research

My papers are also available on my arXiv page.

1. KSp-characteristic classes determine Spin^h cobordism,
   with Jonathan Buchanan.
   Preprint, 2023.
   pdf abstract arXiv code

   A classic result of Anderson, Brown, and Peterson states that the cobordism spectrum MSpin (respectively, MSpin^c) splits as a sum of Eilenberg—Mac Lane spectra and connective covers of real K-theory (respectively, complex K-theory) at 2. We develop a theory of symplectic K-theory classes and use these to build an explicit splitting for MSpin^h in terms of Eilenberg—Mac Lane spectra and spectra related to symplectic K-theory. This allows us to determine the Spin^h cobordism groups systematically. We also prove that two Spin^h-manifolds are cobordant if and only if their underlying unoriented manifolds are cobordant and their KSp-characteristic numbers agree.

2. Bounding the signed count of real bitangents to plane quartics,
   with Mario Kummer.
   manuscripta math.  (2023)
   pdf abstract arXiv

   Using methods from enriched enumerative geometry, Larson and Vogt gave a signed count of the number of real bitangents to real smooth plane quartics. This signed count depends on a choice of a distinguished line. Larson and Vogt proved that this signed count is bounded below by 0, and they conjectured that the signed count is bounded above by 8. We prove this conjecture using real algebraic geometry, plane geometry, and some properties of convex sets.

3. Circles of Apollonius two ways.
   Preprint, 2022.
   pdf abstract arXiv

   Because the problem of Apollonius is generally considered over the reals, it suffers from variance of number: there are at most eight circles simultaneously tangent to a given trio of circles, but some configurations have fewer than eight tangent circles. This issue arises over other non-closed fields as well. Using the tools of enriched enumerative geometry, we give two different ways to count the circles of Apollonius such that invariance of number holds over any field of characteristic not 2. We also pose the geometricity problem for local indices in enriched enumerative geometry.

4. Lifts, transfers, and degrees of univariate maps,
   with Thomas Brazelton.
   Math. Scand.  129(1), 5 — 38 (2023)
   pdf abstract arXiv

   One can compute the local A¹-degree at points with separable residue field by base changing, working rationally, and post-composing with the field trace. We show that for endomorphisms of the affine line, one can compute the local A¹-degree at points with inseparable residue field by taking a suitable lift of the polynomial and transferring its local degree. We also discuss the general set-up and strategy in terms of the six functor formalism. As an application, we show that trace forms of number fields are local A¹-degrees.

5. Conics meeting eight lines over perfect fields,
   with Cameron Darwin, Aygul Galimova, and Miao (Pam) Gu.
   J. Algebra  631, 24 — 45 (2023)
   pdf abstract arXiv

   Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This provides a weighted count of the number of plane conics meeting 8 general lines, where the weight of each conic is determined the geometry of its intersections with the 8 given lines. As a corollary, real conics meeting 8 general lines come in two families of equal size.

6. Bézoutians and the A¹-degree,
   with Thomas Brazelton and Sabrina Pauli.
   Algebra Number Theory  17(11), 1985 — 2012 (2023)
   pdf abstract arXiv code

   We prove that both the local and global A¹-degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that the Bézoutian bilinear form, the Scheja—Storch form, and the A¹-degree for complete intersections are isomorphic. Our global theorem generalizes Cazanave's theorem in the univariate case, and our local theorem generalizes Kass—Wickelgren's theorem on EKL forms and the local degree. This result provides an algebraic formula for local and global degrees in motivic homotopy theory.

7. Rational lines on smooth cubic surfaces.
   Preprint, 2022.
   pdf abstract arXiv

   We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or 27. Over a given field, each of these line counts may or may not be realized by some cubic surface. We give a necessary criterion for each of these line counts in terms of the Galois theory of the base field. Over fields with at least 22 elements, we show that these necessary criteria are also sufficient. As examples, we use these criteria to classify line counts over finitely generated fields, finite transcendental extensions of arbitrary fields, finite fields of order at least 22, real closed fields, and the field of constructible numbers.

8. Bézoutians and injectivity of polynomial maps.
   J. Pure Appl. Algebra  227(6), 107298 (2023)
   pdf abstract arXiv code

   We prove that an endomorphism f of affine space is injective on rational points if its Bézoutian is constant. Similarly, f is injective at a given rational point if its reduced Bézoutian is constant. We also show that if the Jacobian determinant of f is invertible, then f is injective at a given rational point if and only if its reduced Bézoutian is constant.

9. An arithmetic enrichment of Bézout's Theorem.
   Math. Ann.  379(1), 633 — 660 (2021)
   pdf abstract arXiv

   The classical version of Bézout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of Bézout's Theorem over any perfect field by giving a bilinear form-valued count of the intersection points of hypersurfaces in projective space. Over non-algebraically closed fields, this enriched Bézout's Theorem imposes a relation on the gradients of the hypersurfaces at their intersection points. As corollaries, we obtain arithmetic-geometric versions of Bézout's Theorem over the reals, rationals, and finite fields of odd characteristic.

10. All lines on a smooth cubic surface in terms of three skew lines,
    with Daniel Minahan and Tianyi Zhang.
    New York J. Math.  27(1), 1305 — 1327 (2021)
    pdf abstract arXiv

    Jordan showed that the incidence variety of a smooth cubic surface containing 27 lines has solvable Galois group over the incidence variety of a smooth cubic surface containing 3 skew lines. As noted by Harris, it follows that for any smooth cubic surface, there exist formulas for all 27 lines in terms of any 3 skew lines. In response to a question of Farb, we compute these formulas explicitly. We also discuss how these formulas relate to Schläfli's count of lines on real smooth cubic surfaces.

11. The trace of the local A¹-degree,
    with Thomas Brazelton, Robert Burklund, Michael Montoro, and Morgan Opie.
    Homology Homotopy Appl.  23(1), 243 — 255 (2021)
    pdf abstract arXiv

    We prove that the local A¹-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local A¹-degree over the residue field. This fact was originally suggested by Morel’s work on motivic transfers, and by Kass and Wickelgren’s work on the Scheja—Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren relating the Scheja—Storch form and the local A¹-degree.

1. Local contributions in A¹-enumerative geometry.
   pdf description

   My PhD thesis, for which I received the Rudin Prize. The technical contents can be found in "An arithmetic enrichment of Bézout's Theorem" and "Circles of Apollonius two ways." The first chapter provides a non-technical introduction to the subject of enriched enumerative geometry.

2. Heights over finitely generated fields,
   with Soumya Sankar.
   Stacks Project Expository Collection, 222 — 254 (2022)
   pdf abstract

   This is an expository account about height functions and Arakelov theory in arithmetic geometry. We recall Conrad's description of generalized global fields in order to describe heights over function fields of higher transcendence degree. We then give a brief overview of Arakelov theory and arithmetic intersection theory. Our exposition culminates in a description of Moriwaki's Arakelov-theoretic formulation of heights, as well as a comparison of Moriwaki's construction to various versions of heights.