Cohomology theories for algebraic varieties

在V.Voevodsky的开创性工作之后，通过完全拓扑方法处理代数簇成为可能。在这一背景下，所谓的广义上同调理论起着重要的作用。这包括经典的代数K-理论，但也包括相当现代的(更普遍的)代数协边理论。对这类理论及其上同调运算的研究是一个引人入胜的课题。它在代数几何、二次型理论等领域的经典问题中有着广泛的应用。例如，人们可以提到：Rost度公式，代数循环的平滑问题，以及场的u-不变量。这是一个新的快速发展的领域，提供了许多有前途的研究方向。

After the ground breaking works of V. Voevodsky, it became possible to work with algebraic varieties by completely topological methods. An important role in this context is played by the so-called Generalized Cohomology Theories. This includes classical algebraic K-theory, but also a rather modern (and more universal) Algebraic Cobordism theory. The study of such theories and cohomological operations on them is a fascinating subject. It has many applications to the classical questions from algebraic geometry, quadratic form theory, and other areas. One can mention, for example: the Rost degree formula, the problem of smoothing algebraic cycles, and u-invariants of fields. This is a new and rapidly developing area that offers many promising directions of research.