Mathematical Sciences: Operads, Representation Theory and Algebraic Geometry

Kapranov建议利用算子及其对偶性将稳定曲线的模空间的研究扩展到更复杂的稳定映射模空间和Fulton-McPherson紧化构形空间。他建议从一个新的角度来看待艾森斯坦级数和它们的函数方程，从而将两个以前没有联系的表示理论分支联系起来：自同构表示理论和仿射量子群理论。他还建议研究有限域上代数曲面上向量丛上的Hecke算子，以建立与几何朗兰兹对应的类比，这是曲线上向量丛理论中的一个基本原理。与V.Ginzburg合作，Kapranov建议研究二维局部域上的矩阵群的Hecke代数的自然类似，就像有限域上的两个变量的幂级数的域。此外，Kapranov建议将Drinfeld模的理论推广到任意维的代数变体，目的也是为了找到与朗兰兹对应的正确类比。他还建议将字符轴理论推广到涉及堆栈的$p$-进制域的情况。本研究涉及代数几何和表示论两个领域。代数几何是最古老的数学分支之一，最初研究由简单方程定义的平面几何图像，但在最近几十年得到了非常广泛的发展，不仅在数学中找到了应用，而且在粒子物理(微观自由度的几何)、汽车设计(利用代数曲面来开发新的美观形状)、机器人学、理论计算机科学等不同领域都有应用。特别是，有限域上的代数几何(本建议的大部分内容都致力于此)已在构造纠错码、最优网络以及与通信和信息传输有关的其他几个重要问题中得到了应用。

Kapranov proposes to extend the prior study of moduli spaces of stable curves by means of operads and their duality to more sophisticated moduli spaces of stable maps and Fulton-McPherson compactified configuration spaces. He proposes to relate two branches of representation theory which previously were not connected: theory of automorphic representations and theory of affine quantum groups, by looking at Eisenstein series and their functional equations from a new point of view. He also proposes to study Hecke operators on vector bundles on algebraic surfaces over finite fields, in order to establish an analog of the geometric Langlands correspondence, a fundamental principle in the theory of vector bundles on curves. Working with V. Ginzburg, Kapranov proposes to study natural analogs of Hecke algebras for matrix groups over 2-dimensional local fields like the field of power series in two variables over a finite field. In addition, Kapranov proposes to generalize the theory of Drinfeld modules to algebraic varieties of arbitrary dimension, again with the aim of finding the correct analog of the Langlands correspondence. He also proposes to look for a generalization of the theory of character sheaves to the case of $p$-adic fields, which would involve stacks. This research is in the fields of algebraic geometry and representation theory. Algebraic geometry is one of the oldest branches of mathematics which initially studied plane geometric images defined by simple equations but in last few decades has developed very extensively, finding applications not only inside mathematics but in such diverse fields as particle physics (geometry of microscopic degrees of freedom), car design (use of algebraic surfaces to develop new aesthetically appealing shapes), robotics, theoretical computer science and others. In particular, algebraic geometry over finite fields (to which a large part of this proposal is devoted) has found applications in constructing error-correcting codes, optimal net works and in several other important problems related to communications and information transmission.