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KAIST 수리과학과 세미나 



[CMC 정오의 수학산책] AtiyahSinger index theorem 

The AtiyahSinger Index Theorem appeared in 1960’s, which is one of great mathematical achievements in 20th century. It is a far reaching generalization of the GaussBonnetChern Theorem for the Euler characteristic, the Hirzebruch Signature Theorem for the Signature of 4kdimensional compact manifold, the RiemannRochHirzebruch Theorem for the Arithmetic Genus. Hence to understand this magnificent theorem, we need to investigate how the Euler characteristic, Signature and Arithmetic Genus are expressed by the Fredholm Indices for some appropriate geometric operators. In this talk, I will explain briefly the historical background of the Index Theorem, the Fredholm Indices of elliptic operators and discuss how the Index Theorem was motivated from the above classical celebrated theorems. And then, I will go through very briefly the proof of the Index Theorem by using the heat kernel method. If time permits, I will explain the Index Theorem on a compact manifold with boundary, where the etainvariant appears as a boundary correction term. 







