WORKSHOP: Quadratic Forms, Algebraic Groups and Algebraic Cobordism, 26-30 August 2008

代数是一种通用的机制，它允许研究各种性质的数学对象，通过将它们转换成具有乘法或加法等形式运算的对象，从而提供一个可以用广泛的“代数”工具研究的结构。因此，在某种意义上，代数在所有数学中所扮演的角色与数学本身在自然科学中的角色相似。几何是数学的另一个古老分支，它有自己的方法和工具。它起源于对物体形状及其在空间中的位置的研究。通常，当一个数学分支的方法应用于另一个看似无关的分支时，就会取得重大进展。一个例子是代数几何的创建——一个强大的理论，允许通过几何手段研究代数对象，反之亦然。拓扑学或多或少地研究相同的几何对象，但以一种更灵活的方式，允许形状变形，以便只保留最基本的特征。拓扑学的思维方式强烈地影响了20世纪数学的许多分支。在此背景下，一个重要的最新发展是创建了一种称为动机拓扑的理论，它将拓扑的特征与代数几何的特征融合在一起。这一理论起源于2002年获得菲尔兹奖的Voevodsky的作品。会议的重点是两种类型的经典代数对象：二次型和代数群。近年来，由于几何方法的新应用，这些天体的理论经历了巨大的进步。来自动力拓扑学的新技术被证明是特别强大的，可以证明许多长期存在的猜想，并开辟了全新的研究方向。这种新方法的关键工具之一是一种称为代数共坐标的理论，它的灵感来自于拓扑中的经典理论。本次会议的目的是汇集二次型、代数群、代数协数和其他相关数学分支的一些顶尖专家，以展示这些理论的最新进展，并使这些不同背景的专家之间能够进行互动。早期职业研究人员也将有机会展示他们对这些理论的贡献，并从该领域一些最重要的专家的在场中受益。

Algebra is a general machinery which permits to study mathematical objects of various nature by translating them into objects with formal operations such as multiplication or addition, thus providing a structure that can be studied with a wide range of `algebraic' tools. In a certain sense, algebra therefore plays a similar role in all of mathematics as mathematics itself within the natural sciences. Geometry is another ancient branch of mathematics with its very own methods and tools. It originated in the study of the shapes of objects and their position in space. Often, major progress is achieved when methods from one branch of mathematics are applied to another seemingly unrelated branch. One example is the creation of algebraic geometry --- a powerful theory that allows to study algebraic objects via geometric means and vice versa. Topology studies more or less the same geometric objects but in a much more flexible way, allowing the deformation of shape so that only the most essential features remain. The topological way of thinking has strongly influenced many branches of 20th century mathematics. An important recent development in this context is the creation of a theory called motivic topology which fuses features of topology with those of algebraic geometry. This theory originated in the works of Voevodsky who was awarded the Fields Medal in 2002.The conference focuses on two types of classical algebraic objects: quadratic forms and algebraic groups. The theories of these objects have experienced dramatic progress in recent times due to the novel use of geometric methods. The new techniques from motivic topology turned out to be particularly powerful, allowing to prove many long-standing conjectures and opening up completely new directions of research. One of the crucial tools in this new approach is a theory called algebraic cobordism which draws its inspiration from classical theories in topology. The purpose of this conference is to bring together some of the leading experts from quadratic forms, algebraic groups, algebraic cobordism and other related branches of mathematics, in order to present the state of the art of these theories and to enable interaction between specialists with these different backgrounds. Early career researchers will also have an opportunity to present their contributions to these theories and to benefit from the presence of some of the foremost experts on the subject.