Mathematical Sciences: The Geometry of Langlands Duality

主要项目：环格拉斯曼的几何及其在表示理论中的应用：(i)几何朗兰猜想和（ii）模表示。这里(i)是基于我们在几何上构造对偶约群的drinfeld项目的完成中所起的次要作用，以及（ii）基于我们观察到环Grassmannian是任意交换环上分裂约群代数表示局部化的几何设置。主要研究方向：(a)通过p-横向分层研究交同轴和特征旋回；(b)超kahler结构在表征理论中的作用。上述工作的长期目标是将抽象数学中某些高度发达的部分（几何和数论）与理论物理学家（粒子理论）最近发展的思想联系起来。这两种发展都有共同的数学语言表示理论和代数几何的某些部分。预期的统一将大大增加我们对这两个领域的理解。

Abstract Mirkovic The main project: geometry of the loop Grassmannian and applications to representation theory: (i) geometric Langlands conjectures and (ii) modular representations. Here (i) is based on our minor role in the completion of Drinfelds project of constructing the dual reductive group geometricaly, and (ii) on our observation that the loop Grassmannian is the geometric setting for the localization of algebraic representations of split reductive groups over arbitrary commutative rings. Related projects: (a) study of intersection homology sheaves and characteristic cycles via the p-transversal stratifications and (b) the role of the hyperkahler structure in representation theory. A long term goal of the above work is to relate certain highly developed parts of abstract mathematics (geometry and number theory) with the ideas developed recently by theoretical physicist (particle theory). Both of these developments share common mathematical language of representation theory and certain parts of algebraic geometry. The exdpected unification should dramaticaly increase our understanding of both fields.