莫斯科国立大学力学数学系A.S.Mishchenko教授近期访问002全讯白菜网，在002全讯白菜网国际数学暑期学校授课，并于8月1号面向全校做如下学术报告：

报告题目：Noncommutative geometry and topology---a survey

报告时间： 8月1号，下午3:00

报告地点： 002全讯白菜网科学园（航天馆）2H楼201会议室

报告摘要：The Poincare duality, named after Henri Poincare, is a basic result on the structure of the homology and cohomology groups of manifolds.

A form of Poincare duality was first stated, without proof, by Henri Poincare in 1893. It was stated in terms of Betti numbers: the k-th and (n - k)-th Betti numbers of a closed (i.e. compact and without boundary) orientable n-manifold are equal.

H.Poincare has not presented neither accurate concept of the Betti numbers nor strict reasoning the validity of the Poincare duality. This required the creation of new concepts in works by E.Noether (1926), J. Alexander and A.N.Kolmogorov (1935) and many other mathematicians where the homology and cohomology groups  have been defined.

The modern statement of the Poincare duality says that if M is a closed oriented $n$-manifold, and k is an integer, then there is a canonically defined isomorphism from the k-th cohomology group H^k(M) to the (n - k)-th homology group H_{n - k}(M).

In the case of even dimensional manifold, say n=2k, the Poincare duality leads to an isomorphism H^k(M)/cong H_k(M) and to recovering a hidden invariants of the manifolds, namely the signature of the manifold, that turns out to be invariant with respect to orientable bordisms. Therefore the signature can be expressed in the terms of some characteristic classes by the Hirzebruch formula.

The theory of characteristic classes led to broad development of the differential topology, in particular to classification of smooth and combinatorial structures on manifolds, theory of non simply connected manifolds, noncommutative geometry and with their help to solve the major problems in topology.

报告人简介： Alexander S. Mishchenko 教授是国际著名的拓扑学家，莫斯科国立大学力学数学系资深教授，俄罗斯科学院Steklov数学研究所首席研究员，公司数学学科首席国际学术顾问，主要研究方向是拓扑学和非交换几何。Mishchenko教授早年师从Fields奖得主、俄罗斯科学院S. P. Novikov院士学习拓扑学，1965年毕业于莫斯科国立大学力学数学系，1968年获数学科学副博士学位，1973年获数学科学博士学位。他研究兴趣广泛，涉及代数拓扑、流形和复形的拓扑、表示论、算子代数、非交换几何等众多领域，是一位百科全书式的数学大师。迄今，Mishchenko教授已发表150余篇学术论文和专著，引用率均很高。现担任002全讯白菜网首席国际学术顾问,曾获得中国政府友谊奖。