Ostensible contact info:3218 MSB (530) 341-3624 Actual contact info: |
Mailing address:Department of Mathematics One Shields Avenue University of California Davis, CA 95616 USA |

Office hours: T 2-3, W 11-12 | |

Field: algebraic geometry | |

Research interests: moduli spaces, enumerative
geometry, degeneration and deformation techniques, positive-characteristic
phenomena. |

My curriculum vitae.

Some of my (older) photos, and
some more recent (mostly bird)
photos.

Math 150C: Algebra Spring 2015

PE 1: Kokikai Aikido Spring 2015

Math 115A: Number Theory Fall 2014

Math 248B: Algebraic Geometry Winter 2014

Math 21A: Calculus I Fall 2013

Math 248A: Algebraic Geometry Fall 2013

Math 280: Algebraic Curves and Degenerations Winter 2013

Math 150B: Algebra Winter 2013

Math 21A: Calculus I Fall 2012

Math 248B: Algebraic Geometry Winter 2012

Math 250B: Algebra Winter 2011

Math 16B: Short Calculus Winter 2011

Math 248B: Algebraic Geometry Winter 2010

Math 248A: Algebraic Geometry Fall 2009

Math 21B: Calculus II Fall 2009

Math 240A: Differential Geometry Fall 2008

Math 21C: Calculus III Fall 2008

Math 167: Applied Linear Algebra Summer Session II 2008

Math 250C: Algebra Spring 2008

Number theory reading course Winter 2008

Math 21A: Calculus I Fall 2007

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)

Western Algebraic Geometry Symposium February 28-March 1, 2015, at UC Davis

Berkeley-Davis-Stanford Algebraic Geometry Colloquium 2006-2011, at UC Berkeley and Stanford (organizer 2006-2007, 2010-2011)

RFG: The geometry and combinatorics of branched covers 2009-2010, at UC Davis

MSRI Summer Graduate Workshop: Deformation Theory and Moduli in Algebraic Geometry July 23-August 3, 2007, at MSRI

Commutative Algebra and Algebraic Geometry Seminar ongoing, at UC Berkeley (organizer 2005-2007)

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

Recent progress on limit linear series

10/26/2014, AMS Special Session:

A brief survey of my recent work on limit linear series for curves not of compact type, focusing on implications for Brill-Noether theory of graphs.

See also: Dimension counts for limit linear series on curves not of compact type and Limit linear series for curves not of compact type below.

Recent progress on vector bundles with sections

3/18/2014, Utah Algebraic Geometry
Seminar

An overview of my recent work on higher-rank Brill-Noether theory,
including two main branches: modified expected dimensions of moduli
spaces on smooth curves, in the case of special determinant; and
degeneration techniques. The latter is a mix of foundational work on
higher-rank limit linear series, together with applications of this
foundational work to proving new existence results on smooth curves.

See also:
Brill-Noether loci with fixed determinant in
rank 2,
Special determinants in higher-rank Brill-Noether
theory,
Limit linear series moduli stacks in higher rank,
and
Linked symplectic forms and limit linear
series in rank 2 with special determinant (with Montserrat Teixidor
i Bigas) below.

New perspectives on limit linear series

9/5/2013, Yale Algebraic and
Tropical Geometry Seminar

A seminar talk describing a new way to formulate the definition of
Eisenbud-Harris limit linear series, and explaining how this leads to
a definition of limit linear series for curves not of compact type.

See also:
Limit linear series for curves not of compact type
below.

Counting curves on toric surfaces

9/24/2013, Berkeley Commutative
Algebra and Algebraic Geometry Seminar

A seminar talk describing joint work with Fu Liu which uses the work
of Brugalle and Mikhalkin to show that counts of nodal curves on a
certain family of (possibly singular) toric surfaces satisfy
universal formulas similar to the Goettsche-Yau-Zaslow formula for
smooth surfaces. Also gives combinatorial formulas for the coefficients
in the Goettsche-Yau-Zaslow formula.

See also:
Severi degrees on toric surfaces (with Fu Liu)
below.

Limit linear series: constructions and
applications

November 2010, IMPA lecture series

A series of talks describing the Eisenbud-Harris theory of limit linear
series and some applications, and then my own construction of limit
linear series spaces, and some new applications of it to fibers of Abel
maps and higher-rank Brill Noether theory.

Lifting tropical intersections

12/7/2009, MSRI Tropical Colloquium

Describes joint work with Sam Payne studying when tropicalization
commutes with intersection. We describe when this is the case (in a strong
sense taking multiplicities into account) in terms of expected dimension
hypotheses, considering also the case of intersections inside an ambient
subvariety of the torus.

See also:
Lifting tropical intersections (with Sam Payne)
below.

Vector bundles with sections

10/24/2009, Western Algebraic Geometry Seminar

After reviewing classical Brill-Noether theory, discusses the situation
for its higher-rank generalization. Generalizing work of Bertram, Feinberg
and Mukai, presents an attempt to study modified
expected dimensions systematically in the case of vector bundles of rank
2 with fixed special determinant.

See also:
Brill-Noether loci with fixed determinant in rank
2 below.

Beyond Schlessinger: deformation stacks
(or PDF)

4/14/2006, Columbia Algebraic Geometry Seminar

Gives quick introductions to classical deformation theory and Schlessinger's
criterion, as well as to moduli stacks, and then introduces a new framework
for deformation problems, inspired by stack theory, which allows elementary
proofs of very general results relating local and global deformations.

See also:
deformations and automorphisms: a framework for
globalizing local tangent and obstruction spaces below.

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.

Linked determinantal loci and limit linear series, with John Murray (arXiv) Fall 2014

Proves that linked determinantal loci are Cohen-Macaulay, implying Cohen-Macaulayness and flatness for limit linear series spaces. Also yields a comparison theorem between two scheme structures on moduli spaces of limit linear series, necessary for analysis of geometry of Brill-Noether loci via degenerations.

Preprint.

Dimension counts for limit linear series on curves
not of compact type (arXiv) Fall 2014

Investigates the dimension behavior of the new definition of limit linear
for curves not of compact type, in the context of curves of
"pseudocompact type." The first result is that under suitable generality
hypotheses, the dimension will be as expected provided that gluing
conditions impose the maximal codimension. We then analyze these gluing
condition in three specific families, showing that the behavior may
depend on chain structures, on enriched structures, or neither. One
of these families may be seen as generalizing and shedding new light
on the work of Cools, Draisma, Payne and Robeva in the tropical context.

Preprint.

Limit linear series for curves
not of compact type (arXiv) Summer 2014

Introduces a new notion of limit linear series for nodal curves not
of compact type. Constructs moduli spaces, proves a specialization result,
and gives an equivalent definition for a restricted class of curves,
showing in particular that the new definition generalizes that of Eisenbud
and Harris. Finally, proves a smoothing theorem for the same class of
curves, strengthening what was previously known even in the compact type
case.

Preprint.

Limit linear series moduli stacks
in higher rank
(arXiv) Spring 2014

Generalizes the construction of A limit linear series
moduli scheme to higher-rank vector bundles and arbitrary curves
of compact type, and also gives an equivalent description of
Eisenbud-Harris-Teixidor limit linear series, which is new even in the
rank-1 case, and yields a proper moduli space in families in that case.
Proves a comparison theorem which is the starting point for
Linked symplectic forms and limit
linear series in rank 2 with special determinant below.

Preprint.

Linked symplectic forms and limit
linear series in rank 2 with special determinant, with
Montserrat
Teixidor i Bigas
(arXiv) Spring 2014

Generalizes the prior linked symplectic
Grassmannian construction, applying it to to prove smoothing results
for rank-2 limit linear series with fixed special determinant on chains
of curves. Applies the general machinery to prove new results on
nonemptiness and dimension of rank-2 Brill-Noether loci in a range of
degrees.

Preprint.

Severi degrees on toric surfaces,
with
Fu
Liu
(arXiv) Winter 2014

Builds on work of Brugalle and Mikhalkin, Ardila and Block, and Liu to
give universal formulas for the number of nodal curves in a linear system
on a certain family of (possibly singular) toric surfaces. These formulas
are explicitly related to the Goettsche-Yau-Zaslow formula, and are used
to give combinatorial expressions for the coefficients arising in the
latter.

Preprint.

Stability of vector bundles on curves and degenerations
(arXiv) Winter 2014

Introduces a weaker and more robust version of stability for vector
bundles on reducible curves which is well-suited to degeneration
arguments.

Preprint.

Relative dimension of morphisms and dimension for algebraic
stacks
(arXiv) Spring 2013

Motivated by limit linear series and higher-rank Brill-Noether theory,
we introduce a flexible and powerful language for expressing lower bounds
on relative dimension of morphisms of schemes, and more generally of
algebraic stacks. In a complementary direction, we also develop the basic
properties of codimension for algebraic stacks.

Preprint.

Lifting non-proper tropical intersections, with
Joseph
Rabinoff
(arXiv) Summer 2011

For subschemes of the torus with finite intersection, we show that
even if the tropicalizations have non-proper intersection, the stable
intersection points lift within any given connected component of the
tropical intersection, if one works in a suitable partial compactification.
The main technical tool is a moving lemma for analytic spaces.
*Tropical and Non-Archimedean Geometry*, Omid Amini, Matthew
Baker and Xander Faber (eds), Contemporary Mathematics 605, 2013, 15-44.

Special determinants in higher-rank Brill-Noether theory
(arXiv) Summer 2011

Continuing the program begun in Brill-Noether loci
with fixed determinant in rank 2, we set up a general framework to
analyze expected dimensions of higher-rank Brill-Noether loci with fixed
special determinant, and prove dimension bounds for some cases for which
the dimension of the spaces of global sections is small relative to the
rank.
*International Journal of Mathematics* **24** (2013), no. 11,
1350084, 20 pages.

A simple characteristic-free proof
of the Brill-Noether theorem
(arXiv) Summer 2011

We give a proof of the folklore theorem that limit linear series can be
used to give a characteristic-free proof of the Brill-Noether theorem if
one uses a different degeneration from that considered by Eisenbud and
Harris. The theorem allows ramification at up to two points, and we give
a simple, sharp condition for nonemptiness in this setting.
*Bulletin of the Brazilian Mathematical Society* **45** (2014),
no. 4, 807-818 (special issue in honor of Steven Kleiman and Aron Simis).

Abel maps and limit linear series,
with
Eduardo
Esteves
(arXiv) Winter 2011

We explore the relationship between fibers of Abel maps and limit linear
series for reducible curves consisting of two smooth components glued at
a single node. Using the construction of a limit linear
series moduli scheme, we associate to a limit linear series a closed
subscheme of the fiber of the Abel map, which behaves well under
degeneration from a linear series on a family of smooth curves.
*Rendiconti del Circolo Matematico di Palermo* **62** (2013),
no. 1, 79-95 (special issue on algebraic geometry).

Linked alternating forms and linked
symplectic Grassmannians, with
Montserrat
Teixidor i Bigas
(arXiv) Winter 2011

Motivated by applications to higher-rank Brill-Noether theory, we
introduce the concepts of linked alternating and linked symplectic forms
on a chain of vector bundles, and show that the resulting linked symplectic
Grassmannians have good dimensional behavior analogous to that of the
classical symplectic Grassmannian.
*International Mathematics Research Notices* **2014** (2014),
no. 3, 720-744.

Strong nonnegativity and sums of
squares on real varieties, with Mohamed Omar (arXiv) Winter 2011

Motivated by scheme theory, introduces a notion of strong nonnegativity for
real polynomial functions on varieties. Every sum of squares is strongly
nonnegative, so we produce a new obstruction to being a sum of squares. In
convex optimization, we recover and generalize obstructions of Gouveia and
Netzer to convergence of the theta body hierarchy for certain classes of
singular varieties.
*Journal of Pure and Applied Algebra* **217** (2013), no. 5,
843-850.

Linked Hom spaces
(arXiv) Summer 2010

A note describing a Hom version of linked Grassmannians, and applying
it to give a more transparent construction of limit linear series
spaces out of linked Grassmannians.
*Mathematical Research Letters* **18** (2011), no. 2, 329-335.

Lifting tropical intersections,
with Sam Payne
(arXiv) Summer 2010

We use extended tropicalizations and geometry over non-Noetherian valuation
rings to prove general results on lifting tropical intersections to
algebraic intersections, under an expected dimension hypothesis. Our
results apply also to intersections inside ambient subvarieties of the
torus.
*Documenta Mathematica* **18** (2013), 121-175.

Brill-Noether loci with fixed
determinant in rank 2
(arXiv) Spring 2010

Generalizing work of Bertram, Feinberg, and Mukai in the case of fixed
canonical determinant, we produce a range of modified expected dimensions
for Brill-Noether loci of determinants which are special in suitable
senses. Involves a careful study of symplectic Grassmannians and "doubly
symplectic Grassmannians."
*International Journal of Mathematics* **24** (2013), no. 13,
1350099, 24 pages.

Some 4-point Hurwitz numbers in
positive characteristic,
with Irene Bouw
(arXiv) Summer 2009

We compute a family of Hurwitz numbers in characteristic *p*, using
the theory of stable reduction and formulas of Wewers in combination
with degenerations via admissible covers. In the process, we also see that
in most of the cases we consider, every cover has a degeneration to a
(separable) admissible cover.
*Transactions of the AMS* **363** (2011), no. 12, 6685-6711.

Functorial reconstruction theorems for stacks,
with Max Lieblich
(arXiv) Summer 2008

We study the following question: to what extent is a stack determined by
the associated functor of sets of isomorphism classes of objects? The answer
turns out to be that surprisingly often it is completely determined. We show
that a number of classical moduli spaces are uniquely determined as
algebraic stacks by their functors, including moduli spaces of stable curves
with marked points and of stable vector bundles on curves.
*Journal of Algebra* **322** (2009), no. 10, 3499-3541.

Deformations and automorphisms:
a framework for globalizing local tangent and obstruction spaces
(arXiv) Spring 2008

This paper develops a framework inspired by (but not dependent on)
stack theory to treat deformation problems in a uniform but still
relatively elementary manner. The immediate goal is to prove in
generality comparable to Schlessinger's work certain universal results
on global tangent and obstruction spaces, given information on the
corresponding local spaces.
*Annali della Scuola Normale Superiore di Pisa, Classe di Scienze*,
**IX** (2010), no. 3, 581-633.

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* **130** (2008), no. 6,
1687-1708.

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.
*Journal of Pure and Applied Algebra* **213** (2009), no. 9,
1651-1664.

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.

Collected algebraic geometry notes

A single web page containing all my expository notes on algebraic geometry, including both course notes and independent material.

Limit linear series

A book in progress on the Eisenbud-Harris theory of limit linear series,
and its applications. Foundations of the theory are presented in a
completely new manner.

Improving the refereeing process: a simple proposal
Fall 2012

A suggestion for how to reduce the number of rejections which are both
slow and perfunctory.
*Notices of the AMS*, November 2012.

The Weil conjectures (or
PDF) Winter 2007

A brief exposition of the history of the Weil conjectures and their
proof, intended for a very general audience.

In *Princeton Companion to Mathematics*, Princeton University
Press, 2008.

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.