## Algebraic Curves: an Algebraic Approach

Notes from a Fall 2020 graduate course taught at UGA. (pdf) (141 pages)

## Commutative Algebra

In the course of teaching two graduate courses at UGA in 2008, I found the need to refresh and extend my knowledge of "basic" commutative algebra. So I started taking notes. In true epic fashion, although I orginally started with notes on properties of integral extensions (which explains the file name), this section now appears somewhere in the middle of a long set of notes.

Draft (
pdf) (385 pages)

## Complex Analysis

Notes from a Fall 2017 course taught at UGA. (
pdf) (87 pages)

## Elliptic Curves

• Lecture Notes on Elliptic Curves (pdf) (90 pages)

## Field Invariants

• On some elementary invariants of fields. (pdf) (15 pages)

These notes, mostly written after I attended the 2003 Arizona Winter School on model theory and arithmetic, give a sort of introduction to the model theory of fields (assuming, unfortunately, that you know some model theory and some arithmetic geometry and have somehow never managed to combine them!). The point of departure is the search for "invariants" of elementarily equivalent fields. Among other things, I point out that a standard conjecture relating period and index in the Brauer group would play the same role as the Milnor Conjecture did in defining the transcendence degree of an absolutely finitely generated field. But I picked a bad time to try an exposition on the model theory of fields: since this paper has been written, Scanlon has proven the equivalence of elementary equivalence and isomorphism for finitely generated fields of characteristic zero, and Poonen and Pop have presented much stronger definability results than the ones I discuss in Section 2. Perhaps someday I will update the exposition to include these exciting results.

## Field Theory

There is, I believe, a finite amount of "general field theory". Most graduate level algebra courses concentrate on the structure theory of algebraic field extensions, which is of course both beautiful and useful. However, many algebraists need to know more than this. Speaking as an arithmetic algebraic geometer, the structure theory of transcendental extensions -- and especially, the notion of a separable transcendental extension -- inevitably comes up, as do certain other constructions which don't seem to make it into the standard course: e.g. linear disjointness, composita, tensor products of fields, a systematic treatment of norms and traces.

The notes which follow aim to be a "serious" account of all aspects of general field theory. At the moment they cover about half of the material that they should, unfortunately for the most part the better known half. (Update: now about 70%!) Moreover what is present is quite rough: please consider it only a first draft.

Incomplete Draft (pdf) (167 pages)

## General Topology

• Notes accompanying lectures given for Math 4200/6200 (pdf) (291 pages)

## Geometry of Numbers

• Notes accompanying a 2011-2012 VIGRE Research Group (pdf) (159 pages)

## Homological Algebra

• Notes accompanying lectures given for Math 8030 (pdf) (110 pages)

## Honors Calculus / Real Analysis

• Notes accompanying lectures given for Math 2400/2410 ("Spivak Calculus") (pdf) (356 pages)

• 2022 Undergraduate real analysis (notes for Math 4100/6100) (pdf) (107 pages)

• 2023 Undergraduate real analysis (notes for Math 4100/6100) (pdf) (119 pages)

• A textbook-in-progress for Math 3200 (pdf) (175 pages)

## Linear Algebra

• Invariant Subspaces (pdf) (35 pages)

• Supplementary Linear Algebra Notes for Math 3000 (pdf) (97 pages)

## Model Theory

• Notes from a 2010 summer lecture series at UGA (pdf) (59 pages)

## Multivariable Calculus

• Handout 1: More on Dot Products and Cross Products (pdf) (4 pages)
• Handout 2: Kepler's Laws of Planetary Motion. (pdf) (5 pages)
• Handout 3: Geometry of Space Curves. (pdf) (3 pages)
• Handout 4: Differential Calculus on Surfaces. (pdf) (17 pages)
• Handout 5: Vector Fields. (pdf) (9 pages)
• Handout 6: Line Integrals. (pdf) (7 pages)
• Handout 7: Conservative Vector Fields and a Fundamental Theorem. (pdf) (9 pages)
• Handout 8: Green's Theorem. (pdf) (14 pages)
• Handout 9: The Change of Variables Formula. (pdf) (3 pages)
• Review Notes. (pdf) (12 pages)

Total: 83 pages

## Non-Associative Algebra

• Introduction to Nonassociative Algebras (Especially Composition Algebras) (pdf) (22 pages)

## Non-Commutative Algebra

• Notes from a 2011 summer lecture series given at UGA (pdf) (81 pages)

## Number Theory

• Introduction to Number Theory (notes from an under/graduate number theory course taught at UGA in 2007 and 2009) (pdf) (272 pages)

• Algebraic Number Theory I (notes from a graduate number theory course taught at UGA in 2022) (pdf) (116 pages)

• Algebraic Number Theory II: Valuations, Local Fields and Adeles (notes from a graduate number theory course taught at UGA in 2010) (pdf) (157 pages)

• Quadratic Forms: Witt's Theory. (pdf) (34 pages)

• Quadratic Forms: Structure of the Witt Ring. (pdf) (20 pages)

• Quadratic Forms Over Discrete Valuation Fields (pdf) (13 pages)

• Quadratic Forms Over Global Fields (pdf) (16 pages)

## Sequences and Series

• Sequences and Series: a Sourcebook. (pdf) (129 pages)

## Set Theory

• All the set theory I have ever needed to know. (40 pages total)
Most of the time, most of us don't need to know more about set theory than the distinction between finite, countably infinite, and uncountable sets. But once in a while it's nice to know a little bit more: e.g. the least uncountable ordinal comes up in topology, or your colleague asks you for a counterexample to the variant of Zorn's Lemma with "chain" replaced by "countable chain." But it is hard to find a treatment of set theory that goes a little beyond Halmos' Naive Set Theory or Kaplansky's Set Theory and Metric Spaces (both excellent texts) but that isn't off-puttingly foundational and/or axiomatic (i.e., that treats set theory as mathematics, rather than a confusing amalgamation of mathematics and meta-mathematics). For instance, the standard axioms only allow the elements of sets to themselves be sets (so that, e.g., mathematicians do not form a set) and forbid a set from containing itself, although neither of these options seem logically contradictory. But assuming that we are only interested in sets up to equivalence (i.e., bijection), it doesn't matter -- a trick of von Neumann allows us to put any set containing "individuals" and perhaps containing itself into bijection with a "pure" set that does not have either of these properties. I wish someone had told me this a long time ago! So, you guessed it, I am writing up my own notes.

Chapter 1: Finite, countable and uncountable sets. (pdf) (11 pages)
Chapter 2: Order and Arithmetic of Cardinalities. (pdf) (8 pages)
Chapter 3: Ordinalities and their arithmetic; von Neumann's ordinals and cardinals. (pdf) (18 pages)
Chapter 4: Cardinality Questions. (pdf) (3 pages)

## Shimura Curves

Notes from my 2005 ISM course: (106 pages total)
These are all the notes I typed up for my 2005 ISM course on Shimura varieties. In the lectures, I presented more material on Hilbert and Siegel modular varieties, adelic double coset constructions, and strong approximation than has survived in the lecture notes. Most of the omitted material is of a rather standard sort -- it appears in many places -- which is not to say that it shouldn't appear here as well. The reader will notice that the notes are significantly more polished at the beginning and the end than in the middle. I am quite pleased with the very last lecture, which seems to put some of the pieces of the theory together in a new way. I would like to see more detail on arithmetic groups and lots more detail on quaternion orders and trace formulas. Inevitably for notes of this length, the most important results -- like the existence of rational and integral canonical models -- get stated and kicked around a bit but not proved. To remedy this will require significantly more work.

• Lecture 0: Modular curves. (pdf) (6 pages)
• Lecture 1: Endomorphisms of elliptic curves. (pdf) (13 pages)
• Lecture 2: Fuchsian groups. (pdf) (18 pages)
• Lecture 3: More Fuchsian groups. (pdf) (6 pages)
• Lecture 4: Arithmetic Fuchsian groups. (pdf) (6 pages)
• Lecture 4.5: A Crash Course on Linear Algebraic Groups. (pdf) (7 pages)
• Lecture 5: The Adelic Perspective. (pdf) (4 pages)
• Lecture 6: Special points and canonical models. (pdf) (10 pages)
• Lecture 7: Real points. (pdf) (6 pages)
• Lecture 8: Quaternion orders. (pdf) (10 pages)
• Lecture 9: Quaternionic moduli. (pdf) (4 pages)
• Lecture 10: Integral structures, genera and class numbers. (pdf) (16 pages)
(I do not have any good explanation for the bizarre numbering. In actuality there were many more than 12 lectures, and there was nothing exceptional about the lecture I gave on linear algebraic groups, except that when I defined unipotent groups one of the attendees had the guts and honesty to ask, "What is the point of all this?" The point is that you need to know about a whole lot of different things to understand the definition of a Shimura variety!)

## Uniform Distribution

• Some (unpolished) notes on uniform distribution. (pdf) (20 pages)
These are notes on the basics of uniform distribution of sequences, taken on occasion of the Dover republication of the very nice book on this topic by Kuipers and Niederreiter. The notes are incomplete, not including a full-blown treatment of uniform distribution in a compact (or locally compact) group.

Total as of October, 2021: 3110 pages