Course description: The topic for this course is condensed mathematics. This term refers to a collection of techniques starting with an analogue of ordinary set theory in which the role of finite sets is instead played by compact Hausdorff topological spaces (notably including profinite sets). This can then be used to build up a form of commutative algebra for topological rings that smoothly adapts concepts from ordinary commutative algebra, where a more naive approach would run into the basic incompatibility between topology and certain algebraic operations. This proves to be particularly useful in analytic geometry in both its archimedean and nonarchimedean forms.
This course will be offered in a "pandemic-style" hybrid format. Lectures will be given in-person and streamed over Zoom. Course communication (including Zoom links) will take place in a Zulip discussion forum. Office hours will be offered in both in-person and remote formats.
Lecture recordings and detailed lecture notes will be made available publicly via this website. In addition, I will make the livestream and Zulip available for unofficial remote participation; please fill out this Google Form to get access. (Warning: the invite emails from Zulip often get caught in spam filters.)
Environment: I aim to create a conducive learning environment for all students, particularly those who do not see themselves reflected in the mathematical profession at present or have experienced systemic bias affecting their mathematical education. I insist that all participants do their part to maintain this environment. I also pledge all reasonable effort to address accessibility issues that come to my attention.
Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera].
Lectures: MWF 9-10:20am in B402A, plus a Zoom livestream. Note the disagreement with the officially posted lecture times; in practice we will meet roughly twice per week, but the schedule of lectures will be a bit irregular (see below). Recordings will be posted to this website. (Warning to remote viewers: the time zone in San Diego is UTC-7 through November 1, and thereafter UTC-8.)
Office hours: Each lecture will be followed by 30 minutes of hybrid office hours, in the same physical and Zoom rooms. Additional office hours will be announced over Zulip (timings of these will vary).
Homework: Detailed handwritten/typed notes on any one lecture or a short writeup of a topic not to be covered in lecture. In both cases, about 2-3 pages are expected in any legible format. Unofficial participants are welcome but not expected to contribute as well.
Final exam: None. Disregard any information from the UCSD Registrar to the contrary.
Grading: 100% homework.
Textbook: None. I will post lecture notes in HTML and PDF formats (generated using PreTeXt); these will be updated throughout the course.
Prerequisites: None; however, familiarity with algebra at the level of Math 200A-C (particularly homological algebra), and global and local fields at the level of Math 204A, is strongly recommended. At times, familiarity with algebraic geometry at the level of Math 203A-C may also be helpful. I will also make liberal use of standard point-set topology.
Topics by date (with videos): Note: the course meets only on the indicated dates. The distribution of topics across dates, and the correspondence to sections, will remain in flux until the course concludes. The videos will eventually all be available via a single playlist.