 # WEEKLY ACTIVITIES

The coincidence Lefschetz Theorem
 Người báo cáo: Jean Paul BRASSELET (Institut de Mathématiques de Luminy)Thời gian: 9h sáng thứ 5, 23/2 Địa điểm: Phòng 4 nhà A14, Viện Toán học Tóm tắt:Â Given two maps \$f\$ and \$g\$ between compact oriented manifolds \$M\$ and \$N\$Â of the same dimension,Â the coincidence points are defined as points \$xin M\$ such that \$f(x)=g(x)\$. At these points,Â one defines the coincidence index (intersection number of the graphs).Â Lefschetz coincidence Theorem says that the sum of indices iis equal toÂ alternating sum of suitable matrixÂ  traces. The Lefschetz fixed point formula is just the caseÂ \$M=N\$ and \$g\$ is the identity map \$1_M\$ on \$M\$.Â The result has been generalized inÂ  the case of manifolds with different dimensions.Â M. Goresky and R. MacPherson have extendedÂ the Lefschetz fixed point theorem for singular varietiesÂ  in the context of intersection homology and with suitable hypothesis on the varieties and the maps.Â In this lecture, I will recall the main definitionsÂ and results concerning the Lefschetz coincidence indices and classesÂ in the case of manifolds with same and possibly different dimensions.Â In the case of possibly singular varieties, I will recall the situationÂ of the Goresky-MacPherson Lefschetz fixed point theorem for singular varieties.Â That leads to the Lefschetz coincidence Theorem in the case of singularÂ varieties. I will give examplesÂ  to illustrate the results.

### Highlights

 31/10/22, Conference:International school on algebraic geometry and algebraic groups 08/08/23, Conference:Đại hội Toán học Việt Nam lần thứ X