Практическая информация

C 30 июня по 4 июля 2025 в Центре Фундаментальной Математики МФТИ пройдет летняя школа "Фундаментальные аспекты геометрии и топологии". Курсы лекций прочитают Кристофер Брав, Борис Шойхет, Yingdi Qin и Анна Нордскова. Адрес проведения школы: Радиотехнический корпус МФТИ, Долгопрудный, Институтский пер. 9, стр. 1, аудитория 113.


Раписание

Время	Понедельник	Вторник	Среда	Четверг	Пятница
11:00 - 12:30	Шойхет	Шойхет	Шойхет	Qin	Qin
12:30 - 14:00	Обед
14:00 - 15:30	Брав	Брав	Брав	Нордскова	Нордскова
15:30 - 16:00	Чай
16:00 - 17:30	Нордскова	Нордскова	Qin	Шойхет	—

Аннотации курсов

Introduction to solid algebraic geometry

Chris Brav

In the first lecture, we review the Hopkins-Neeman theorem for a commutative Noetherian ring, which reconstructs the Zariski spectrum from the structure of the derived category as a tensor category, specifically in terms of idempotent algebra objects in the derived category. This motivates thinking of algebraic geometry in terms of the geometry of tensor categories. In the second lecture, we introduce the tensor category of solid modules over a commutative ring, in the sense of Clausen-Scholze, and begin analysing its structure in terms of certain idempotent algebra objects. In the final lecture, we review the theory of valuation rings and of the valuative spectrum of a ring, after Zariski and Huber, and explain a weak analogue of the Hopkins-Neeman theorem for solid modules in terms of the valuative spectrum.

Sato Grassmannians and condensed mathematics

Yingdi Qin

Condensed mathematics is a framework that aims to provide a more convenient way to treat algebraic objects equipped with topology, such as topological abelian groups or topological vector spaces. It has been developed to combine algebra and topology in a new way, by introducing the abelian category of 'condensed abelian groups', which contains (compactly generated) topological abelian groups as a full subcategory.

Once playing with the tensor products of condensed modules, one immediately realizes that we need the concept of 'complete' condensed modules and 'complete' tensor products to produce the desirable results of certain tensor products. For example, the desirable tensor product of function rings on two spaces are supposed to be the function rings on the product of the two spaces. We have several different versions of completeness in condensed mathematics. Among them are solid modules in the non-Archimedean case, and liquid/gaseous modules in the Archimedean case.

These lectures will introduce solid and liquid/gaseous modules in condensed mathematics. And talk about its applications in Sato-Segal-Wilson Grassmannians, which finds applications in integrable systems, moduli of curves/sheaves and geometric representation theory.

Вокруг теоремы Барратта-Придди-Квиллена

Борис Шойхет

Мы обсудим Гамма-пространства Сигала и его доказательство теоремы Барратта-Придди-Квиллена, а также групповое пополнение несвязных топологических моноидов и плюс-конструкцию Квиллена в алгебраической К-теории. Мы также обсудим взаимосвязи между этими вещами.

Non-commutative crepant resolutions

Анна Нордскова

Non-commutative crepant resolutions (NCCR) were introduced by Michel Van den Bergh over 20 years ago and have since found many applications and interconnections in various areas of algebraic geometry, representation theory, mirror symmetry and even string theory. In this course I will give a gentle introduction to this notion and try to motivate the definition by sketching some of the context in which it appeared (including the non-commutative algebra approach to Bridgeland’s proof of a conjecture due to Bondal and Orlov, tilting theory and McKay correspondence). We will consider various examples and discuss how NCCRs relate to classical (commutative) crepant resolutions as well as some other notions such as Kuznetsov’s categorical (weakly and strongly) crepant resolutions. In the last lecture I will talk about NCCRs of cones over del Pezzo surfaces. This part will contain some new results obtained jointly with Van den Bergh.