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SUMMARY:Resurgence and Partial Theta Series - David Sauzin (CNRS (Centre n
 ational de la recherche scientifique))
DTSTART:20221102T091000Z
DTEND:20221102T100000Z
UID:TALK185204@talks.cam.ac.uk
DESCRIPTION:\n\n\n\nFollowing hal.archives-ouvertes.fr/hal-03502404v3 I wi
 ll show how partial theta series\, i.e. functions of the form$\\Theta(\\ta
 u) := \\sum_{n>0} f(n) e^{i\\pi n^2 \\tau/M}$ with $f : Z \\to C$ an $M$-p
 eriodic function (or the product of a power of $n$ by such function)\, giv
 e rise to divergent asymptotic series at every rational point of the bound
 ary of their domain of definition $\\{Im\\tau>0\\}$. I will discuss the su
 mmability and resurgence properties of these series by means of explicit f
 ormulas for their formal Borel transforms\, and the consequences for the m
 odularity properties of $\\Theta$\, or its "quantum modularity'' propertie
 s in the sense of Zagier's recent theory. &Eacute\;calle's "Alien calculus
 " allows one to encode this phenomenon in a kind of "Bridge equation". Int
 eresting examples stem from the study of Gukov-Pei-Putrov-Vafa invariants 
 and Witten-Reshetikhin-Turaev invariants for the Poincar&eacute\; homology
  sphere (cf. [Gukov-Putrov-Marino\, arXiv:1605.07615]) or more generally S
 eifert homology 3-spheres ([Andersen-Misteg&aring\;rd\, J. Lond. Math. Soc
 . 2022]).\n\n\n\n
LOCATION:Seminar Room 1\, Newton Institute
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