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SUMMARY:Lothar Goettsche (https://www.ictp.it/phonebook/person?id=2630)
DTSTART:20220422T120000Z
DTEND:20220422T133000Z
DTSTAMP:20260404T110914Z
UID:ICTP-IGAP/1
DESCRIPTION:Title: Computation of vertical Vafa-Witten invariants\nby Lothar Goe
ttsche (https://www.ictp.it/phonebook/person?id=2630) as part of ICTP/IGAP
Algebraic Geometry Seminar\n\n\nAbstract\nLast time we introduced nested
Hilbert schemes and how one can express vertical Vafa-Witten as intersecti
on numbers on nested Hilbert schemes.\n\nAfter reviewing this we will the
sketch of proof of Laarakker's structure theorem for the vertical Vafa-Wi
tten invariants\, expressing them in terms of universal generating functio
ns and Seiberg-Witten invariants. The proof uses the cobordism invariants
of intersection\nnumbers on Hilbert schemes of points.\n\nThen we will sho
w how one can use this to explicitely determine the generating function fo
r the Vertical-Vafa-Witten invariants\, by first reducing to the case of t
oric surfaces and then localizing on Hilbert schemes of points on toric su
rfaces.\n\nIn the remaining two (2) lectures we will\n\n(1) finish explain
ing the ingrediends of this computation\, and present the formulas for th
e vertical-Vafa-Witten invariants\, in terms of modular forms\n\n(2) state
Mochizuki's formula for computing the horizontal Vafa-Witten invariants a
nd use it to compute horizontal Vafa-Witten invariants.\n\nPrevious Lectur
e held on 6th of April:\n\nTitle: Vertical Vafa-Witten invariants and nest
ed Hilbert schemes\n\nAbstract: We state the structure theorem of Laarakke
r for the vertical Vafa-Witten invariants of a projective surface S. We in
troduce nested Hilbert schemes (an incidence variety in products of Hilber
t schemes of points and curves on the surface S)\, and their relation to v
ertical components of Vafa-Witten moduli spaces.\n\nWe describe how the ve
rtical Vafa-Witten invariants can be computed in terms of nested Hilbert s
chemes. We sketch the proof of Laarakker's structure theorem.\n\nRegister
in advance for this meeting:\nhttps://zoom.us/meeting/register/tJEpfuCvpjg
uHdKtzcES0xzHegVySNQN0hC3 \nAfter registering\, you will receive a confirm
ation email containing information about joining the meeting.\n\nThis will
be a hybrid seminar. All are very welcome to join either online or in per
son (if provided with a green pass). Venue: Euler Lecture Room (ICTP Leona
rdo Da Vinci Building)\, for those wishing to attend in person.\n
LOCATION:https://stable.researchseminars.org/talk/ICTP-IGAP/1/
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BEGIN:VEVENT
SUMMARY:Lothar Goettsche (ICTP)
DTSTART:20220504T140000Z
DTEND:20220504T150000Z
DTSTAMP:20260404T110914Z
UID:ICTP-IGAP/2
DESCRIPTION:Title: Computation of the vertical Vafa-Witten invariants II: localizati
on on Hilbert schemes of points\nby Lothar Goettsche (ICTP) as part of
ICTP/IGAP Algebraic Geometry Seminar\n\n\nAbstract\nLast time we used nes
ted Hilbert schemes to express vertical Vafa-Witten invariants of a surfac
e S in terms of (unknown) universal power series\, which are generating fu
nctions for intersection numbers on Hilbert schemes of points on S.\n\nIn
this lecture we will reduce to the case that the surface S is toric\, and
then explain how to explicitly compute the generating functions via locali
zation. We will present the formulas obtained.\n\nNext time we will sketch
how to use Mochizuki's formula to do the same for the horizontal Vafa-Wit
ten invariants.\n\nPRESENCE: ICTP Leonardo Da Vinci Building\, Luigi Stasi
Lect. Room\n\nONLINE:\nRegister in advance for this meeting:\nhttps://zoo
m.us/meeting/register/tJEtd-GvqjorE9BtDAqiw4qFvW-xmtQQ0wOO \n\nAfter regis
tering\, you will receive a confirmation email containing information abou
t joining the meeting.\n
LOCATION:https://stable.researchseminars.org/talk/ICTP-IGAP/2/
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BEGIN:VEVENT
SUMMARY:Lothar Göttsche (ICTP)
DTSTART:20220518T143000Z
DTEND:20220518T153000Z
DTSTAMP:20260404T110914Z
UID:ICTP-IGAP/3
DESCRIPTION:Title: Mochizuki's formula and the horizontal Vafa-Witten invariants
\nby Lothar Göttsche (ICTP) as part of ICTP/IGAP Algebraic Geometry Semin
ar\n\n\nAbstract\nIn this lecture compute the horizontal Vafa-Witten invar
iants\, by using Mochizuki's formula\, which allows to reduce the computat
ion to Hilbert schemes of points.\n\nThen one can localization in a very s
imilar way as for the vertical Vafa-Witten invariants. This computation co
nfirms some of the predictions of S-duality in ranks 2 and 3.\n\nFurthermo
re\, combining with the computations of $S$-duality we predict the generat
ing functions for the horizontal Vafa-Witten invariants in rank at most 5.
\n\nRegister in advance for this meeting:\nhttps://zoom.us/meeting/registe
r/tJAldOuoqTkiEtDRPyfZcKmD4WXPD29fNcow \nAfter registering\, you will rece
ive a confirmation email containing information about joining the meeting.
\n\nThis will be a hybrid seminar. All are very welcome to join either onl
ine or in person.\nVenue: Luigi Stasi Lecture Room (ICTP Leonardo Da Vinci
Building)\, for those wishing to attend in person.\n
LOCATION:https://stable.researchseminars.org/talk/ICTP-IGAP/3/
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