- 행사명(국문): 고등과학원 개원25주년 기념강연I,II (Zoom)
- 행사명(영문): The Quadranscentennial KIAS Lecture I,II (Zoom)
- 행사 일자: 2021-09-14 ~ 2021-09-15
- 장소: Zoom 강연
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.
For countably many special values of the argument of t one obtains 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.