Title: Linear Algebra Done Wrong: A Naive introduction to Fourier-Mukai Transform
Abstract: At the beginning of the talk, I will recall the definition of Fourier transform in analysis and introduce the concept of "kernel construction". Then I will show by an example that "kernel construction" in linear algebra gives us the "wrong" definition of linear map between vector spaces. When we apply "kernel construction" in homological algebra, we obtain the desired Fourier-Mukai transform. Several examples of Fourier-Mukai transforms will be given to stress their importance. If time permits, an important application of Fourier-Mukai transform will be discussed, which is Gabriel's result: every smooth projective variety is determined by its category of coherent sheaves.