This is a repository of notes I’ve made that people might find useful. Nothing here is original – I have tried as much as possible to give due references. Comments are welcomed.

Motivic/Étale Homotopy Theory

• I gave a talk on a concrete example, due to Po Hu, of a Nisnevich square in homotopy theory that displays an exotic element of the Picard group.
• NOT SAFE FOR CONSUMPTION: this is an ongoing sequence of notes/outline/brain dump on Steenrod operations and the Milnor conjectures following Voevodsky, Rioü, Powell, Mazza-Voevodsky-Weibel and the guidance of Suslin. Comments hugely appreciated and I will say so when it is actually entirely safe for public consumption.
• I passed my qualifying exam on étale homotopy theory and Friedlander’s proof of the Adams conjecture and made extensive notes on related topics and apparatus (completion of spaces, homotopy theory of pro-spaces, base change theorems in étale cohomology, examples of étale homotopy types etc.) – the notes are quite messy.
• On Morel’s Stable Connectivity Theorem: this is an exposition of Morel’s proof of the stable connectivity theorem which, hopefully, contains accurate statements and proofs.

Algebraic Topology

• Here’s an outline of May’s proof of the recognition principle so you can look through the monograph in a geodesic fashion.
• I gave a talk on the Atiyah-Segal completion theorem (following Adams, Haeberly, Jackowski and May) way back when. It also has quick recollections on equivariant $K$-theory

Algebraic Geometry

• I gave a talk on étale cohomology emphasizing on how to compute the example of a curve. This was part of the pre-talbot seminar on motivic homotopy theory.
• Coming up: a note on Bridgeland Stability
• I have written up notes on some major basic theorems in algebraic geometry. These include:
1. The degree-genus formula (without mentioning cohomology).
2. Completed cohomology, formal functions and Zariski’s main theorem.
3. Following Görtz and Wedhorn, I wrote up a proof of Bezout’s theorem on curves

Topology of Varieties

1. How do you understand a morphism of $k$-schemes in terms of generic points?
2. Connectedness and connectivity in algebraic geometry (to come)

Higher Category Theory

• As part of WCATSS 2013, I gave a broad overview on how the language of $\infty$-categories creeps up naturally in homotopical algebra.

Places I’ve learned things from:

Boss-man:

JNKF

AA, AK, AM, BA, BG, BKBW, DC, DG, DW, GV, GW, HLT, JBJC, JH, JM, JK, MHoy, MHill, ML, MM, MXY, PAØPVK, QCH, SG, SK, TJF, TBJW ,YFZ

Mathoverflow: