kms 대한수학회

Quadranscentennial KIAS Lecture (Zoom)

Professor Maxim Kontsevich (IHES)

2021 September 14(Tue) 4-5 pm, Zoom Link: https://us02web.zoom.us/j/89897256260

      September 15(Wed) 4-5 pm, Zoom Link: https://us02web.zoom.us/j/83303259051

Title: "Morse-Novikov theory for holomorphic 1-forms"

Abstract: It is well known that a generic function on a compact smooth manifold has only Morse critical points (i.e. locally equivalent to a non-degenerate quadratic form). Counting gradient lines between critical points (saddle connections) one obtains a differential on so called Morse complex, giving a way to calculate cohomology of the underlying  manifold with integer coefficients.

About 40 years ago S.P.Novikov proposed a generalization of Morse theory when the function is replaced by a closed 1-form. 

  I will talk about new developments in Morse-Novikov theory  when the 1-form is real part of a holomorphic 1-form rescaled by a non-zero complex parameter t.

A remarkable feature of the holomorphic situation is that for a generic value of the argument of t there is no saddle connections at all, and one obtains a canonical basis in Morse-Novikov cohomology, represented by "infinite chains" which are typically everywhere dense.

a canonical change of the basis, giving a new elementary example of so-called Wall-Crossing structure (originally discovered in much more complicated theory of Donaldson-Thomas invariants). In concrete terms, one obtains a relation between some explicit rational matrix-valued functions in several variables, and some counting problems in dynamical systems. I will also review the connection of new theory to resurgent series, like e.g. Stirling formula for the asymptotic of Gamma function at infinity.