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SUMMARY:CURRENT TRENDS IN THE CATEGORICAL APPROACH TO ALGEBRAIC AND SYMPLE
CTIC GEOMETRY
DTSTART;VALUE=DATE-TIME:20230204T000000Z
DTEND;VALUE=DATE-TIME:20230206T080000Z
DTSTAMP;VALUE=DATE-TIME:20230603T071114Z
UID:indico-event-421@indico.ipmu.jp
DESCRIPTION:Dates: February 4 -6\, 2023\n\nVenue: Kavli IPMU\n\nSpeakers:\
n\nAgnieszka Bodzenta\nAlexey Bondal\nWill Donovan\nMikhail Kapranov\nTats
uki Kawagaki\nOkawa Shinnosuke\nYukinobu Toda\n\n \n\n\n\nSCHEDULE\nSatur
day 4 FEB 2023\nBalcony room A\, 4-th floor:\n10:00 - 11:00 Free discussio
n (optional)\nSeminar room A\, ground floor:\n14:00 - 15:00 Mikhail Kapran
ov (Kavli IPMU)\,\nPerverse sheaves on symmetric products and Ran spaces\n
15:30 - 16:30 Tatsuki Kuwagaki (Kyoto University)\,\nAn introduction to sh
eaf quantization\nSeminar room A\, Balcony 4-th floor:\n17:00 - 19:00 Free
discussion\n\n \n\nSunday 5 Feb 2023\nBalcony room A\, 4-th floor:\n10:0
0 - 11:00 Free discussion (optional)\nSeminar room A\, ground floor:\n14:0
0 - 15:00 Will Donovan (Tsinghua University)\,\nHomological comparison of
resolution and smoothing\n15:30 - 16:30 Shinnosuke Okawa (Osaka University
)\,\nBlowing down noncommutative cubic surfaces\nSeminar room A\, Balcony
4-th floor:\n17:00 - 19:00 Free discussion\n\n \n\n \n\nMonday 6 Feb 202
3\nBalcony room A\, 4-th floor:\n10:00 - 11:00 Free discussion (optional)\
nConference Hall\, ground floor:\n14:00 - 15:00 Agnieszka Bodzenta (Warsaw
University)\,\nReconstruction of a surface from reflexive sheaves\n15:30
- 16:30 Yukinobu Toda (Kali IPMU)\,\nQuasi BPS categories for C^3\nSeminar
room A\, Balcony 4-th floor:\n17:00 - 19:00 Free discussion\n\n \n\n\n\n
\n\n\n\nABSTRACTS\n\nAgnieszka Bodzenta (Warsaw University)\n\nReconstruct
ion of a surface from reflexive sheavesConsider the category D of normal s
urfaces and open embeddings with complement of codimension two. I will sho
w that any connected component of D contains a unique codim-2-saturated su
rface\, i.e. such X that any morphism in D with domain X is an isomorphism
. I will call such X the codim-2-saturated model of any surface X_0 in its
connected component.The category Ref of reflexive sheaves is constant on
a connected component of D. I will recover X from Ref(X_0). The main tool
in the reconstruction will be the right abelian envelope of Ref(X_0) with
its canonical exact structure. It will allow us to pass from Ref(X_0) to R
ef(U) for any open U in X_0. A characterization of quasi-affine surfaces i
n terms of their categories of reflexive sheaves will finally allow us to
define X as the colimit of spectra of centers of Ref(U) over all quasi-aff
ine open U in X. This is based on a joint work with A. Bondal.\n\n \n\nWi
ll Donovan (Tsinghua University)\n\nHomological comparison of resolution a
nd smoothing\n\nA singular space often comes equipped with (1) a resolutio
n\, given by a morphism from a smooth space\, and (2) a smoothing\, namely
a deformation with smooth generic fibre. I will discuss existing results
and work in progress on how these may be related homologically\, starting
with the threefold ordinary double point as a key example.\n\n \n\nMikhai
l Kapranov (Kali IPMU)\n\nPerverse sheaves on symmetric products and Ran s
paces\n\nI will discuss the results\, joint with V. Schechtman\, describin
g perverse sheaveson two closely related types of spaces: $Sym^n(C)$\, the
symmetric products of the complex line $C$ and the Ran space $Ran(C)$. Th
ey are given in terms of the formal theory of Hopf algebras of certain typ
e\, i.e.\, of the PROBs of universal (co)operations in such algebras. The
space $Ran(C)$ being infinite-dimensional\, the next step of incorporating
actual Hopf algebras (rather than the PROB of (co)operations) into our pi
cture requires extending the concept of perverse sheaves to include object
s not supported on any finite skeleton. Such an extension with good proper
ties is not clear at the present.\n\n\n\n\n\n \n\nTatsuki Kuwagaki (Kyoto
University)\n\nAn introduction to sheaf quantization\n\nAbstract: Sheaf q
uantization of Lagrangian submanifold plays important role in symplectic t
opology and algebraic analysis. In symplectic topology\, it is a model of
A-branes. In algebraic analysis\, it is an enrichment of constructible she
aves/perverse sheaves. In this talk\, I’d like to give an introduction t
o the recent development of the theory of sheaf quantizations.\n\n \n\n
\n\n\n\nShinnosuke Okawa (Osaka University) \n\nBlowing down noncommutativ
e cubic surfaces\n\nIn 2001 Van den Bergh established the notion of blowup
of noncommutative surfaces and proved that the blowup of a noncommutative
$\\mathbb{P}^2$ in $6$ points is a cubic surface of a noncommutative $\\m
athbb{P}^3$. In this talk we identify the morphism from the configuration
space of points to the moduli of noncommutative $\\mathbb{P}^3$s by invest
igating the (global) monodromy of lines on noncommutative cubic surfaces.
As a corollary we prove the converse to the aforementioned theorem under m
ild genericity conditions. If time permits\, we also explain how it was he
lpful to consider the corresponding Poisson geometry. This is a joint work
in progress with Ingalls\, Sierra\, and Van den Bergh.\n\n \n\nYukinobu
Toda (Kavli IPMU)\n\nQuasi BPS categories for $\\mathbb{C}^3$Abstract: \n\
nThe quasi BPS categories are some subcategories ofderived categories of d
erived commuting stacks\, and appear as semiorthogonal summands of categor
ies of matrix factorizations which categorify Donaldson-Thomas invariants
on $\\mathbb{C}^3$. In this talk\, I will discuss several properties of qu
asi BPS categories in relation to BPS cohomology in CoDT theory. Among oth
er things\, I will discuss a conjecture on 3-dimensional analogue of Bridg
eland-King-Reid-Haiman derived McKay correspondence for Hilbert schemes of
points on surfaces via quasi BPS categories. This is a part of my series
of joint works with Tudor Padurariu.\n\n\n\n \n\nOrganizers:Alexey Bondal
(Kavli IPMU\, Steklov Math Institute) \n\n \n\n \n\n \n\nSponsor/Cospo
nsor:\n\nKakenhi\nKavli IPMU\n\n \n\n\n\n \n \n \n\n \n\n\n \n\n\n
\n\n \n\n\n\n\n\nhttps://indico.ipmu.jp/event/421/
LOCATION:Media Hall in Kashiwa Library
URL:https://indico.ipmu.jp/event/421/
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