If you click on the links below, and if the classes were made public, you will find notes from previous classes.
Fall 2017 Math 122: Algebra I: Theory of Groups and Vector Spaces
Fall 2017 Math 221: Algebra
Fall 2016 Seminar on Derived Geometry and Floer Theory
Fall 2015 Math 277: Fukaya categories, sheaves, and cosheaves
Fall 2015 Math 230a: Differential Geometry
Fall 2014 Math 122: Algebra I: Theory of Groups and Vector Spaces
Fall 2014 Math 230a: Differential Geometry
Fall 2013 Math 231a: Algebraic Topology
Fall 2013 Math 280x: Bridgeland Stability
Fall 2013 Reading Seminar on oo-Categories
As I Take My Notes (Hiro's translation of the Kodaira quote. Original below.)
To me, there is nothing harder to read than a math book (papers included). To read through a math book of hundreds of pages, from beginning to end, is a Herculean task. When I open a math book, there are first axioms and definitions; then there are theorems and proofs. I know that mathematics is a thing which becomes incredibly easy and clear once you understand it, so I try to read only the theorems and somehow understand. I try to think of proofs on my own. Most of the time, I don't get it even after I think about it. Having no other choice, I try reading the proof in the book. But even after reading it once or twice, I still don't feel like I understand it. So I try copying the proof into my notebook. Then I notice a part of the proof I dislike. I try to think if there must be another proof. It's great if I find one right away, but otherwise it takes a long time until I give up. And if I go about in this way, after a month finally arriving at the end of the first chapter, I forget the content toward the beginning. Having nothing else I can do, I review the chapter from the start again. Then the entire structure of the chapter begins to bother me. I think things like, it seems better to take care of Theorem Seven before proving Theorem Three. So I create another notebook where I reorganize the whole chapter. I finally feel like I understand the first chapter, and I feel at peace, but it's troublesome that it took so terribly long. To get to the last chapter of a book with hundreds of pages is near impossible. I would very much appreciate it if somebody could teach me a quick way to read mathematical texts.
私にとって数学の本（論文も含めて）ほど読みにくいものはない。 数百ページもある数学の本をはじめから終わりまで読み通すことは至難の業である。 数学の本を開いてみると、まずいくつか定義と公理があって、それから定理と証明が書いてある。 数学というものは、わかってしまえば何でもない簡単で明瞭な事柄であるから、定理だけ読んで何とかわかろうと努力する。 証明を自分で考えてみる。 たいていの場合は考えてもわからない。 仕方がないから本に書いてある証明を読んでみる。 しかし一度や二度読んでもなかなかわかったような気がしない。 そこで証明をノートに写してみる。 すると今度は証明の気に入らないところが目につく。 もっと別な証明がありはしないかと考えてみる。 それがすぐに見つかればよいが、そうでないと諦めるまでにだいぶ時間がかかる。 こんな調子で一ヶ月もかかってやっと一章の終わりに達した頃には、初めの方を忘れてしまう。 仕方がないからまた初めから復習する。 そうすると今度は章全体の排列が気になり出す。 定理三よりも定理七を先に証明しておく方がよいのではないか、等と考える。 そこで章全体をまとめ直したノートを作る。 これでやっと第一章がわかった気がして安心するのであるが、それにしてもひどく時間がかかったので困る。 数百ページある本の終章に達するのは時間的にも不可能に近い。 何か数学書を早く読む方法があったら教えて貰いたいと思う。
A quote from Grothendieck
Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle -- while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects. In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of 30 or 35 years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birth-right, as it was mine: the capacity to be alone.
A quote from G.K. Chesterton
A man must love the thing very much if he not only practices it without any hope of fame or money, but even practices it without any hope of doing it well.
A quote from Maryam Mirzakhani (on her work with Eskin) (Source)
...Even Mirzakhani herself is amazed, in retrospect, that the two stuck with it. “If we knew things would be so complicated, I think we would have given up,” she said. Then she paused. “I don't know; actually, I don't know,” she said. “I don't give up easily.”
Here I've compiled a list of good sources for learning about various topics. It's often hard to find the right source to learn from; I hope this list will help you out.
Math Overflow Posts. Here are some Math Overflow posts I found interesting. They will probably give you a good idea of the kinds of things I think about.
- Derived Functors vs. Spectral Sequences
- Topologists' loops vs Algebraists' loops.
- Riemann-Hilbert at the dg level
- Spaces with no topological monoid structure which are homotopy equivalent to topological monoids.
- Algebras over the little disks operad
- Deligne's Conjecture about little 2-discs
- Kontsevich Formality via an exlicit homotopy
- What does actually being a CW complex provide in algebraic topology?
- Computing homotopy (co)limits in a nice simplicial model category?
- Transversality in the proof of the Blakers-Massey Theorem. Is it necessary?
- What is the intuition behind the Freudenthal suspension theorem
- Is there a high concept explanation for why "simplicial" leads to "homotopy-theoretic?"
- Lagrangian Submanifolds in Deformation Quantization
- Fourier Mukai for Fukaya Categories
More Traditional Resources, by topic:
- Mathematics and Rigor.
"Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics."An opinion piece by Arthur Jaffe and Frank Quinn.
Atiyah et al's response to "Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics."
- Basic Symplectic Geometry
Symplectic geometry : an introduction based on the seminar in Bern in 1992. B. Aebischer et al.
Dynamical Systems IV: Symplectic Geometry and its Applications. Arnol'd et al.
- Homological Algebra (Abelian categories, triangulated categories, derived categories, derived functors, cohomology.)
Chapter One of Kashiwara and Shapira, Sheaves on Manifolds
- Sheaves and Cohomology of Sheaves
Chapter Two of Kashiwara and Shapira, Sheaves on Manifolds
- Model Categories
Dwyer and Spalinski
- Perturbative Chern-Simons, Chord Diagrams, and Knot Invariants
Dror Bar-Natan's Thesis
- Mapping Class Groups
Farb and Margulis's in-progress book.
- Basic Algebraic Topology
- Differential Topology
Milnor, Topology from a Differentiable Viewpoint
Guillemin and Pollack (I've heard this is now out of print, but it is still a great book.)
- The Casson Invariant
Akbulut and McCarthy, Casson's Invariant for Oriented Homology Three-Spheres: An Exposition
- Morse Theory
Milnor, Morse Theory
The table below contains notes prepared by various Northwestern participants of 2012 Simons Center workshop on QFTs. This pre-seminar was organized by Chris Elliott. Oh, and here is a link to the Simons Center Workshop itself. I've also included links to the actual talk and notes from the workshop itself. The notes from the workshop were taken chiefly by Dmitri Pavlov, though some were provided by the speakers themselves.
|Chris Elliott, Classical Field Theories and Relativity||Notes|
|Michael Couch, Basics of Super-Symmetric Algebras|
|Yuan Shen, Super-Symmetric Actions||Notes|
|Jesse Wolfson. Spinors, Dirac Operators, K-Theory||Notes by Yuan, Notes by Jesse. Here we have video and notes from the workshop.|
|Kevin Costello. Twistor spaces in supersymmetry||Notes by Yuan|
|Ian Le. Representations of the Super-Poincare Group||Here are the video and notes from the workshop.|
|Chris Elliott. SUSY Lagrangians and Classical Theories, I.||Here are the video and notes from the workshop. Notes by Chris.|
|Michael Couch. SUSY Lagrangians and Classical Theories, II.||Here are the video and notes from the workshop.|
|Sam Gunningham. Twisting N=2 Super Yang-Mills||Here are the video and notes from the workshop.|
|Yuan Shen. Twisting N=2 Super Yang-Mills||Here are the video and notes from the workshop.|
|Philsang Yoo. Relativistic Quantum Field Theory|
|Philsang Yoo. Gauge Theory in two dimensions.||Here are the video and notes from the workshop.|
|Hiro Lee Tanaka. Supersymmetric Yang-Mills Theory||Here are the video and notes from the workshop. More notes.|
Below are some things I wrote up in graduate school for various purposes here and there. Be warned that these were written when I was still a student; some of these notes are still embarrassing.
- My notes on a talk I gave about the non-abelian Hodge theorem. For David Nadler's seminar.
- My notes on Lurie's take on spectra. Very brief.
- My Notes on the Casson Invariant, prepared for a talk given at NU for Josh's Rozansky-Witten Seminar.
- My Notes on BG as a stack, prepared for David Nadler's seminar.
- My Notes on quasicategories and topological categories, hand-written. This was in preparation for a talk in David Nadler's seminar.
- Part I and Part II of a talk I gave on model categories and homotopy categories. (Part I is notes taken by Alison Smith, and Part II are my own notes I prepared for the talk.)