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SUMMARY:Fourier transform: from abelian schemes to Hitchin systems I - Jun
 liang Shen (Yale University)
DTSTART:20240520T130000Z
DTEND:20240520T140000Z
UID:TALK216748@talks.cam.ac.uk
DESCRIPTION:This is a series of 3 talks\, where we will focus on geometry 
 and topology of abelian fibrations --- these are maps whose general fibers
  are complex tori but special fibers may be highly singular and complicate
 d. The decomposition theorem of Beilinson\, Bernstein\, Deligne\, and Gabb
 er (BBDG) and the support theorem of Ng&ocirc\; provide powerful tools for
  studying these maps\; Corti-Hanamura further predicted that the sheaf-the
 oretic BBDG decomposition is governed by algebraic cycles. In recent years
 \, the study of Hitchin system predicts a list of surprising properties co
 ncerning the cohomological shadow of the BBDG decomposition theorem for th
 e Hitchin system and related geometries.\nIn my talks\, I will explain a g
 eometric tool\, a theory of Fourier transform\, which helps us to understa
 nd various questions and conjectures for abelian fibrations.&nbsp\;I will 
 start with the case of an abelian scheme (i.e. an abelian fibration withou
 t singular fiber)\, where the Fourier theory has been established by Beauv
 ille and Deninger-Murre more than 30 years ago. Then I will discuss the ca
 se with singular fibers. Our ultimate goal is to explain how to use the Fo
 urier transform to construct the desired algebraic cycles for Hitchin's in
 tegrable system as predicted by Corti-Hanamura. If time permits\, I will d
 iscuss further applications of the Fourier transform and the algebraic cyc
 les constructed from it\; this includes connections to the P=W conjecture\
 , \\chi-independence phenomanon etc. Based on joint work (in progress) wit
 h Davesh Maulik and Qizheng Yin.
LOCATION:Seminar Room 2\, Newton Institute
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