My curriculum vitae (or PDF)

Some recent photos I took.

Number theory reading course  Winter 2008, at UC Davis

Math 21A: Calculus I  Fall 2007, at UC Davis

Math 256B: Algebraic Geometry  Spring 2007, at UC Berkeley

Math 256A: Algebraic Geometry  Fall 2006, at UC Berkeley

Math H113: Honors Abstract Algebra  Spring 2006, at UC Berkeley

Math 254A: Number Theory  Fall 2005, at UC Berkeley

Berkeley-Stanford Algebraic Geometry Colloquium  ongoing, at UC Berkeley and Stanford

Commutative Algebra and Algebraic Geometry Seminar  ongoing, at UC Berkeley

Western Algebraic Geometry Seminar  March 18-19, 2006, in Berkeley

Graduate Number Theory Seminar: Algebraic Number Theory and Elliptic Curves  Fall 2000, at MIT

Graduate Number Theory Seminar: Kolyvagin's Euler System  Spring 2000, at MIT

The representation theory, geometry, and combinatorics of branched covers (or PDF)
    10/25/2006, Berkeley Representation Theory, Geometry and Combinatorics Seminar
Discusses the general relationship between branched covers of the Riemann sphere and combinatorial group theory, and presents new results on connectedness of some classical Hurwitz spaces (joint with Fu Liu), and on a version of the Riemann existence theorem in characteristic p.
See also: the irreducibility of certain pure-cycle Hurwitz spaces (with Fu Liu), and linear series and existence of branched covers below.

Progress on Riemann existence via degenerations (or PDF)
    4/23/2006, AMS Special Session: Galois Groups in Arithmetic and Geometry, Durham, NH
Discusses two sharp results on existence and non-existence of tame genus-0 covers of the projective line. These are obtained by shifting to the point of view of linear series and using degeneration techniques. Note that Question 11 has since been answered affirmatively, in the irreducibility of certain pure-cycle Hurwitz spaces (with Fu Liu).
See also: linear series and existence of branched covers below.

Limit linear series: an overview (or PDF)
    2/25/2006, Bellingham Algebraic Geometry Seminar (joint UBC/UW)
Gives an introduction to the Eisenbud-Harris theory of limit linear series, and proceeds to discuss a functorial compactification, as well as some more recent results on the good properties of this compactification.
See also: a limit linear series moduli scheme, linked Grassmannians and crude limit linear series and flatness of the linked Grassmannian (with David Helm) below.

Rational functions in one variable with given ramification (or PDF)
    11/9/2004, Berkeley Commutative Algebra and Algebraic Geometry Seminar
Discusses the number of rational functions on the projective line with prescribed ramification in positive characteristic, giving a formula when the ramification indices are all smaller than p.
See also: rational functions with given ramification in characteristic p below.

Transversality of non-general Schubert cycles (or PDF)
    10/23/2004, AMS Special Session: Modern Schubert Calculus, Evanston, IL
Discusses enumerative questions on maps from curves to projective space with prescribed ramification, and a reduction to a question on transversality of Schubert cycles associated to rational normal curves. This question has been answered affirmatively by Mukhin, Varchenko, and Tarasov in this preprint. Touches also on issues of reality.
See also: the number of linear series on curves with given ramification and linear series over real and p-adic fields below.

Elementary applications of a technical theorem (or PDF)
    10/18/2004, Brown Algebraic Geometry Seminar
Discusses some applications of a technical theorem of Mochizuki to counting lattice points in certain polytopes, and to studying rational functions with prescribed ramification.
See also: Mochizuki's crys-stable bundles: a lexicon and applications and Mochizuki's indigenous bundles and Ehrhart polynomials (with Fu Liu) below.

Frobenius-unstable vector bundles and the generalized Verschiebung (or PDF)
    4/23/2004, Kyoto University Algebraic Geometry Seminar
Surveys results on the Verschiebung rational map induces on moduli spaces of vector bundles on curves by pulling back under Frobenius. Then discusses in more detail the case of genus 2 curves, and vector bundles of rank 2, including a sketch of the degeneration argument for the number of Frobenius-unstable bundles in this case.
See also: the generalized Verschiebung map for curves of genus 2 and Frobenius-unstable bundles and p-curvature below.

Linear series and existence of branched covers (arXiv)  Summer 2005, revised Fall 2006
A paper giving new results on existence and, more notably, non-existence of tamely branched covers. Degeneration techniques are used heavily, primarily from the point of view of limit linear series, and a relationship with Mochizuki's indigenous bundles is exploited as well. This has now been heavily revised, with the main result of the irreducibility of certain pure-cycle Hurwitz spaces below allowing for a greatly simplified proof of a stronger version of the main theorem.
Compositio Mathematica 144 (2008), no. 1, 89-106.

The irreducibility of certain pure-cycle Hurwitz spaces, with Fu Liu (arXiv)  Fall 2006
We use a combination of geometric and group-theoretic techniques to prove that Hurwitz spaces of genus-0 covers of the projective line having a single ramified point over each branch point are irreducible.
American Journal of Mathematics, to appear.

Linked Grassmannians and crude limit linear series (arXiv)  Spring 2006
A paper comparing the crude limit linear series of a limit linear series moduli scheme below to those of Eisenbud-Harris, showing that the compactification is well-behaved, and giving a new, very direct proof of the Brill-Noether theorem.
International Mathematics Research Notices 2006, no. 25, article ID 25782, 1-27.

Flatness of the linked Grassmannian, with David Helm (arXiv)  Spring 2006
A note proving that the linked Grassmannian schemes introduced in a limit linear series moduli scheme are Cohen-Macaulay, reduced, and flat.
Proceedings of the AMS 136 (2008), no. 10, 3383-3390.

Deformations of covers, Brill-Noether theory, and wild ramification (arXiv)  Fall 2004
A note explaining the deformation theory of covers with branching partially specified. Applications are given to positive-characteristic Brill-Noether theory for ramified one-dimensional linear series, with some preliminary investigation into wildly ramified rational functions.
Mathematical Research Letters 12 (2005), no. 4, 483-491.

Mochizuki's crys-stable bundles: a lexicon and applications (arXiv)  Fall 2004
A lexicon translating Mochizuki's work into the language of vector bundles with connection, and giving applications to Frobenius-unstable vector bundles and rational functions with prescribed ramification.
Publications of RIMS 43 (2007), no. 1, 95-119.

The generalized Verschiebung map for curves of genus 2 (arXiv)  Fall 2004
Edited version of chapter IV of my thesis. Discusses the degree and geometry of the Verschiebung map for rank 2 vector bundles on curves of genus 2.
Mathematische Annalen 336 (2006), no. 4, 963-986.

Frobenius-unstable bundles and p-curvature (arXiv)  Fall 2004
Edited version of chapters III and VI of my thesis. Uses techniques of connections with vanishing p-curvature to study vector bundles of rank 2 destabilized by Frobenius on curves of genus 2.
Transactions of the AMS 360 (2008), no. 1, 273-305.

Logarithmic connections with vanishing p-curvature (arXiv)  Fall 2004
Edited version of chapter V of my thesis. Relates logarithmic connections with vanishing p-curvature on the projective line to rational maps with prescribed ramification.
Submitted for publication.

A limit linear series moduli scheme (arXiv)  Summer 2004
Edited version of chapter II of my thesis. Gives a functorial compactification of the Eisenbud-Harris space of limit linear series.
Annales de l'Institut Fourier, 56 (2006), no. 4, 1165-1205.

Rational functions with given ramification in characteristic p (arXiv)  Summer 2004
Edited version of chapter I of my thesis. Studies the number of rational functions on the projective line with prescribed ramification in positive characteristic, giving a formula when the ramification indices are all smaller than p.
Compositio Mathematica 142 (2006), no. 2, 433-450.

Mochizuki's indigenous bundles and Ehrhart polynomials, with Fu Liu (or PDF)  Spring 2004
A short paper applying certain finite-flatness results of Mochizuki to obtain identities for numbers of lattice points in different polytopes, and conversely applying the theory of Ehrhart polynomials to show that Mochizuki's indigenous bundles are counted by polynomials in the characteristic of the base field.
Journal of Algebraic Combinatorics 23 (2006), no. 2, 125-136.

Linear series over real and p-adic fields (or PDF)  Spring 2004
A short note following up on implications for reality (and p-adicality) of the arguments of the below paper.
Proceedings of the AMS 134 (2006), no. 2, 989-993.

The number of linear series on curves with given ramification (or PDF)  Spring 2003
A simple paper coming out of, but standing alone from my thesis. Uses the Eisenbud-Harris theory of limit linear series.
International Mathematics Research Notices 2003, no. 47, 2513-2527.

Limit linear series in positive characteristic and Frobenius-unstable vector bundles on curves (or PDF)  Winter 2004
My PhD thesis, completed at MIT under the direction of Johan de Jong. Also available in default tex formatting (PDF), with a smaller number of pages. These are corrected versions. Finally, there is a list of mathematical errata detailing corrections to the submitted version.

The Riemann Hypothesis for elliptic curves, with Jasbir Chahal (or PDF)  Fall 2006
An exposition of zeta functions and Riemann hypotheses for global fields, including an elementary proof of the Riemann hypothesis for elliptic curves over finite fields.
American Mathematical Monthly 115 (2008), no. 5, 431-442.

Two degeneration techniques for maps of curves (or PDF)  Spring 2005
Slightly expanded writeup of a talk on admissible covers and limit linear series at the Snowbird Joint Summer Research Conference Algebraic Geometry: Presentations by Young Researchers, in July 2004.
Snowbird Lectures in Algebraic Geometry, 137-143, Contemporary Mathematics 388, AMS, 2005.

Note on cohomology and base change (or PDF)
A rough note collecting general results on the behavior of cohomology sheaves under base change. Primarily intended to help track down references for statements more general than those found in Hartshorne.

Note on dimension theory (or PDF)
A rough note discussing the pathologies of dimension theory even for Noetherian schemes, and ways in which one can phrase dimension-theoretic arguments so that they go through nonetheless.

Connections, curvature, and p-curvature (or PDF)
Rough lecture notes on algebraic connections from both the classical and Grothendieckian point of view, including Mochizuki's definition of p-curvature in the latter setting.