Гравитация в теории гладких гомотопических типов

В этом курсе мы погрузимся в важный аспект фундаментальной физики - теории гравитации и в связанные с ней проблемы. Почему гравитация была и остается проблемой на протяжении уже нескольких веков.

-Быстро сформулируем сеттинг теории бесконечность-топоcов, в котором записывается физика, сцепления (или когезии, cohesion), инфинитезимальность в терминах дифференциально-когезивных топосов.
-Дальше обсудим базу - классическую теорию поля в современной формулировке: предквантовые локальные теории поля, геометрии Картана и определение теорий гравитации. Примеры теории поля: Heterotic, type II, DualHeterotic, Chern-Simons 3d, 7d и др.
-Что такое процедура квантования. Примеры квантования простых механических систем, сигма-моделей и калибровочных теорий поля.
-Рассмотрим башню Уайтхеда ортогональной группы и поймем как из нее получается весь так называемый спектр М-теории.
Узнаем, что такое общековариантный контекст и как Эйнштейн пришел к своей общей теории относительности.
-Поймем, что гравитация и включение взаимодействия теорией поля с ней имеет место в старших геометриях Картана, за счет чего мы опишем множество других теорий гравитаций: супергравитация гетеротической теории струн в 11d, (Анти-) де Ситтеровская гравитация, гравитация Эйнштейна и др.
-Обсудим известный вопрос о «квантовании гравитации», и почему уже 100 проблема не дошла до завершения, несмотря на многие прорывы в виде М-теории и теории струн. Что произошло в последние пару лет в этом направлении, и почему наоборот имеет смысл смотреть на гравитацию как на что-то в себе, к чему не нужно применять известные процедуры квантования для согласования с квантовой теорией.
-В курсе лекций сделаем вклад в это направление.
Поставим и решим непростую задачу о поиске формализма для гравитации внутренне в дифференциальных сцепленных топосах. Это приведет к тому, что например уравнения Эйнштейна станут условием на равенство коциклов в дифференциальных когомологиях в слайс топосе над инфинитезимальной окрестностью многообразия (или стэка).
И на таком уровне абстракции станет лучше видно, что теории гравитации действительно особенны и требуют особого подхода.

In this course, we will dive into an important aspect of fundamental physics - the theory of gravity and the problems associated with it. Why gravity has been and continues to be a problem for several centuries now.

-We will quickly formulate the setting of infinity-topos theory, in which the fundamental physics internalises, cohesion, and infinitesimality in terms of differentially-cohesive topos.
-Further we will discuss the base, classical field theory in its modern formulation: pre-quantum local field theories, Cartan geometries, and the definition of gravitational theories. Examples of field theory: Heterotic, type II, DualHeterotic, Chern-Simons 3d, 7d, etc.
-What is a quantization procedure. Examples of quantization on simple mechanical systems, sigma-models and gauge field theories.
-We will examine the Whitehead tower of the orthogonal group and understand how the whole so-called spectrum of M-theory is derived from it.
We will learn what a general covariant context is and how Einstein arrived at his general theory of relativity.
-We will understand gravity and the inclusion of field theory interactions with it in Cartan's higher geometries, at the expense of which we will describe many other gravity theories: supergravity of heterotic string theory in 11d, (Anti)-de Sitter gravity, Einstein gravity, etc.

-We will discuss the well-known question about "quantization of gravitation", and why even after 100 years since foundation of General Relativity this problem has not yet come to completion, in spite of many breakthroughs in the form of M-theory and string theory. What has happened in the last couple of years in this direction, and why on the contrary it makes sense to look at gravity as something unique, to which it is not necessary to apply known quantization procedures to agree with quantum theory.
-In the course of lectures we will make a contribution to this direction.
We will put and solve a difficult problem about a search of a formalism for gravitation internally in differential coupled toposes. This will lead to the fact that, for example, Einstein's equations become a condition for equality of cocycles in differential cohomology in a slice topos over an infinitesimal neighborhood of a manifold (or stack).
And at such level of abstraction it will be better seen that theories of gravitation are really special and require a special approach.

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