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UID:078957e2cc2ff830acc81af80a18be90
CATEGORIES:Colloquia
CREATED:20221218T213619
SUMMARY:Arnold Conjecture Over Integers
LOCATION:Zoom Meeting; Contact a faculty member for link
DESCRIPTION:<p style="line-height: 1.284; text-align: justify; padding: 0pt 0pt 8pt;">A
 rnold's famous conjecture on the numbers of fixed points of Hamiltonian dif
 feomorphisms on symplectic manifolds is widely regarded as the major drivin
 g force for the emergence of the subject called “symplectic topology”; it a
 lso motivated numerous important developments in geometry and topology, mos
 t notably the invention of Floer homology. Recently Shaoyun Bai and I prove
 d a version of Arnold conjecture for all compact symplectic manifolds which
  is stronger than all previously proved versions. Our proof relies on a tec
 hnical breakthrough made in an earlier paper of ours based on the original 
 idea of Fukaya-Ono, which allows one to define integer-valued counts of pse
 udoholomorphic curves. This new technique applies not only to Floer theory 
 and Arnold conjecture, but also to other directions such as Gromov-Witten t
 heory, leading to new integer-valued curve counting invariants.</p><p style
 ="margin-bottom: 7px; line-height: 1.284; text-align: justify;">In this tal
 k I will review the background and history of Arnold conjecture, the protot
 ypical argument using Floer homology, and the subtlety about counting pseud
 oholomorphic curves with symmetries. Then I will talk about our technical b
 reakthrough about integral counts of pseudoholomorphic curves and applicati
 ons, including both the definition of integer-valued Gromov-Witten invarian
 ts and the proof of the integral version of Arnold conjecture. Some prospec
 tive works will also be sketched at the end of the talk. This talk is based
  on the joint works with Shaoyun Bai (arxiv: 2201.02688, 2209.08599).</p>
CONTACT:Guangbo Xu, Texas A&amp;M
DTSTAMP:20260827T053836
DTSTART;TZID=America/New_York:20221220T140000
DTEND;TZID=America/New_York:20221220T150000
SEQUENCE:0
TRANSP:OPAQUE
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