Pete  L. Clark
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  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. After I reached 100 pages, I found it psychologically necessary to post what I already had, although what I have at the moment is clearly a very rough draft. One aspect of this is that I am continually moving the sections in an attempt to have the material appear in a linear logical order. This sometimes has the Bourbakistic effect of postponing some important and relatively easy concept until late in the game, e.g. principal ideal domains. There is also an aspect of wishful thinking to these notes in that some of the more difficult results are stated as yet without proof.

The reader should also be(a )ware of the fact that the following important topics are not covered at all in these notes: completions (and filtrations, graded rings,...), the Hilbert polynomial, depth, regular sequences, the Koszul complex, regular rings, differentials.

Draft (pdf) (269 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 58%!) Moreover what is present is quite rough: please consider it only a first draft.

    Early Draft (pdf) (86 pages)

      General Topology

    These notes are a work in very slow progress.

    Chapter I: Basics (16 pages)
    Section 1: From Metric Spaces to Topological Spaces (
    pdf)
    Section 2: The Notion of a Topological Space (pdf)
    Section 3: Alternative Characterizations of Topological Spaces (pdf)
    Section 4: The Lattice of Topologies on a Given Set (pdf)
    Section 5: (Neighborhood) Sub/bases (pdf)

    Chapter II: Convergence (pdf) (28 pages)

      Honors Calculus

  • Notes accompanying lectures given for Math 2400/2410 ("Spivak Calculus") (pdf) (175 pages)

      Local Fields

  • Introduction to Local Fields (written in Fall 1999, retexed with some comments in June 2009) (pdf) (14 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) (80 pages)

      Number Theory

  • Introduction to Number Theory (notes from an under/graduate number theory course taught at UGA in 2007 and 2009) (pdf) (262 pages)

      Quadratic Forms

  • Part 1: Quadratic Forms Chapter I: Witt's Theory. (pdf) (29 pages)

  • Part 2: Quadratic Forms Over Fields: Structure of the Witt ring. (pdf) (18 pages)

      Real Analysis II (Math 243, McGill, 2005) (Total: 94 pages)

    It was great fun teaching a semester of real analysis at McGill. I was initially surprised at how elementary the syllabus was -- one variable, no explicit mention of compactness, metric spaces, no Lebesgue integration -- but I ended up not missing any of that stuff much: it meant that almost every result was due to Cauchy, Riemann or Weierstrass. The course text was Russell Gordon's Real Analysis: A First Course, but I ended up writing extensive lecture notes. One nontraditional feature was that the homework problems are directly embedded in the notes -- not even necessarily at the end of each section. So the lecture notes really had to be read, and this worked out rather well. In fact, even the guy who gave me my only frowny-faced rating on you-know-what infamous website -- he called me arrogant and sneaky [sneaky, at least, rings false] -- followed by admitting that I made ``good concise course notes.'' Here they are with some comments.
    Final thoughts: The course was packed full, contentwise: I wouldn't want to cover much more material in a single semester. However, one might imagine getting to a bit more by having covered at least some of the material on series in the first semester; indeed, this seems more traditional. I did have time to say once or twice, ``No, not good enough; you need to go back over that material and I'll test you again later," but one should have the luxury of doing this in any undergraduate course. Also the homework was quite an integral part of the course, and the students found it very challenging: I myself led a weekly problem session, in which the vast majority of the solutions were presented by me. It was telling that all in all it was the top half of the class who showed up most consistently for office hours, including some of the very best students. I should also mention that the first semester was taught (not by me) commonly to a group of 60 students who then split up into two sections for the second semester: mine was the non-honors section (although I had a couple of ``ringers.'') If you attended both sections you would certainly be able to tell which was which -- the honors section covered things like metric spaces, and had trickier problems -- but I think that, although my course was easier, I was offering almost as much ``content.'' I say this in part as a warning (and partly to myself, at that): I worked the students long and hard; several of them said that the homework took at least 10 hours a week. In general, I found that Canadian students responded quite well to large work loads (which is not to say that this was the norm in all of their classes; sometimes the course syllabi struck me as old-fashioned and unambitious compared to what I was used to from American universities). I was frankly rather horrified at how little most of the students knew at the beginning of the course (e.g. I assumed that the first batch of convergence tests would be familiar from calculus, but this really seemed not to be the case) and I can honestly say that almost everyone improved steadily and significantly throughout. The final exam I gave was (truly) hard, and though they looked traumatized as they walked out most of them did quite well, in some cases turning in their best performance. All in all this was an outcome I will be trying to replicate the next time I teach such a course.

      Semigroups

    Introduction to Semigroups and Monoids. (pdf) (11 pages)

      Sequences and Series

  • Some (quite rough, in places) notes on infinite sequences and series. (pdf) (114 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

    (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 February, 2012: 1541 pages