Lunch Seminar on Condensed Mathematics

Condensed mathematics is an exciting new technology, due to Clausen-Scholze, for dealing with topologies on algebraic objects.

For example, objects such as the p-adic integers or the real numbers are classically considered as topological abelian groups. This category has some shortcomings: it is not abelian, making it difficult to do homological algebra with it. Condensed abelian groups are closely related to topological abelian groups, but circumvent these problems.

In this seminar, we want to learn the basics of this theory.

Talks

We will follow the notes here. Conveniently, the chapters correspond to actual lectures, so we will try to distribute each chapter as one talk.

• Talk I: Condensed Sets (Thomas Nikolaus). This talk introduces the basic idea of condensed objects. Notes
• Talk II: Condensed Abelian Groups (Konrad Bals). This talk discusses the basic structure of the category of condensed abelian groups, like being an abelian category.
• Talk III: Cohomology (Leon Hendrian). Since condensed objects are encoded as sheaves on a certain site, sheaf cohomology naturally appears. This behaves very interestingly, and some crucial examples are computed in this talk.
• Talk IV: Locally compact abelian groups (Achim Krause). A topological abelian group gives rise to a condensed abelian group. In this talk, this is discussed in the special case of locally compact abelian groups, where all possible Ext groups are computed.
• Talk V: Solid abelian groups (Mirko Stappert). Tensor products don't quite behave as desired in the world of condensed abelian groups. This talk introduces a certain localisation of this category, solid abelian groups, which fixes some of these problems.
• Talk VI: Solid abelian groups II (Jakob Scholbach). In this talk, we prove that the category of solid abelian groups is well-behaved.
• Talk VII: Analytic rings (Felix Janssen). The pair of a condensed ring together with a suitable notion of free modules over it is called analytic ring. This talk introduces this notion. The motivating example is the discrete ring of integers together with the free solid abelian groups, but other important examples are established.
• Talk VIII: Solid A-modules (Maximilian Tönies). The example of analytic ring given in the previous talk, given by the integers together with solid abelian groups, gets generalized to an arbitrary finitely generated base ring instead of the integers.
• Talk IX: Globalization (Stefano Ariotta). The notion of analytic rings gets related to a suitable theory of adic spaces.
• Talk X: Globalization II (Timo Richarz). This talk finishes the work started in the previous talk and establishes a good (derived) category of quasicoherent modules on adic spaces.
• Talk XI: Coherent duality (Jonas McCandless). This talk establishes a six-functor-formalism in the adic setting, and proves the analogue of Serre duality.

Format and Schedule

We will use zoom for talks by participants at

• Wednesday, 14-16,

starting in the week of April 22nd. There are multiple options to give a talk this way, including:

• Writing on a tablet or similar, while sharing the screen.
• Preparing slides or hand-written scanned notes ahead of time, and then sharing the screen while clicking through them and giving the talk.
• Using a webcam or phone, pointed at a sheet of paper on a desk, as a kind of physical whiteboard.

Of course, if you want to give a talk but are not sure what options might work for you, I am happy to help.

Interested participants please email me ahead of time, I will then distribute instructions on how to join the meetings.