|B類(講究) C類(実習)/Category B Category C
|Seminar on Geometry 1
Seminar on Geometry 2
Seminar on Geometry 3
Seminar on Geometry 4
Practical Class on Geometry 1
Practical Class on Geometry 2
Practical Class on Geometry 3
Practical Class on Geometry 4
|See English text.
|Scholze became an instant mathematical superstar with his 2011 thesis that introduced "Perfectoid Spaces." His theory has vastly expanded the reach of geometrical and analytical methods in p-adic geometry. Condensed mathematics holds the promise of accomplishing the same in a broader framework that, in addition to p-adic geometry, includes complex geometry and analysis. Such a unification is entirely new and may well supply the geometrical underpinning to make it possible to attack some of the most important conjectures in mathematics. The goal of the course is to study this new framework, as it is being developed.
|In many situations, topology and algebra do not interact well. For example, the category of topological abelian groups is not an abelian category. To wit, the identity map from the real numbers with the discrete topology to the real numbers with the usual topology does not have a kernel or a cokernel. Condensed mathematics is a replacement of the notion of a topological space that, on the one hand, does not loose any information, and, on the other hand, interacts as well with algebra as one could possibly hope. By definition, a condensed set is a sheaf of sets on the category of profinite sets with finite mutually surjective families of maps as coverings. A topological space X gives rise to a condensed set that to a profinite set S assigns the set of continuous maps from S to X. However, there are many more condensed sets, including the "missing" kernel and cokernel of the map above. Moreover, condensed sets form a topos, so condensed sets behave just like sets, except that the axiom of choice does not hold in general. So we can define condensed abelian groups, condensed rings, etc. as usual. The category of condensed abelian groups is an abelian category, so there are no problems doing homological algebra therein. In fact, among abelian categories, the category of condensed abelian groups is particularly easy to work with, since it has enough compact projective generators.
Some knowledge of homological algebra is necessary, and knowledge of some algebraic topology and algebra geometry would be helpful.
|Algebraic geometry, algebraic topology.
|Evaluation is based on lectures given in class and written notes.
|Clausen and Scholze gave twenty lectures about condensed mathematics at an online masterclass organized by the University of Copenhagen in November 2020. The lectures are available at
https://www.youtube.com/playlist?list=PLAMniZX5MiiLXPrD4mpZ-O9oiwhev-5Uq and notes from the lectures are available at https://www.math.ku.dk/english/calendar/events/condensed-mathematics
|The material is cutting-edge and somewhat demanding, so you should expect to spend a good amount of time on it.
|Please write email to ask questions.
|It is possible.
|Please contact me ahead of time to confirm level of background knowledge.
|Condensed mathematics, analytic geometry, six-functor formalism.
|Some familiarity with Lurie's theory of infinity-categories is helpful, since this is the modern way of doing homological algebra and derived algebraic geometry.
|Participants take turns to give lectures and prepare lecture notes. The notes should be helpful for all participants.