My research is in arithmetic geometry, algebraic geometry, and tropical geometry. I am interested in a variety of topics within these subjects, most of which are at a basic level questions about geometry over fields with a non-Archimedean valuation. I am also interested in the interplay between tropical and arithmetic geometry. Tropical methods can be used to prove theorems about canonical subgroups of abelian varieties and component groups of Néron models of Jacobians; conversely, arithmetic methods can be applied to prove tropical lifting theorems and structure theorems about tropical curves. At the moment I am thinking about explicitly constructing semistable models of curves, the tropical consequences of the model theory of valued fields, and the relationship between algebraic and tropical intersection theory, among other things.

Papers and Preprints:

• Joseph Rabinoff, Higher-level canonical subgroups for p-divisible groups, to appear in the Journal of the Institute of Mathematics of Jussieu.

  Let R be a complete rank-1 valuation ring of mixed characteristic (0,p), and let K be its field of fractions. A g-dimensional truncated Barsotti-Tate group G of level n over R is said to have a level-n canonical subgroup if there is a K-subgroup of G\tensor_R K with geometric structure (\Z/p^n\Z)^g consisting of points "closest to zero". We give a nontrivial condition on the Hasse invariant of G that guarantees the existence of the canonical subgroup, analogous to a result of Katz and Lubin for elliptic curves. The bound is independent of the height and dimension of G.

• Joseph Rabinoff, Tropical analytic geometry, Newton polygons, and tropical intersections, to appear in Advances in Mathematics.

  In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler-Kapranov-Lind and Gubler. We have written this paper to be as readable as possible both to tropical and arithmetic geometers.

• Matthew Baker, Sam Payne, and Joseph Rabinoff, Nonarchimedean geometry, tropicalization, and metrics on curves, submitted.

  We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity formula in tropical elimination theory to the case of a nontrivial valuation. For curves, we explore in detail the relationship between skeletal metrics and lattice lengths on tropicalizations and show that the maps from the analytification of a curve to the tropicalizations of its toric embeddings stabilize to an isometry on finite subgraphs. Other applications include generalizations of Speyer's well-spacedness condition and the Katz-Markwig-Markwig results on tropical j-invariants.

• Brian Osserman and Joseph Rabinoff, Lifting non-proper tropical intersections, submitted.

  We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) \cap Trop(X') lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X(\Delta) and its associated extended tropicalization N_R(\Delta); the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the closure of C in N_R(\Delta). The proof involves a result on continuity of intersection numbers in the context of non-Archimedean analytic spaces.

Older Papers:

• Joseph Rabinoff, Hybrid grids and the Homing Robot, Discrete Appl. Math 140 (2004), no. 1-3, 155-168.
  
  (From the Duluth Research Experience for Undergraduates.)

Expository Articles:

• The theory of Witt vectors, a write-up of various "well-known" results about Witt vectors.
• The Bruhat-Tits building of a p-adic Chevalley group and an application to representation theory, my senior thesis at Harvard University.

Please see my CV for a full list of my teaching experience. I am currently on an NSF fellowship and am not teaching this year.