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SUMMARY:Alexander Kuznetsov (Steklov Mathematical Institute)
DTSTART;VALUE=DATE-TIME:20201102T160000Z
DTEND;VALUE=DATE-TIME:20201102T170000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/1
DESCRIPTION:Title: Rationality and derived categories of some Fano threefo
lds over non-closed fields\nby Alexander Kuznetsov (Steklov Mathematical I
nstitute) as part of BIRS workshop: Derived\, Birational\, and Categorical
Algebraic Geometry\n\n\nAbstract\nIn a joint work with Yu.Prokhorov we es
tablished rationality criteria for geometrically rational Fano threefolds
over non-closed fields of characteristic zero such that their geometric P
icard number is one. I will report on similar results for Fano threefolds
whose geometric Picard number is higher but the Picard number over the bas
e field is one. I will also describe the derived categories of these varie
ties over the base field and discuss the relation between their structure
and rationality properties.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rina Anno (Kansas State University)
DTSTART;VALUE=DATE-TIME:20201102T170000Z
DTEND;VALUE=DATE-TIME:20201102T180000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/2
DESCRIPTION:Title: Generalized braid group actions\nby Rina Anno (Kansas S
tate University) as part of BIRS workshop: Derived\, Birational\, and Cate
gorical Algebraic Geometry\n\n\nAbstract\nConsider a diagrammatic category
whose objects are partitions of n and whose morphisms are braids with mul
tiplicities where strands are allowed to merge and come apart\, so topolog
ically such a braid is a trivalent graph with boundary. In addition\, we a
dd framing on edges with multiplicities greater than 1. The usual (type A)
braid group is then the group of automorphisms of (1\,1\,...\,1). We prov
e that any DG enhanceable triangulated category D with a braid group actio
n (of which there are numerous examples in algebraic geometry) can be comp
leted to a representation of this diagrammatic category. We do this by con
structing a monad over D that is best described as the nil Hecke algebra g
enerated by the generators of the braid group action\, and considering sui
table categories of modules over its "block subalgebras". If D=D(X)\, thos
e modules would be complexes of sheaves on X with additional data. Similar
structures have been known before\, but they satisfy stronger conditions
(i.e. the twist of framing on a multiple strand being a shift\, which in o
ur construction is not the case). This is joint work in progress with Timo
thy Logvinenko.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ludmil Katzarkov (University of Miami)
DTSTART;VALUE=DATE-TIME:20201102T193000Z
DTEND;VALUE=DATE-TIME:20201102T203000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/3
DESCRIPTION:Title: New Birational Invariants\nby Ludmil Katzarkov (Univers
ity of Miami) as part of BIRS workshop: Derived\, Birational\, and Categor
ical Algebraic Geometry\n\nAbstract: TBA\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ana Maria Castravet (University of Paris-Saclay\, Versailles)
DTSTART;VALUE=DATE-TIME:20201103T160000Z
DTEND;VALUE=DATE-TIME:20201103T170000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/4
DESCRIPTION:Title: Exceptional collections on moduli spaces of pointed sta
ble rational curves\nby Ana Maria Castravet (University of Paris-Saclay\,
Versailles) as part of BIRS workshop: Derived\, Birational\, and Categoric
al Algebraic Geometry\n\n\nAbstract\nI will report on joint work with Jeni
a Tevelev answering a question of Orlov. We prove that the Grothendieck-K
nudsen moduli spaces of pointed stable rational curves with n markings adm
it full\, exceptional collections which are invariant under the action of
the symmetric group $S_n$ permuting the markings. In particular\, a conseq
uence is that the K-group with integer coefficients is a permutation $S_n$
-lattice.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Wemyss (University of Glasgow)
DTSTART;VALUE=DATE-TIME:20201103T170000Z
DTEND;VALUE=DATE-TIME:20201103T180000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/5
DESCRIPTION:Title: Stability conditions via Tits cone intersections\nby Mi
chael Wemyss (University of Glasgow) as part of BIRS workshop: Derived\, B
irational\, and Categorical Algebraic Geometry\n\n\nAbstract\nI will expla
in that stability conditions for general Gorenstein terminal 3-fold flops
can be described as a covering map over something reasonable. Basically\,
part of the description comes from the movable cone\, and its image under
tensoring by line bundles. Alas\, there is more. This extra stuff is no
t immediately "birational" information\, and it is a bit mysterious\, but
it does have a very natural noncommutative interpretation\, with geometric
corollaries. In the process of this\, I'll describe some of the new hype
rplane arrangements that arise\, which visually are very beautiful. If ti
me allows\, I will also explain some applications to autoequivalences and
to curve counting. This is joint work with Yuki Hirano\, and with Osamu Iy
ama.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Izzet Coskun (UIC)
DTSTART;VALUE=DATE-TIME:20201103T193000Z
DTEND;VALUE=DATE-TIME:20201103T203000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/6
DESCRIPTION:Title: Brill-Noether Theorems for moduli spaces of sheaves on
surfaces\nby Izzet Coskun (UIC) as part of BIRS workshop: Derived\, Birati
onal\, and Categorical Algebraic Geometry\n\n\nAbstract\nIn this talk\, I
will discuss the problem of computing the cohomology of the general sheaf
in a moduli space of sheaves on a surface. I will concentrate on the case
of rational and K3 surfaces. The case of rational surfaces uses the stack
of prioritary sheaves and is joint work with Jack Huizenga. The case of K3
surfaces uses Bridgeland stability and is joint work with Howard Nuer and
Kota Yoshioka.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Rizzardo (University of Liverpool)
DTSTART;VALUE=DATE-TIME:20201104T160000Z
DTEND;VALUE=DATE-TIME:20201104T170000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/7
DESCRIPTION:by Alice Rizzardo (University of Liverpool) as part of BIRS wo
rkshop: Derived\, Birational\, and Categorical Algebraic Geometry\n\nAbstr
act: TBA\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Barbacovi (UCL)
DTSTART;VALUE=DATE-TIME:20201104T170000Z
DTEND;VALUE=DATE-TIME:20201104T180000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/8
DESCRIPTION:Title: A geometric presentation of the flop-flop autoequivalen
ce as a(n inverse) spherical twist\nby Federico Barbacovi (UCL) as part of
BIRS workshop: Derived\, Birational\, and Categorical Algebraic Geometry\
n\n\nAbstract\nThe homological interpretation of the Minimal Model Program
conjectures that flips should correspond to embeddings of derived categor
ies\, and flops to equivalences. Even if the conjecture doesn’t provide
us with a preferred functor\, there is an obvious choice: the pull-push vi
a the fibre product. When this approach work\, we obtain an interesting au
toequivalence of either side of the flop\, known as the “flop-flop autoe
quivalence”. Understanding the structure of this functor (e.g. does it s
plit as the composition of simpler functors?) is an interesting problem\,
and it has been extensively studied. In this talk I will explain that ther
e is a natural\, i.e. arising from the geometry\, way to realise the “fl
op-flop autoequivalence” as the inverse of a spherical twist\, and that
this presentation can help us shed light on the structure of the autoequiv
alence itself.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emanuele Macri (Université Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20201104T193000Z
DTEND;VALUE=DATE-TIME:20201104T203000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/9
DESCRIPTION:Title: Antisymplectic involutions on projective hyperkähler m
anifolds\nby Emanuele Macri (Université Paris-Saclay) as part of BIRS wor
kshop: Derived\, Birational\, and Categorical Algebraic Geometry\n\n\nAbst
ract\nAn involution of a projective hyperkähler manifold is called antisy
mplectic if it acts as (-1) on the space of global holomorphic 2-forms. I
will present joint work with Laure Flapan\, Kieran O'Grady\, and Giulia Sa
ccà on antisymplectic involutions associated to polarizations of degree 2
. We study the number of connected components of the fixed loci and their
geometry.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Booth (University of Antwerp)
DTSTART;VALUE=DATE-TIME:20201105T160000Z
DTEND;VALUE=DATE-TIME:20201105T170000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/10
DESCRIPTION:Title: Topological Hochschild cohomology for schemes\nby Matt
Booth (University of Antwerp) as part of BIRS workshop: Derived\, Biration
al\, and Categorical Algebraic Geometry\n\n\nAbstract\nHochschild cohomolo
gy behaves well over a field\, and its derived analogue Shukla cohomology
behaves well over any base commutative ring. Both are intimately related t
o deformation theory. To study `nonlinear' deformations (e.g. Z/p^2 over Z
/p)\, one wants to study Mac Lane cohomology\, which introduces nonadditiv
e features. Mac Lane cohomology ought to be the same thing as topological
Hochschild cohomology\; the analogue for homology is known by work of Pira
shvili and Waldhausen. I'll give a quick recap on topological Hochschild c
ohomology\, which is morally just Shukla cohomology with base `ring' the s
phere spectrum. I'll then give a definition of THH^* for schemes\, along w
ith some comparison theorems showing that for reasonable schemes\, any of
the `obvious' definitions that one might make all agree. I'll give some (e
asy!) computations of THH^* for P^1 and P^2 over a finite field.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Addington (University of Oregon)
DTSTART;VALUE=DATE-TIME:20201105T170000Z
DTEND;VALUE=DATE-TIME:20201105T180000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/11
DESCRIPTION:Title: A categorical sl_2 action on some moduli spaces of shea
ves\nby Nicolas Addington (University of Oregon) as part of BIRS workshop:
Derived\, Birational\, and Categorical Algebraic Geometry\n\n\nAbstract\n
We study certain sequences of moduli spaces of sheaves on K3 surfaces\, bu
ilding on work of Markman\, Yoshioka\, and Nakajima. We show that these s
equences can be given the structure of a geometric categorical sl_2 action
in the sense of Cautis\, Kamnitzer\, and Licata. As a corollary\, we get
an equivalence between derived categories of some moduli spaces that are
birational via stratified Mukai flops.\n\nI'll spend most of my time on a
nice example. This is joint with my student Ryan Takahashi.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Huizenga (Pennsylvania State University)
DTSTART;VALUE=DATE-TIME:20201105T193000Z
DTEND;VALUE=DATE-TIME:20201105T203000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/12
DESCRIPTION:Title: The cohomology of general tensor products of vector bun
dles on the projective plane\nby Jack Huizenga (Pennsylvania State Univers
ity) as part of BIRS workshop: Derived\, Birational\, and Categorical Alge
braic Geometry\n\n\nAbstract\nUsing recent advances in the Minimal Model P
rogram for moduli spaces of sheaves on the projective plane\, we compute t
he cohomology of the tensor product of general semistable bundles on the p
rojective plane. More precisely\, let V and W be two general stable bund
les\, and suppose the numerical invariants of W are sufficiently divisible
. We fully compute the cohomology of the tensor product of V and W. In pa
rticular\, we show that if W is exceptional\, then the tensor product of V
and W has at most one nonzero cohomology group determined by the slope an
d the Euler characteristic\, generalizing foundational results of Drézet\
, Göttsche and Hirschowitz. We also characterize when the tensor product
of V and W is globally generated. Crucially\, our computation is canonical
given the birational geometry of the moduli space\, providing a roadmap f
or tackling analogous problems on other surfaces. This is joint work with
Izzet Coskun and John Kopper.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Inbar Klang (Columbia University)
DTSTART;VALUE=DATE-TIME:20201106T170000Z
DTEND;VALUE=DATE-TIME:20201106T180000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/13
DESCRIPTION:Title: Hochschild homology for C_n -equivariant things\nby Inb
ar Klang (Columbia University) as part of BIRS workshop: Derived\, Biratio
nal\, and Categorical Algebraic Geometry\n\n\nAbstract\nAfter an overview
of Hochschild homology and topological \nHochschild homology\, I will talk
about about the twisted versions of \nthese that can be defined in the pr
esence of an action of a finite \ncyclic group. I will discuss joint work
with Adamyk\, Gerhardt\, Hess\, \nand Kong in which we develop a theoretic
al framework and computational \ntools for these twisted Hochschild homolo
gy theories.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sofia Tirabassi (Stockholm University)
DTSTART;VALUE=DATE-TIME:20201106T193000Z
DTEND;VALUE=DATE-TIME:20201106T203000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/14
DESCRIPTION:Title: The Brauer group of bielliptic surfaces\nby Sofia Tirab
assi (Stockholm University) as part of BIRS workshop: Derived\, Birational
\, and Categorical Algebraic Geometry\n\n\nAbstract\nWe study the behavior
of the pullback map between the Brauer group of a bielliptic surface and
that of its canonical cover. This is joint work with E. Ferrari and. Vodr
up.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barbara Bolognese (University of Sheffield)
DTSTART;VALUE=DATE-TIME:20201106T160000Z
DTEND;VALUE=DATE-TIME:20201106T170000Z
DTSTAMP;VALUE=DATE-TIME:20210124T144229Z
UID:BIRS_20w5176/15
DESCRIPTION:Title: A partial compactification of the stability manifold\nb
y Barbara Bolognese (University of Sheffield) as part of BIRS workshop: De
rived\, Birational\, and Categorical Algebraic Geometry\n\n\nAbstract\nBri
dgeland stability manifolds of Calabi-Yau categories are of noticeable int
erest both in mathematics and in physics. By looking at some of the known
example\, a pattern clearly emerges and gives a fairly precise description
of how they look like. In particular\, they all seem to have missing loci
\, which tend to correspond to degenerate stability conditions vanishing o
n spherical objects. Describing such missing strata is also interesting fr
om a mirror-symmetric perspective\, as they conjecturally parametrize inte
resting types of degenerations of complex structures. All the naive attemp
ts at constructing modular partial compactifications show how elusive and
subtle the problem in fact is: ideally\, the missing strata would correspo
nd to stability manifolds of quotient triangulated categories\, but establ
ishing such correspondence on geometric level and viewing stability condit
ions on quotients of the original triangulated category as suitable degene
rations of stability conditions is not straightforward. In this talk\, I w
ill present method to construct such partial compactifications if some add
itional hypoteses are satisfied\, by realizing our space of interest as a
suitable metric completion of the stability manifold.\n
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