publications

• Homology of spaces of curves on blowups

  with Philip Tosteson
  2024 · arxiv

  abstract

  We consider the space of holomorphic maps from a compact Riemann surface to a projective space blown up at finitely many points. We show that the homology of this mapping space equals that of the space of continuous maps that intersect the exceptional divisors positively, once the degree of the maps is sufficiently positive compared to the degree of homology. The proof uses a version of Vassiliev's method of simplicial resolution. As a consequence, we obtain a homological stability result for rational curves on the degree \(5\) del Pezzo surface, which is analogous to a case of the Batyrev–Manin conjectures on rational point counts.

• The Mumford conjecture (after Bianchi)

  with Dan Petersen
  2024 · arxiv

  abstract

  We give a self-contained and streamlined rendition of Andrea Bianchi's recent proof of the Mumford conjecture using moduli spaces of branched covers.

• Homological stability for the space of hypersurfaces with marked points

  with Alexis Aumonier
  2023 · arxiv

  abstract

  We study the space of marked smooth hypersurfaces in a given linear system. Specifically, we prove a homology h-principle to compare it with a space of sections of an appropriate jet bundle. Using rational models, we compute its rational cohomology in a range of degrees, and deduce a homological stability result for hypersurfaces of increasing degree. We also describe the Hodge weights on the stable cohomology, and thereby connect our topological result to motivic results of Howe.

• Cohomological and motivic inclusion-exclusion

  with Sean Howe
  2024 · Compositio Mathematica, 160, 9, pp. 2228–2283
  journal · arxiv

  abstract

  We categorify the inclusion-exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.

• Zeta statistics and Hadamard functions

  with Margaret Bilu and Sean Howe
  2022 · Advances in Mathematics, 407
  journal · arxiv

  abstract

  We introduce the notion of 'Hadamard convergence' for zeta functions of a family of algebraic varieties. This is stronger than both convergence of point counts over finite fields and convergence in dimension of clases in a suitable Grothendieck ring. We make the meta-conjecture that any 'natural' sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.

• Cohomology of the universal smooth cubic surface

  2021 · The Quarterly Journal of Mathematics, 72, 3, pp. 795–815
  journal · arxiv

  abstract

  We compute the rational cohomology of the universal family of smooth cubic surfaces using Vassiliev's method of simplicial resolution. Modulo embedding, the universal family has cohomology isomorphic to that of \(\mathbb{P}^2\). A consequence of our theorem is that over the finite field of order \(q\), away from finitely many characteristics, the average number of points on a smooth cubic surface is \(q^2 + q + 1\).

• The space of cubic surfaces equipped with a line

  2021 · Mathematische Zeitschrift, 298(1), pp. 653–670
  journal · arxiv

  abstract

  The Cayley–Salmon theorem implies the existence of a 27-sheeted covering space parametrizing the lines contained in smooth cubic surfaces over complex numbers. We compute the rational cohomology of the total space of this cover, using the spectral sequence in the method of simplicial resolution developed by Vassiliev. The covering map is an isomorphism in cohomology (in fact of mixed Hodge structures) and the cohomology ring is isomorphic to that of \(PGL(4,\mathbb{C})\). We derive as a consequence of our theorem that over the finite field of order \(q\) the average number of lines on a cubic surface equals 1 (away from finitely many characteristics); a priori this average is \(1 + O(q^{-1/2})\) by a standard application of the Weil conjectures.

• Configurations of noncollinear points in the projective plane

  with Ben O'Connor
  2021 · Algebraic and Geometric Topology, 21(4), pp. 1941–1972
  journal · arxiv

  abstract

  We consider the space \(F_n\) of configurations of \(n\) points in \(P^2\) satisfying the condition that no three of the points lie on a line. For \(n = 4, 5, 6\), we compute \(H^*(F_n; \mathbb{Q})\) as an \(S_n\)-representation. The cases \(n = 5, 6\) are computed via the Grothendieck–Lefschetz trace formula in étale cohomology and certain 'twisted' point counts for analogous spaces over \(\mathbb{F}_q\).

• Arithmetic statistics on cubic surfaces

  2020 · Research in the Mathematical Sciences, 7(3), p. 23
  journal · arxiv

  abstract

  We compute the distributions of various markings on smooth cubic surfaces defined over a finite field, for example the distribution of pairs of points, 'tritangents' or 'double sixes'. We also compute the (rational) cohomology of associated bundles and covers over complex numbers.

• Coxeter transformation groups and reflection arrangements in smooth manifolds

  with Priyavrat Deshpande
  2016 · Journal of Homotopy and Related Structures, 11(3), pp. 571–597
  journal · arxiv

  abstract

  We prove a version of Salvetti's theorems for Coxeter groups acting on manifolds with each reflection fixing a codimension-\(1\) submanifold. Then the fundamental groups of the tangent bundle complement and its quotient by the Coxeter group serve as the analogue of the (pure) Artin group. We construct a combinatorial cell complex which is equivariantly homotopy equivalent to the tangent bundle complement.

talks

Hypersurfaces and inclusion-exclusion

• 24 Oct 2023 · Topology seminar · Florida State University

• 28 Jun 2023 · Mathematics Seminar · Chennai Mathematical Institute

• 2 Nov 2022 · Mathematics and Applied Mathematics seminar · Mälardalens University, Västerås

Cohomological and motivic inclusion-exclusion

• 3 Mar 2022 · Algebraic Geometry and Number Theory Seminar · IST Austria

Stability of zeta statistics

• 26 Nov 2021 · Algebraic Geometry Seminar · Tata Institute of Fundamental Research

• 15 Jan 2021 · Algebra/Topology Seminar · University of Copenhagen

• 17 Nov 2020 · Algebra & Geometry Seminar · Stockholm University + KTH Royal Institute of Technology

Noncollinear points in the projective plane

• 8 Jul 2020 · CMI Online Seminar Series · Chennai Mathematical Institute · video · notes

Points and lines on cubic surfaces

• 7 Dec 2022 · Mathematics Colloquium · Kuwait University

• 9 Feb 2021 · Algebra, Geometry and Number Theory seminar · University of Bath

• 9 Nov 2019 · Special Session on Arithmetic Geometry in Finite Characteristic · AMS Sectional at University of California, Riverside

• 16 Sep 2019 · Number Theory Seminar · University of Utah

• 14 Sep 2019 · Special Session on Geometry and Topology in Arithmetic · AMS Sectional at University of Wisconsin–Madison

• 10 Jun 2019 · Workshop on Arithmetic Topology · PIMS, University of British Columbia · slides

• 19 Apr 2019 · Algebraic Geometry Seminar · Stanford University

• 29 Mar 2019 · Geometric Topology Seminar · Columbia University

• 17 Mar 2019 · Midwest Algebraic Geometry Graduate Conference · The University of Illinois at Chicago

• 20 Feb 2019 · lightning talk · Braids, Resolvent Degree and Hilbert’s 13th Problem Workshop · IPAM, University of California, Los Angeles · slides

• 6 Feb 2019 · Algebraic Geometry Seminar · The University of Illinois at Chicago

The space of cubic surfaces equipped with a line

• 13 Jun 2018 · lightning talk · Roots of Topology Workshop · University of Chicago · slides

• 8 Jun 2018 · Topology Students Workshop · Georgia Institute of Technology · slides

• 23 May 2018 · Algebraic Geometry Seminar · University of Chicago

Single cohomology classes in your area — this last weird trick will shock you!

• 23 Aug 2017 · colloquium · Chennai Mathematical Institute

Simplicial resolution a la Vassiliev and counting points

• 21 Aug 2017 · mathematics seminar · Chennai Mathematical Institute

Artin groups related to reflections on manifolds

• 11 Sep 2014 · research lecture series · Chennai Mathematical Institute · slides

et cetera

t-shirt designs

most of my t-shirts are of my own design, though the jokes are often not original
I use spreadshirt for the actual printing
png files with background · email me for printing-friendly versions

thesis defense crossword

pdf · spreadsheet copy

beer skits

annual comedy show written and performed by the second year graduate students at UChicago
YouTube playlist for beer skits 2017