Preprints and Papers
Papers (recent to old, click for abstract)
5. Compactly supported A1-Euler characteristics of symmetric powers of cellular varieties, joint with Anna M. Viergever and Herman Rohrbach. (2024). arXiv link (submitted)
Abstract: The compactly supported A1-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an anologue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties taking values in the Grothendieck-Witt ring of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, Pajwani and Pál construct a power structure on GW(k) and show that the compactly supported A1-Euler characteristic respects these two power structures for 0-dimensional varieties, or equivalently finite dimensional étale k-algebras. In this paper, we define the class Sym_k of symmetrisable varieties to be those varieties for which the compactly supported A1-Euler characteristic respects the power structures and study the algebraic properties of K_0(Sym_k). We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported A1-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.
4. Galois invariants of finite abelian descent and Brauer sets, joint with Brendan Creutz and Felipe Voloch. (2024). arXiv link (submitted)
Abstract: For a variety over a global field, one can consider subsets of the set of adelic points of the variety cut out by finite abelian descent or Brauer-Manin obstructions. Given a Galois extension of the ground field one can consider similar sets over the extension and take Galois invariants. In this paper, we study under which circumstances the Galois invariants recover the obstruction sets over the ground field. As an application of our results, we study finite abelian descent and Brauer-Manin obstructions for isotrivial curves over function fields and extend results obtained by the Creutz-Voloch for constant curves to the isotrivial case.
3. Power structures on the Grothendieck--Witt ring and the motivic Euler characteristic, joint with Ambrus Pál. (2023) arXiv link (submitted)
Abstract: For k a field, we construct a power structure on the Grothendieck--Witt ring of k which has the potential to be compatible with symmetric powers of varieties and the motivic Euler characteristic. We then show this power structure is compatible with the power structure when we restrict to varieties of dimension 0.
2. The valuative section conjecture, étale homotopy, and Berkovich spaces. (2023) arXiv link (submitted)
Abstract: We reinterpret a result of Pop and Stix on the p-adic section conjecture in terms of Berkovich spaces and fixed points. In doing this, we see a version of the result extends to larger classes of fields, which in turn allows us to prove a valuative section conjecture type result for a larger class of varieties. This adds to the programme to reinterpret anabelian geometry results in terms of étale homotopy types.
Abstract: We prove an arithmetic refinement of the Yau-Zaslow formula by replacing the classical Euler characteristic in Beauville's argument by a "motivic Euler characteristic", related to the work of Levine. Our result implies similar formulas for other related invariants, including a generalisation of a formula of Kharlamov and Rasdeaconu on counting real rational curves on real K3 surfaces, and Saito's determinant of cohomology.
Miscellaneous writing
Two problems in arithmetic geometry inspired by homotopy theory: enriched curve counting for the Yau-Zaslow formula and a homotopic approach to the section conjecture. (2023). PhD Thesis.
Note that this is in two chapters: chapter 1 contains the work of the "Arithmetic Yau-Zaslow Formula" and "Power Structures on ___" papers, and Chapter 2 was made into "The valuative section conjecture" paper. The version of Chapter 2 in the paper above is largely unchanged, but chapter 1 is reworked so that it reads more as one narrative rather than 2 papers. This means that Chapter 1 is almost exclusively a treatment of the motivic Euler characteristic for the first 100 pages, and then there is an application to the Yau-Zaslow formula at the end once the theory is all built up. This probably makes for better bedtime reading than reading the two papers separately, but it also means that if you're interested in actual results and applications, it's probably better to read the individual papers, as they're more geared towards the results than the narrative.
Below are my LSGNT Mini projects as well as my Masters' dissertation. These are largely expository, and especially the mini projects may have mistakes in them.
A computation of the algebraic K theory of finite fields using Galois cohomology
Master's dissertation, written for the MMath at the University of Oxford in Spring 2019. Supervised by Dr Martin Gallauer. Largely expository.
An equivariant motivic Euler characteristic
First year mini project written for the London School of Geometry and Number Theory in Spring 2020. Supervised by Dr Ambrus Pál. Link removed as this is pretty much makes up the "Equivariant Euler characteristic" section of the preprint "An arithmetic Yau-Zaslow formula".
First year mini project written for the London School of Geometry and Number Theory in Summer 2020. Supervised by Professor Alexei Skorobogatov. Almost all expository.
Seminars organised
Junior étale homotopy study group (Autumn 2020)
