Infinite-dimensional Lie algebras (such as Kac-Moody, Virasoro etc.) govern, in many ways, various moduli spaces associated to algebraic curves. To pass from curves to higher-dimensional varieties, it is necessary to work in the setup of derived geometry. This is because many feature of the classical theory seem to disappear in higher dimensions but can be recovered in the derived (cohomological) framework. The lectures consist of 3 parts:

1. (1)

   Review of derived geometry and of the phenomenon of “recovery of missing features”.

2. (2)

   The derived analog of the field of Laurent series in n variables (“with poles at a single point”). The corresponding higher current algebras and their relation to derived moduli spaces of G-bundles (based on joint work with G. Faonte and B. Hennion).

3. (3)

   Derived Lie algebras of vector fields, their central extensions and cohomology. Role of factorization algebras in studying such cohomology (based on joint with B. Hennion and work in progress with B. Hennion and A. Khoroshkin).

无限维李代数（例如Kac-Moody，Virasoro等）以多种方式控制与代数曲线相关的各种模空间。要从曲线传递到更高维的变体，必须在派生几何的设置中进行工作。这是因为古典理论的许多特征似乎在更高维度上消失了，但可以在派生的（同调）框架中得到恢复。讲座分为三部分：

1. （1）

   审查派生的几何形状和“丢失特征的恢复”现象。

2. （2）

   Laurent级数场在n个变量中的派生类比（“极点在单个点”）。相应的更高电流代数及其与G束的导出模空间的关系（基于与G. Faonte和B. Hennion的共同研究）。

3. （3）

   向量场的李式代数，它们的中心扩展和同调。分解代数在研究此类同调性中的作用（基于与B. Hennion的联合以及B. Hennion和A. Khoroshkin的研究工作）。