doi.org/10.1016/j.aim.2018.09.040

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

in Mathematical Proceedings of the Cambridge Philosophical Society, 1-24 (October 2017).

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arXiv: arxiv.org/pdf/1603.07212.pdf

Journal: doi.org/10.1017/S0305004117000767

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

arXiv preprint.

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arXiv: arxiv.org/pdf/1603.03931.pdf

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

in Israel Journal of Mathematics,Volume 220, Issue 2, pp 739-762 (June 2017).

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arXiv: arxiv.org/pdf/1506.02189.pdf

Journal: doi.org/10.1007/s11856-017-1534-7

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

with Ryo Takei, Zachary Clawson, Slav Kirov, and Alex Vladimirsky.

in SIAM Journal on Control and Optimization 53/2: 712–744 (2015).

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arXiv: arxiv.org/pdf/1110.6221.pdf

Journal: doi.org/10.1137/110853182

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.