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  • 고등과학원 개원25주년 기념강연I,II (Zoom) 안내
  • 작성자 관리자 등록일 2021-09-02 조회수 164
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    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

     

    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.

     

    For countably many special values of the argument of  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.

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