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. I am currently a senior lecturer (tenure-track) at the Einstein Institute of Mathematics at The Hebrew University of Jerusalem.
Papers/Preprints (the most up-to-date versions of all papers may be found at this webpage):
Quasi-connected Reductive Groups (arXiv version, to appear in Archiv der Mathematik): (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, submitted): 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, to appear in Algebra and Number Theory): 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, to appear in Memoirs of the American Mathematical Society): 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?