北京雁栖湖应用数学研究院 北京雁栖湖应用数学研究院

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清华大学 "求真书院"
清华大学丘成桐数学科学中心
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上海数学与交叉学科研究院
BIMSA > Deformation theory and rational homotopy \(ICBS\)
Deformation theory and rational homotopy
In one sentence, the teaching of deformation theory is that, in characteristic zero, every equation describing deformation of some object can be rewritten as an equation $(d+a)^2=0$ with an appropriate understanding of what precisely $d$ and $a$ are. These are Maurer-Cartan equations in differential graded (or dg) Lie algebras. The other, seeminly unrelated reason to study dg Lie algebras is rational homotopy theory --- by works of Quillen and Sullivan, the homotopy category of (nilpotent and connected) dg Lie algebras is the so-called rational homotopy category, which is, roughly, category that knows everything about homotopy theory of topological spaces "modulo torsion". In this course we will study the homotopy category of dg Lie algebras with the goal to application to these two subjects --- deformations and rational homotopies, and with the emphasis on examples. At the culmination point of a course we are going to apply the techniques studied to an algebraic problem that is seemingly unrelated to homotopy theory at all --- we will prove a theorem that a nilpotent Lie algebra could be reconstructed from knowing its universal enveloping as just an associative algebra.
Professor Lars Aake Andersson
讲师
格里戈里·帕帕亚诺夫
日期
2025年03月04日 至 06月03日
位置
Weekday Time Venue Online ID Password
周二 17:05 - 18:40 A3-3-201 ZOOM 13 637 734 0280 BIMSA
周四 16:10 - 17:50 A3-3-201 ZOOM 13 637 734 0280 BIMSA
修课要求
Some algebraic topology --- homotopy groups and cohomology algebras; some differential geometry --- connections on manifolds, de Rham complex Laplace operators, complex structures; related algebra --- it would be good to know beforehand what a derived functor is, but not necessary.
课程大纲
1. Example problems of deformation theory. Deformations of differential in a complex, associative algebras, Lie algebras, flat connections. Maurer-Cartan equation.
2. Homotopy theory of dg-Lie algebras. L-infinity algebras and homotopy transfer theorem.
3. Analysis of homotopy transfer formulas. Kuranishi space.
3. Deformations of complex manifolds. Kodaira-Spencer algebra. Formality.
4. Superconnections. Deformations of coherent sheaves on smooth complex manifolds.
5. Period maps. Bogomolov-Tian-Todorov theorem. Artamkin-Mukai theorem.
5. Deformations of commutative algebras. Hochschild and Harisson cohomology. Euler idempotents. Hochschild-Kostant-Rosenberg theorem.
6. Koszul duality. Sullivan and Quillen models for rational homotopy types. Reconstruction theorem for nilpotent universal enveloping algebras.
7. Curved dg-Lie algebras. Fedosov quantization.
8*. If time permits: Kontsevich deformation quantization.
9*. If time permits: Descent of deformation functors.
参考资料
M. Manetti "Lectures on deformations of complex manifolds"
M. Kontsevich, Y. Soibelman "Deformation theory I"
K. Fukaya "Deformation theory, homological algebra and mirror symmetry"
Y. Felix, S. Halperin, J.-C. Thomas "Rational Homotopy Theory"
听众
Advanced Undergraduate , Graduate , 博士后 , Researcher
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
I am mostly interested in the applications of homological algebra to the problems of geometry, in the broad sense.
北京雁栖湖应用数学研究院
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No. 544, Hefangkou Village Huaibei Town, Huairou District Beijing 101408

北京市怀柔区 河防口村544号
北京雁栖湖应用数学研究院 101408

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