I am a mathematician working in number theory and algebraic geometry, especially over imperfect fields. I received my Ph.D. in mathematics from Stanford University, where my adviser was Brian Conrad.
Papers/Preprints:
The Restricted Picard Functor (arXiv Version): We prove in significant generality the (almost-)representability of the Picard functor when restricted to smooth test schemes. The novelty lies in the fact that we prove such (almost-)representability beyond the proper setting.
Non-Extendability of Abel-Jacobi Maps (arXiv Version): We investigate the "natural" locus of definition of Abel-Jacobi maps. In particular, we show that, for a proper, geometrically reduced curve C -- not necessarily smooth -- the Abel-Jacobi map from the smooth locus C^{sm} into the Jacobian of C does not extend to any larger (separated, geometrically reduced) curve containing C^{sm} except under certain particular circumstances which we describe explicitly. As a consequence, we deduce that the Abel-Jacobi map has closed image except in certain explicitly described circumstances, and that it is always a closed embedding for irreducible curves not isomorphic to the projective line.
Algebraic Groups with Torsors That Are Versal for All Affine Varieties (arXiv Version, joint with Uriya First and Mathieu Florence): We show that G is unipotent if and only if it admits a G-torsor over a quasi-compact base that is weakly versal for all finite type regular affine k-schemes. Our proof is characteristic-free and it also gives rise to a quantitative statement: If G is a non-unipotent subgroup of GLn, then a G-torsor over a quasi-projective k-scheme of dimension d is not weakly versal for finite type regular affine k- schemes of dimension n(d + 1) + 2. This means in particular that every such G admits a nontrivial torsor over a regular affine (n + 2)-dimensional variety. When G contains a nontrivial torus, we show that nontrivial torsors already exist over 3-dimensional smooth affine varieties (even when G is special), and this is optimal in general.
Rigidity And Unirational Groups (arXiv Version): We prove a rigidity theorem for morphisms from products of open subschemes of the projective line into solvable groups not containing a copy of the additive group (for example, wound unipotent groups). As a consequence, we deduce several structural results for unirational group schemes, including that unirationality for group schemes descends through separable extensions. We also apply the main result to prove that permawound unipotent groups are unirational and -- when wound -- commutative.
Permawound Unipotent Groups (arXiv Version): We introduce the class of permawound unipotent groups, and show that they simultaneously satisfy certain "ubiquity" and "rigidity" properties that in combination render them very useful in the study of general wound unipotent groups. As an illustration of their utility, we present two applications: We prove that nonsplit smooth unipotent groups over (infinite) finitely-generated fields have infinite first cohomology; and we show that every commutative p-torsion wound unipotent group over a field of degree of imperfection 1 is the maximal unipotent quotient of a commutative pseudo-reductive group, thus partially answering a question of Totaro.
Mikhail Borovoi, Criterion For Surjectivity Of Localization In Galois Cohomology Of A Reductive Group Over A Number Field, with an appendix by Zev Rosengarten (arXiv Version, to appear in Comptes Rendus - Série Mathématique): Let G be a connected reductive group over a number field F , and let S be a set (finite or infinite) of places of F . We give a necessary and sufficient condition for the surjectivity of the localization map from H1(F,G) to the “direct sum” of the sets H^1(Fv,G) where v runs over S. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.
On The Galois And Flat Cohomology Of Unipotent Algebraic Groups Over Local And Global Function Fields II (Michigan Mathematical Journal Advance Publication, 1--21 (2023)): (joint with Nguyễn Duy Tân and Nguyễn Quốc Thắng) Let k be a field of positive characteristic p with degree of imperfection 1 and a non-trivial discrete valuation. We show that the Galois and flat cohomology of unipotent k-groups of dimension < p-1 are finite (in fact trivial) if and only if they are k-split. Some examples are given to show that the bound p-1 is best possible, and several applications to local-global principles are also presented.
Picard Groups of Algebraic Groups And An Affineness Criterion (arXiv version, Comptes Rendus Mathématique, Vol.\,361 (2023), 559--564): We prove that an algebraic group over a field k is affine precisely when its Picard group is torsion, and show that in this case the Picard group is finite when k is perfect, and the product of a finite group of order prime to p and a p-primary group of finite exponent when k is imperfect of characteristic p.
Quasi-connected Reductive Groups (arXiv version, Archiv der Mathematik, Vol. 118 (2022), 27-38): (joint with Mikhail Borovoi and Andrei A. Gornitskii) By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set H^1(R,G) of a quasi-connected reductive group G over the field R of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.
Moduli Spaces of Morphisms Into Solvable Algebraic Groups (arXiv version, Algebra and Number Theory, Vol.\,16, No.\,10 (2022), 2493--2531): We construct (in significant generality) moduli spaces representing the functor of morphisms from a scheme into a solvable algebraic group.
Pathological Behavior of Arithmetic Invariants of Unipotent Groups (arXiv version, Algebra and Number Theory, Vol. 15, Issue 7 (2021), 1593-1626): We show that the nice behavior of Tamagawa numbers, Tate-Shafarevich sets, and other arithmetic invariants of pseudo-reductive groups over global function fields fails in general for non-commutative unipotent groups. We also give some positive results which show that Tamagawa numbers do exhibit some reasonable behavior for arbitrary connected linear algebraic groups over global function fields.
Tamagawa Numbers and Other Invariants of Pseudo-reductive Groups Over Global Function Fields (arXiv version, Algebra and Number Theory, Vol.\,15 (2021), No.\,8, 1865--1920): We study Tamagawa numbers and other invariants (especially Tate-Shafarevich sets) attached to commutative and pseudo-reductive groups over global function fields. In particular, we prove a simple formula for Tamagawa numbers of commutative groups and pseudo-reductive groups. We also show that the Tamagawa numbers and Tate-Shafarevich sets of such groups are invariant under inner twist, as well as proving a result on the cohomology of such groups which extends part of classical Tate duality from commutative groups to all pseudo-reductive groups. Finally, we apply this last result to show that for suitable quotient spaces by commutative or pseudo-reductive groups, the Brauer--Manin obstruction is the only obstruction to strong (and weak) approximation.
Translation-Invariant Line Bundles on Linear Algebraic Groups (arXiv version, Journal of Algebraic Geometry, 30 (2021), 433-455): We study the Picard groups of connected linear algebraic groups, and especially the subgroup of translation-invariant line bundles. We prove that this subgroup is finite over every global function field. We also utilize our study of these groups in order to construct various examples of pathological behavior for the cohomology of commutative linear algebraic groups over local and global function fields.
Tate Duality in Positive Dimension (arXiv version, Memoirs of the American Mathematical Society, Vol.\,290, No.\,1444): This manuscript generalizes classical Poitou-Tate duality (local duality, the nine-term exact sequence, etc.) from finite group schemes to arbitrary affine commutative group schemes of finite type. The motivation for doing so was to study Tamagawa numbers of linear algebraic groups over global function fields. This study is taken up in the papers "Tamagawa Numbers and Other Invariants of Pseudo-reductive Groups Over Global Function Fields" and "Pathological Behavior of Arithmetic Invariants of Unipotent Groups" above. The results of this work are also used in the paper "Translation-Invariant Line Bundles on Linear Algebraic Groups" above.
Some undergrad papers (some of these are rather old, with suboptimal use of LaTex):
An Erdős–Turán Inequality For Compact Simply-Connected Semisimple Lie Groups: The classical Weyl criterion gives a criterion for the equidistribution of points on the circle in terms of the smallness of various exponential sums. The Erdős–Turán inequality is a quantitative version of the Weyl Criterion: it bounds a suitable measure of how far a sequence is from being equidistributed (the discrepancy) in terms of various exponential sums. This paper gives an analogous inequality for compact simply-connected semisimple Lie groups, bounding the failure of a sequence of elements of such a group G to be equidistributed among the conjugacy classes of G in terms of suitable sums of characters of representations of G along the sequence.
The Grunwald-Wang Theorem: We give a proof of the classical Grunwald-Wang Theorem, which answers the question: to what extent is it true/false that an element of a number field K is a global nth power if it is so everywhere locally?
Yet Another Proof of Quadratic Reciprocity: What it sounds like. Note: Ehud de Shalit tells me that this proof was actually already found by Richard Swan.
The Restricted Picard Functor (arXiv Version): We prove in significant generality the (almost-)representability of the Picard functor when restricted to smooth test schemes.