学术报告（李冠宇 6.9）

Abstract:

Representations play a crucial role in modern algebra. Surprisingly, all representations can be parameterized by an affine scheme known as the representation variety. Despite being useful, representation varieties have certain inherent limitations. Firstly, these varieties tend to have singularities that make it challenging to understand their geometry. Secondly, when we apply them to take the fundamental group of a topological space, we lose higher-order information about the space. To address these issues, we introduce Derived Algebraic Geometry (DAG). DAG offers several reasons for its usefulness. Many classical objects, such as the cotangent complex and the intersection formula, find a more natural representation in the derived setting. Additionally, DAG can be seen as approximating the underlying space through a sequence of smooth spaces, providing additional tools for studying scheme geometry.In this talk, I will delve into the foundations of DAG, explaining how they can be used to derive the representation variety. The discussion will cover intriguing properties, selected computations, and potential applications if time permits.