Programs and Abstracts

Programs and Abstracts
Topology of torus actions and applications
to geometry and combinatorics
August 7 – 11, 2014, Daejeon, Korea
Organizers
Scientific Committee
• Anthony Bahri (Rider University, U.S.A)
• Victor Buchstaber (Steklov Institute of Mathematics, Russia)
• Megumi Harada (McMaster University, Canada)
• JongHae Keum (KIAS, Korea)
• Zhi Lu (Fudan University, China)
• Mikiya Masuda (Osaka City University, Japan)
• Taras Panov (Moscow University, Russia)
• Nigel Ray (Manchaster University, U.K.)
• Dong Youp Suh (KAIST, Korea)
Local Committee
• Suyoung Choi (Ajou University, Korea)
• Tomoo Matsumura (KAIST, Korea)
• Seonjeong Park (NIMS, Korea)
• Dong Youp Suh (KAIST, Korea)
1
Program
Date
August 7 (Thu) – 11 (Mon), 2014
Venue
Daejeon Convention Center(DCC), Daejeon, Korea
Official webpage
http://toric.kaist.ac.kr
Scope of the conference
The main topic of the conference is the interplay of torus actions with topology, geometry, and the combinatorics of manifolds, orbifolds, and algebraic
varieties with torus actions. Toric manifolds, symplectic toric manifolds,
and quasitoric manifolds illustrate the correspondence between topological
and geometric objects with torus actions and, combinatorial objects such as
moment polytopes or fans. Moreover, equivariant cohomology theory and
homotopy theory play important roles in the study of such topics. The rank
of the acting torus is half of the dimension of the space mentioned above,
but in this conference the focus is on torus actions of arbitrary rank, and on
the correspondence of such spaces with combinatorial objects. Also covered,
are geometric objects without torus actions, on which can be associated
combinatorial structures such as Okounkov bodies. All those working in
the fields of algebaric topology, transformation group theory, toric geometry, symplectic geometry, and combinatorics are encouraged to participate
in the conference.
This conference is a satellite conference of ICM 2014, Seoul. One can
find the list of all satellite conferences at
http://www.icm2014.org/en/program/satellite/satellites.
2
Speakers
Invited speakers
• Anthony Bahri (Rider University, U.S.A)
• Frédéric Bosio (Universite de Poitiers, France)
• Suyoung Choi (Ajou University, Korea)
• Alastair Darby (University of Manchester, U.K.)
• Matthias Franz (University of Western Ontario, Canada)
• Megumi Harada (McMaster University, Canada)
• Hiroaki Ishida (RIMS, Japan)
• Kiumars Kaveh (University of Pittsburgh, U.S.A.)
• Valentina Kiritchenko (National Research University “Higher School
of Economics”, Russia)
• Amalendu Krishna (Tata Institute, India)
• Andrei Kustarev (Moscow State University, Russia)
• Bernd Sturmfels (UC Berkeley, U.S.A., and KAIST, Korea)
• Alexandru Suciu (Northeastern University, U.S.A.)
• Tatsuru Takakura (Chuo University, Japan)
• Svetlana Terzić (University of Montenegro, Montenegro)
• Misha Verbitsky (National Research University “Higher School of Economics”, Russia)
• Li Yu (Nanjng University, China)
3
Contributed speakers
• Hiraku Abe (OCAMI, Japan)
• Anton Ayzenberg (Osaka City University, Japan)
• Yumi Boote (University of Manchester, U.K.)
• Li Cai (Kyushu University, Japan)
• Graham Denham (University of Western Ontario, Canada)
• Yury Eliyashev (Higher School of Economics, Moscow, Russia)
• Nikolay Erokhovets (Lomonosov Moscow State University, Russia)
• Hajime Fujita (Japan Women’s University, Japan)
• Saibal Ganguli (Institute of Mathematical Sciences, India)
• Miho Hatanaka (Osaka City University, Japan)
• Tatsuya Horiguchi (Osaka City University, Japan)
• Thomas Huettemann (Queen’s University, Canada)
• Taekgyu Hwang (KIAS, Korea)
• Shintaro Kuroki (University of Tokyo, Japan)
• Eunjeong Lee (KAIST, Korea)
• Changzheng Li (Kavli IPMU, Japan)
• Ivan Limonchenko (Moscow State University, Russia)
• Tomoo Matsumura (KAIST, Korea)
• Hanchul Park (KIAS, Korea)
• Seonjeong Park (NIMS, Korea)
• Andrés Pedroza (Universidad de Colima, Mexico)
• Soumen Sarkar (Univeristy of Regina, Canada)
• Christopher Seaton (Rhodes College, U.S.A.)
• Evgeny Smirnov (Higher School of Economics, Moscow, Russia)
• Jongbaek Song (KAIST, Korea)
• Yusuke Suyama (Osaka City University, Japan)
4
• Jihyeon Jessie Yang (McMaster University, Canada)
• Takahiko Yoshida (Meiji University, Japan)
• Qibing Zheng (Nankai University, China)
5
Time table
6
August 7 (Thursday)
9:00 - 9:40
opening
9:50 – 10:30
Anthony Bahri
Recent results about polyhedral products and toric spaces
13
Coffee break
11:00 – 11:30
11:40 – 12:10
Yury Eliyashev
Complex geometry of moment-angle manifolds
38
Tatsuya Horiguchi
The equivariant cohomology rings of Peterson varieties in all
Lie types
44
Li Cai
A family of polytopal moment-angle manifolds
36
Changzheng Li
On equivariant Pieri rule of isotropic Grassmannians
49
Lunch break
14:00 – 14:40
Matthias Franz
Big polygon spaces
17
14:50 – 15:30
Alexandru Suciu
Combinatorial covers, abelian duality, and propagation of
resonance
26
Coffee break
16:00 – 16:30
16:40 – 17:10
Miho Hatanaka
Spin toric manifolds associated to graphs
43
Evgeny Smirnov
Schubert polynomials and pipe dreams
57
Seonjeong Park
Betti numbers of toric origami manifolds
53
Tomoo Matsumura
Equivariant Giambelli formula for isotropic flag varieties
51
7
August 8 (Friday)
9:00 - 9:40
Hiroaki Ishida
Complex manifolds with maximal torus actions
19
9:50 – 10:30
Valentina Kiritchenko
Demazure operators and geometric mitosis
21
Coffee break
11:00 – 11:30
11:40 – 12:10
Soumen Sarkar
Explicit triangulation of complex projective spaces
55
Jongbaek Song
The integral cohomology ring of toric orbifolds and weighted
projective towers
58
Thomas Huettemann
Toric vatieties and finite domination of chain complexes
45
Yumi Boote
The cohomology ring structure of the symmetric square of
HPn
35
Lunch break
14:50 – 15:30
Tatsuru Takakura
Vector partition functions and the topology of multiple
weight varieties
27
Coffee break
16:00 – 16:30
16:40 – 17:10
17:30 – 18:10
Nikolay Erokhovets
Simple polytopes and simplicial complexes with Buchstaber
number 2
39
Saibal Ganguli
McKay correspondence in quasitoric orbifolds
42
Ivan Limonchenko
Combinatorics of simple polytopes, their Stanley-Reisner
rings and moment-angle manifolds
50
Jihyeon Jessie Yang
Okounkov bodies, toric degenerations, and Bott-Samelson
varieties
60
Bernd Sturmfels
The convex hull of a space curve
25
Conference Dinner
August 9 (Saturday)
9:00 - 9:40
Megumi Harada
Newton-Okounkov bodies, symplectic geometry, and representation theory
18
9:50 – 10:30
Kiumars Kaveh
Integrable systems,
Okounkov bodies
20
toric degenerations and Newton-
Intermission
11:00 – 11:30
Yusuke Suyama
Examples of toric manifolds whose orbit spaces by the compact torus are not simple polytope
59
Eunjeong Lee
Grossberg-Karshon twisted cubes and hesitant walk avoidance
48
Lunch break
Excursion
9
August 10 (Sunday)
9:00 - 9:40
Misha Verbitsky
Complex subvarieties in homogeneous complex manifolds
29
9:50 – 10:30
Frédérick Bosio
On diffeomorphic moment-angle manifolds
14
Coffee break
11:00 – 11:30
11:40 – 12:10
Hiraku Abe
On the toric manifolds arising from the root systems of type
B and C
33
Takahiko Yoshida
Torus fibrations and localization of index
61
Qibing Zheng
The cohomology algebra of polyhedral product spaces
62
Hajime Fujita
On well-definedness of the local index
41
Lunch break
14:00 – 14:40
Amalendu Krishna
On contractivility of Koras-Russell threefolds
22
14:50 – 15:30
Andrey Kustarev
Monomial equivariant emveddings of quasitoric manifolds
and the problems of existence of invariant almost complex
structures
23
Coffee break
16:00 – 16:30
16:40 – 17:10
Anton Ayzenberg
Homology of manifolds with locally standrad torus actions
34
Andrés Pedroza
Hamiltonian loops on symplectic blow ups
54
Shintaro Kuroki
Complexity one GKM graph with symmetries and an obstruction to be a torus graph
47
Taekgyu Hwang
Symplectic capacities from Hamiltonian circle actions
46
August 11 (Monday)
9:00 - 9:40
Alastair Darby
Geometric T k -equivariant complex bordism
16
9:50 – 10:30
Svjetlana Terzić
The theory of (2n, k)-manifolds
28
Coffee break
11:00 – 11:30
Graham Denham
Elliptic braid groups are duality groups
37
11:40 – 12:10
Hanchul Park
Torsions of cohomology of real toric manifolds
52
Christopher Seaton
Orbifold and non-orbifold linear symplectic quotients
56
Lunch break
14:00 – 14:40
Li Yu
On a class of quotient spaces of moment-angle complexes
30
14:50 – 15:30
Suyoung Choi
On the classification of toric varieties
15
Coffee break
11
Invited talks
12
Recent results about polyhedral products
and toric spaces
Anthony Bahri
A report on the work of Ali Al-Raisi and on joint work with Martin Bendersky, Fred Cohen and Sam Gitler. The topics to be discussed will include
the equivariance properties of the stable splitting of polyhedral products,
the quotients or certain real moment-angle manifolds by the action of cyclic
groups and the cohomology of polyhedral products.
If time allows I shall try to say something about the cohomology of the
free loop spaces of certain toric spaces.
Name : Anthony Bahri
Address : Rider university
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 9:50 – 10:30. Room 105
13
On diffeomorphic moment-angle manifolds
Frédérick Bosio
The geometry of a moment-angle manifold (MAM) over a polytope is
determined by the combinatorics of this last one, but two different polytope combinatorics may produce the same geometry of MAMs. We present
here some examples and counterexamples of this phenomenon, with different
reinforcements of the notion of diffeomorphism between MAMs.
Name : Frédérick Bosio
Address : Université de Poitiers
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 9:50 – 10:30. Room 105
14
On the classification of toric varieties
Suyoung Choi
A fundamental result of toric geometry is that there is a bijection between toric varieties and fans. More generally, it is known that some class of
manifolds having well-behaved torus actions, called topological toric manifolds M 2 n, whose second Betti numbers are m − n can be classified in terms
of combinatorial data containing simplicial complexes K with m vertices.
We remark that topological toric manifolds are a generalization of smooth
toric varieties. The number m − n is known as the Picard number when
M 2n is a compact smooth toric variety, say a toric manifold.
The classification problems for toric manifolds are important and interesting. As varieties, Kleinschmidt and Batyrev have classified toric manifolds of Picard number 2 and 3, respectively. Recently, H. Park and I have
attempted to classify them in different ways which use the relationship between the topological toric manifolds over a simplicial complex K and those
over the complex obtained by simplicial wedge operations from K.
In this talk, we discuss about the classification of toric manifolds up to
Picard number 4. Furthermore, we also discuss about the classification of
toric objects including topological toric manifolds and real topological toric
manifolds up to m − n = 4.
This is a joint work with Hanchul Park.
Name : Suyoung Choi
Address : Department of mathematics, Ajou University, San 5, Woncheondong, Yeongtong-gu, Suwon, 443-749, Rep. of Korea
e-mail : [email protected]
Talk schedule : August 11 (Mon), 2014. 14:50 – 15:30. Room 105
15
Geometric T k -Equivariant Complex Bordism
Alastair Darby
We begin by studying manifolds M 2n with a smooth effective T k -action
and a stably complex T k -structure, where k ≤ n. We use a variant of
GKM/torus graphs to encode certain properties of these manifolds. These
graphs, considered as combinatorial objects in their own right, are then
classified using a boundary operator on a class of exterior polynomials. The
manifolds in question represent equivalence classes of the geometric T k U :T k of manifolds M 2n with a smooth
equivariant complex bordism groups Z2n
effective T k -action. We then show, using equivariant K-theory characteristic
numbers, that the graphs and their related exterior polynomials encode
exactly the normal data around the fixed point set of the manifold. This
allows us to give some structure to the equivariant bordism groups.
Name : Alastair Darby
Address : University of Manchester
e-mail : [email protected]
Talk schedule : August 11 (Mon), 2014. 9:00 – 9:40. Room 105
16
Big polygon spaces
Matthias Franz
Let T = (S 1 )r be a torus. We present a new class of compact orientable T -manifolds, called “big polygon spaces” [2]. They generalize a
“mutant” previously constructed by Franz–Puppe [3] and are related to
polygon spaces, which appear as their fixed point sets. Like for the latter, the definition of a big polygon space involves a length vector ` ∈ Rr .
Big polygon spaces are never equivariantly formal. Nevertheless, for
suitable ` their equivariant cohomology can be computed by means of the
Chang–Skjelbred sequence (or “GKM method”), and their equivariant Poincaré pairing is perfect. No examples of compacted orientable T -manifolds X
with these properties were known so far.
More generally, one can realize syzygies of any order less than r/2 as
the equivariant cohomology of a big polygon space. This shows that the
bound on the syzygy order of HT∗ (X) obtained by Allday–Franz–Puppe [1]
is sharp.
[1] C. Allday, M. Franz, V. Puppe, Equivariant cohomology, syzygies and
orbit structure, arXiv:1111.0957, to appear in Trans. Amer. Math. Soc.
[2] M. Franz, Maximal syzygies in equivariant cohomology, arXiv:1403.4485
[3] M. Franz, V. Puppe, Freeness of equivariant cohomology and mutants of
compactified representations, pp. 87–98 in: M. Harada et al. (eds.), Toric
topology (Osaka, 2006), Contemp. Math. 460, AMS, Providence, RI, 2008
Name : Matthias Franz
Address : University of Western Ontario, London, Ontario N6A 5B7, Canada
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 14:00 – 14:40. Room 105
17
Newton-Okounkov bodies, symplectic geometry, and representation theory
Megumi Harada
The celebrated Bernstein-Kushnirenko theorem from Newton polyhedra
theory relates the number of solutions of a system of polynomial equations
with the volumes of their corresponding Newton polytopes. This motivated
developments in the theory of toric varieties, which connects the combinatorics of a convex integral polytope ∆ with the (equivariant) geometry of
the associated toric variety X(∆). In the more general setting of symplectic manifolds and Hamiltonian actions, the Atiyah/Guillemin-Sternberg and
Kirwan convexity theorems link equivariant symplectic and algebraic geometry to the combinatorics of moment map polytopes. In the case of a toric
variety X(∆), the moment map polytope ∆ fully encodes the geometry of
X(∆), but this fails in general. Okounkov constructed for an (irreducible)
projective variety X ⊆ P(V ) equipped with an action of a reductive algebraic
˜ and a natural projection from ∆
˜ to the moment
group G, a convex body ∆
map polytope ∆ of X. The volumes of the fibers of this projection encode
the so-called Duistermaat-Heckman measure, and in particular, one recovers
˜ In recent work, Kavehthe degree of X (i.e. the symplectic volume) from ∆.
Khovanskii and Lazarsfeld-Mustata generalized Okounkov’s ideas; given the
data of a variety X and a (big) divisor D on X, they construct a convex body
˜
˜
∆(X,
D) with dimR (∆(X,
D)) = dimC (X) (called a Newton-Okounkov body
or Okounkov body) even without presence of any group action. Thus the
constructions of Kaveh-Khovanskii and Lazarsfeld-Mustata show that there
are combinatorial objects of ‘maximal’ dimension associated to X in great
generality. This talk will be an introduction to this relatively recent theory,
with a focus on its relationships with symplectic geometry (in particular, of
integrable systems) and representation theory.
Name : Megumi Harada
Address : McMaster University
e-mail : [email protected]
Talk schedule : August 9 (Sat), 2014. 9:00 – 9:40. Room 105
18
Complex manifolds with maximal torus actions
Hiroaki Ishida
We say that an effective action of a compact torus G on a connected
smooth manifold M is maximal if there exists a point x ∈ M such that
dim G dim Gx = dim M . In this talk, we give a one-to-one correspondence
between the family of connected closed complex manifolds with maximal
torus actions and the family of certain combinatorial objects, which is a
generalization of the correspondence between complete nonsingular toric
varieties and complete nonsingular fans.
Name : Hiroaki Ishida
Address : Research Institute for Mathematical Science, Kyoto University
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 9:00 – 9:40. Room 105
19
Integrable systems, toric degenerations and
Newton-Okounkov bodies
Kiumars Kaveh
Let X be a smooth projective variety of dimension n over C, with a given
embedding in a projective space. Using the theory of Newton-Okounkov
bodies and an associated toric degeneration, we construct – under a mild
technical hypothesis on X – an integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued
functions {H1 , . . . , Hn } on X which are continuous on all of X, smooth on
an open dense subset U of X, and pairwise Poisson-commute on U . Moreover
H1 , . . . , Hn generate a (real) torus action on U . The image of the ‘moment
map’ µ = (H1 , . . . , Hn ) : X → Rn is precisely the Newton-Okounkov body
∆ = ∆(R, v) associated to the homogeneous coordinate ring R of X, and an
appropriate choice of a valuation v on R. This work provides a rich source
of new examples of integrable systems. Examples include elliptic curves,
flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties. I will also discuss application to finding lower
bounds for Gromov width of projective varieties. This is a joint work with
Megumi Harada.
Name : Kiumars Kaveh
Address : University of Pittsburgh
e-mail : [email protected]
Talk schedule : August 9 (Sat), 2014. 9:50 – 10:30. Room 105
20
Demazure operators and geometric mitosis
Valentina Kiritchenko
In [K], a convex-geometric algorithm was introduced for building new
analogs of Gelfand–Zetlin polytopes for an arbitrary reductive group G.
Similarly to the Gelfand–Zetlin polytopes, there is a polytope Pλ for every
dominant weight λ of G. The exponential sum over the lattice points inside the polytope Pλ coincides with the Weyl character of the irreducible
representation of G with the highest weight λ. I describe an algorithm (geometric mitosis) for finding a collection of faces in Pλ that represents the
Demazure character for any element w of the Weyl group of G (i.e. the
exponential sum over the lattice points in these faces coincides with the Demazure character corresponding to w and λ). For GLn and Gelfand–Zetlin
polytopes, this algorithm reduces to a geometric version of Knutson–Miller
mitosis considered in [KST]. I also describe a combinatorial realization of
geometric mitosis for symplectic groups.
[K] V.Kiritchenko, Divided difference operators on convex polytopes, arXiv:1307.7234
[math.AG], to appear in Adv. Studies in Pure Math.
[KST] V. Kiritchenko, E. Smirnov, V. Timorin, Schubert calculus and
Gelfand–Zetlin polytopes, Russian Math. Surveys, 67 (2012), no.4, 685–719
Name : Valentina Kiritchenko
Address : National Research University Higher School of Economics
Vavilova St. 7, 112312 Moscow, Russia
and
Institute for Information Transmission Problems, Moscow, Russia
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 9:50 – 10:30. Room 105
21
On contractibility of Koras-Russell threefolds
Amalendu Krishna
The Koras-Russell threefold is an affine three-fold over the field of complex numbers. This three-fold is equipped with a Gm -action with unique
fixed point. It is known that this three-fold is topologically contractible. It
is an open question if is also algebraically contractible. We shall show that
the Koras-Russell threefold is algebraically stably contractible. This is joint
work with Marc Hoyois and Paularne Ostvaer.
Name : Amalendu Krishna
Address : School of mathematics, Tata institute of fundamental research,
Mumbai, India
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 14:00 – 14:40. Room 105
22
Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures
Andrey Kustarev
A well-known theorem by Mostow and Pale states that for any manifold with smooth action of torus T n there exists representation of T n in
Euclidean linear space RN and T n -equivariant smooth embedding of the
manifold into RN . In general setting one can say nothing about minimizing
the dimension of RN . We consider quasitoric manifolds equipped with action of half-dimensional torus and modelled with combinatorial data (P, Λ),
where P is a simple convex polytope of dimension n with m faces of codimension one, Λ is a characteristic (n × m)-integer matrix. Our first task is
to introduce an explicit constuction of equivariant embedding for quasitoric
manifolds in terms of (P, Λ).
Every simple convex polytope defines a smooth manifold ZP known as
moment-angle manifold. We utilize the construction of that manifold as
(m)-dimensional real algebraic T m -invariant manifold ZP ⊂ Cm with T m
acting on Cm as usual. In this constuction ZP comes as complete intersection of real quadratic hypersurfaces in Cm . The quasitoric manifold M is
now defined as orbit space of free action of toric subgroup K on ZP , where
K ⊂ T m is a kernel of map l : T m → T n defined by Λ.
Using this construction we describe a family of equivariant monomial
maps Cm → C s.t. their restrictions on ZP are constant on orbits of K ⊂ T m
and therefore are well-defined on M . We will show that one can choose N
monomial maps s.t. equivariant map M → Rn × CN is an embedding of M .
Here M → Rn is a moment map for M coinciding with projection map on
orbit space P ⊂ Rn and action of T n on Rn is assumed to be trivial.
Condider set S of codimension one subgroups in T n that appear as stationary subgroups of points of M . Every subgroup H ⊂ S defines an ndimensional integer weight vector of representation T n → T n /H = S 1 . By
taking composition with map T m → T n determined by Λ we obtain mdimensional integer vector that in turn defines a monomial map ϕH : Cm →
C. The restriction ϕH |ZP is factorized through M .
We will show that maps ϕH , H ∈ S, are enough to generate an equivariant smooth real algebraic embedding M → Rn × Cq , where q = |S|.
It follows that in case of quasitoric manifolds one can explicitly construct
an equivariant embedding to linear space of relatively low dimension using
23
combinatorical data (P, Λ).
A necessary and sufficient condition for existence of T n -invariant almost
complex structure on a quasitoric manifold was obtained by author in 2009,
also in terms of combinatorial data (P, Λ). In this talk we will present
another solution of this problem that uses construction of equivariant embedding described above.
Name : Andrey Kustarev
Address : Moscow State University
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 14:50 – 15:30. Room 105
24
The convex hull of a space curve
Bernd Sturmfels
The boundary of the convex hull of a compact algebraic curve in real
3-space defines an algebraic surface. For general curves, that boundary
surface is reducible, consisting of tritangent planes and stationary bisecants.
We express the degree of this surface in terms of the degree, genus and
singularities of the curve. We present methods for computing their defining
polynomials, we show colorful pictures, and we discuss extension to higher
dimensions. This is based on articles with Kristian Ranestad.
Name : Bernd Sturmfels
Address : UC Berkeley and KAIST
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 17:30 – 18:10. Room 105
25
Combinatorial covers, abelian duality, and
propagation of resonance
Alexandru Suciu
We revisit and generalize a result of Eisenbud, Popescu, and Yuzvinsky,
which says that the resonance varieties of a hyperplane arrangement complement propagate. It turns out that the topological underpinning for this
phenomenon is a certain abelian duality property, coupled with the minimality property enjoyed by many spaces, such as complements of linear, elliptic,
and toric arrangements, as well as Cohen-Macaulay toric complexes. The
key ingredient in the proof is a general, cohomological vanishing result for
spaces that admit suitable ‘combinatorial’ covers.
This is joint work with Graham Denham and Sergey Yuzvinsky.
Name : Alexandru Suciu
Address : Northeastern University, Boston
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 14:50 – 15:30. Room 105
26
Vector partition functions and the topology of multiple weight varieties
Tatsuru Takakura
A multiple weight variety is, by definition, a symplectic quotient of a
direct product of several coadjoint orbits of a compact Lie group G, with
respect to the diagonal action of the maximal torus. Its geometry and the
topology are closely related to the combinatorics concerned with the weight
space decomposition of a tensor product of irreducible representations of G.
For example, when we consider the Riemann-Roch index and the symplectic volume of a multiple weight variety, we are naturally lead to the study
of vector partition functions with multiplicities and the associated volume
functions.
In this talk, we discuss some formulas to describe vector partition functions and volume functions, especially a generalization of the formula of
Brion-Vergne. Also, we make it more explicit in the case of the root system of type A. Then, by using them, we investigate the structure of the
cohomology of certain multiple weight varieties in detail.
Name : Tatsuru Takakura
Address : Department of Mathematics, Chuo University, 1-13-27 Kasuga,
Bunkyo-ku, Tokyo 112-8551, Japan
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 14:50 – 15:10. Room 105
27
The theory of (2n, k)-manifolds
Svjetlana Terzić
Our theory of (2n, k)-manifolds is devoted to the extending of the well
known results about the connection between the combinatorics of a torus action and algebraic topology of the underlying manifolds to the wide class of
manifolds with a compact torus action. We assign to our manifold M 2n the
convex polytope P k using the analogous of the moment map, where k is the
dimension of the compact torus which effectively acts on a smooth manifold
M 2n with isolated fixed points. In general case P k is not a simple polytope
and the orbit space M 2n /T k is not homeomorphic to P k . The new in our
approach is that for the purpose of the description of the combinatorics of
the torus action we introduce the so-called admissible polytopes which are
convex polytopes spanned by some subsets of vertices of the polytope P k .
We obtain CW-complex whose open cells are interiors of admissible polytopes.
In terms of the height function for one-dimensional skeleton of this complex we obtain combinatorial description of the Betti numbers of M 2n as
well as the equivariant cohomology of M 2n . It will be described the construction of the models of M 2n and of the orbit space M 2n /T k . In the
case of M 2n /T n it is generalization of the well known model of the toric
manifolds. The case of M 2n /T k is new. The applications is directed to the
solution of the well known problem of the structure of G(p, q)/T p for the
complex Grassmann manifolds.
The talk is based on the joint results with Victor M. Buchstaber.
Name : Svjetlana Terzić
Address : Faculty of Science, Univesity of Montenegro, Podgorica, Montenegro
e-mail : [email protected]
Talk schedule : August 11 (Mon), 2014. 9:50 – 10:30. Room 105
28
Complex subvarieties in homogeneous complex manifolds
Misha Verbitsky
A principal torus fibration over a Kahler manifold is called positive if a
pullback of a Kahler form from the base is exact. In this case it is never
Kahler. By Borel-Remmert-Tits theorem, any simply connected compact
complex homogeneous manifold is a principal torus fibration over a partial
flag space. They are positive in most examples.
Name : Misha Verbitsky
Address : National Research University “Higher School of Economics”
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 9:00 – 9:40. Room 105
29
On a class of quotient spaces of momentangle complexes
Li Yu
For a simplicial complex K and a partition α of the vertex set of K, we
define a quotient space of the (real) moment-angle complex of K by some
(not necessarily free) torus action determined by α. We obtain an analogue
of Hochster’s formula to compute the cohomology groups of such a space
with any coefficients. Moreover, we show that their cohomology rings with
Z2 -coefficients are isomorphic as multigraded Z2 -modules (or algebras) to
the cohomology of some multigraded differential algebra determined by K
and α. This generalizes the isomorphism between the cohomology ring of
a moment-angle complex ZK and the Tor algebra of the face ring of K. In
addition, we explain how to extend these results to a wider range of spaces.
[1] A. Bahri, M. Bendersky, F. Cohen and S. Gitler, The Polyhedral Product
Functor: a method of computation for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010), 1634–1668.
[2] V. M. Buchstaber and T. E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, 24. American Mathematical Society, Providence, RI, 2002.
[3] V. M. Buchstaber and T. E. Panov, Toric topology, arXiv:1210.2368.
[4] X. Cao and Z. Lü, Möbius transform, moment-angle complexes and
Halperin-Carlsson conjecture, J. Algebraic Combin. 35 (2012), no. 1, 121–
140.
[5] M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds
and torus actions, Duke Math. J. 62 (1991), no.2, 417–451.
[6] Z. Lü and T. E. Panov, Moment-angle complexes from simplicial posets,
Cent. Eur. J. Math. 9 (2011), no. 4, 715–730.
[7] M. Masuda and T. E. Panov, On the cohomology of torus manifolds,
Osaka J. Math. 43 (2006), 711–746.
Name : Li Yu
Address : Department of Mathematics and IMS, Nanjing University, Nanjing, 210093, P.R.China
e-mail : [email protected]
Talk schedule : August 11 (Mon), 2014. 14:00 – 14:40. Room 105
30
Contributed talks
32
On the toric manifolds arising from the
root systems of type B and type C
Hiraku Abe
We compare the toric manifolds associated with the Weyl chambers of
the root systems of type Bn and type Cn , through the computations of
intersection numbers of invariant divisors.
Name : Hiraku Abe
Address : Osaka City University Advanced Mathematical Institute (OCAMI)
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 11:00 – 11:30. Room 107
33
Homology of manifolds with locally standard torus actions
Anton Ayzenberg
Let M 2n be a manifold with locally standard action of compact torus T n .
The orbit space Q = M 2n /T n is a nice manifold with corners. If Q is acyclic
and all its faces are acyclic, Masuda and Panov proved that cohomology
of M are described similar to cohomology of (quasi)toric manifolds. More
precisely, cohomology is concentrated in even degrees, dim H 2i (M ) = hi (Q),
and H ∗ (M ) is the quotient algebra of the face ring by the linear system of
parameters.
I consider the case when every proper face of Q is acyclic, but Q itself
is arbitrary. To calculate (co)homology of M in this case I use spectral
sequence associated to filtration by orbit types. In this generality h0 - and
h00 -vectors of Buchsbaum simplicial posets come in play.
Name : Anton Ayzenberg
Address : Osaka City Univeristy
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 16:00 – 16:30. Room 107
34
The cohomology ring structure of the symmetric square of HPn
Yumi Boote
This talk concerns the symmetric square X of a quaternionic projective
space HPn . By definition, X underlies the global quotient orbifold associated
to the involution that interchanges the factors of HPn ×HPn ; it has dimension
8n, and admits an action of the quaternionic torus (S 3 )n . I shall describe
the geometry of X in terms of the Thom space of a certain real 4-plane
bundle, by analogy with results of James, Thomas, Toda, and Whitehead
for the symmetric square of a sphere. This viewpoint leads to a calculation
of the multiplicative structure of the integral cohomology ring of X, which
is quite delicate. The mod 2 cohomology ring and the action of the Steenrod
algebra follow rather more straightforwardly. I shall make comparisons with
the integral and mod 2 equivariant cohomology of the global quotient, which
are easier to compute and provide important input to the main calculation.
In order to be as clear as possible I shall focus on the case of HP3 , which has
dimension 24 and represents all major features of the general case. My talk
describes work in progress, which I expect will form part of my PhD thesis
in 2015.
Name : Yumi Boote
Address : School of Mathematics, University of Manchester, Oxford Road,
Manchester M13 9PL
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 11:40 – 12:10. Room 108
35
A family of polytopal moment-angle manifolds
Li Cai
In this talk, we shall illustrate that the intersection of the unit sphere
and the space of minima of certain Siegel leaves, with respect to the pnorm, provides a family of polytopal moment-angle manifolds; the respective
moment-angle complex appears as the limit set as p approaches infinity.
Name : Li Cai
Address : Graduate school of Mathematics, Kyushu University, 744, Motooka, Nishi-Ku, Fukuoka-city 819-0395, Japan
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 11:40 – 12:10. Room 107
36
Elliptic braid groups are duality groups
Graham Denham
The elliptic braid group is the fundamental group of a configuration space
of n points in 2-dimensional torus. We show that such groups are duality
groups, extending the known result for classical braid groups. The method
is an instance of a more general cohomological vanishing construction which
also has applications to torus arrangements, right-angled Artin groups, and
hyperplane complements. This is joint work with Alex Suciu (Northeastern)
and Sergey Yuzvinsky (Oregon).
Name : Graham Denham
Address : University of Western Ontario
e-mail : [email protected]
Talk schedule : August 11 (Mon), 2014. 11:00 – 11:30. Room 105
37
Complex geometry of moment-angle manifolds
Yury Eliyashev
The general construction of moment-angle manifold was introduced in
the toric topology setting, but particular cases were appeared as a part of
symplectic reduction construction for toric manifolds. The fact that some
moment-angle manifolds admits a complex structure was discovered quite
recently, in the last decade the series of papers on this topic was published.
Moment-angle provides a class of examples on non-Kähler manifolds. This
talk is devoted to complex geometry of these manifolds. We will study Dolbeault cohomology, complex subvarieties, vector bundles and other related
objects on these manifolds.
Name : Yury Eliyashev
Address : Higher School of Economics, Moscow, Russia
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 11:00 – 11:30. Room 107
38
Simple polytopes and simplicial complexes
with Buchstaber number 2
Nikolay Erokhovets
With each simplicial (n − 1)-complex K on m vertices toric topology
associates an (m + n)-dimensional moment-angle complex ZK with a canonical action of a torus T m . If K is a boundary complex of a polytope polar to
a simple polytope P , then the space ZP = ZK is a smooth manifold. The
equivariant topology of ZK depends only on the combinatorics of K, which
gives a tool to study the combinatorics of simple polytopes and simplicial
complexes in terms of the algebraic topology of moment-angle complexes
and vice versa. A Buchstaber invariant s(K) is equal to the maximal dimension of torus subgroups H ⊂ T m , H ' T k , that act freely on ZK . In
2002 V.M. Buchstaber stated the problem to find an effective description
of s(K) in terms of the combinatorics of K. In 2012 he reformulated the
problem in two ways: 1) To characterize the polytopes and complexes with
fixed value of s(K); 2) To calculate s(K) for fixed dimension n. There is an
n-dimensional analog RZK for the Zm
2 -action. The corresponding number
sR (K) is called a real Buchstaber invariant. We have s(K) 6 sR (K) 6 m−n.
The most convenient cases to study are cases of n 6 3 and s(K) 6 2,
since for these cases s(K) = sR (K). Let N (K) be the set of missing faces
of K.
Theorem 1 I. s(K) = 1 iff either |N (K)| = 1, or N (K) = {τ1 , τ2 }, τ1 ∩
τ2 6= ∅, or |N (K)| > 3 and any three missing faces intersect.
II. s(K) = 2 iff there exist either two or three missing facets with empty
intersection and N (K) does not contain any of the following subsets:
1) {τ1 , τ2 , τ3 }: τ1 ∩ τ2 = τ1 ∩ τ3 = τ2 ∩ τ3 = ∅;
2) {τ1 , τ2 , τ3 , τ4 }: τ1 ∩ (τ2 ∪ τ3 ∪ τ4 ) = τ2 ∩ τ3 ∩ τ4 = ∅;
3) {τ1 , τ2 , τ3 , τ4 , τ5 }: τ1 ∩ τ2 = τ1 ∩ τ5 = τ1 ∩ τ3 ∩ τ4 = τ2 ∩ τ3 ∩ τ5 =
τ2 ∩ τ4 ∩ τ5 = ∅;
4) {τ1 , τ2 , τ3 , τ4 , τ5 , τ6 }: τ1 ∩ τ3 = τ1 ∩ τ2 ∩ τ4 = τ1 ∩ τ2 ∩ τ5 = τ1 ∩ τ4 ∩ τ6 =
τ1 ∩ τ5 ∩ τ6 = τ2 ∩ τ3 ∩ τ6 = τ3 ∩ τ4 ∩ τ5 = ∅;
5) {τ1 , τ2 , τ3 , τ4 , τ5 , τ6 , τ7 }: τ1 ∩ τ2 ∩ τ4 = τ1 ∩ τ3 ∩ τ5 = τ1 ∩ τ6 ∩ τ7 =
τ2 ∩ τ3 ∩ τ6 = τ2 ∩ τ5 ∩ τ7 = τ3 ∩ τ4 ∩ τ7 = τ4 ∩ τ5 ∩ τ6 = ∅.
For simple polytopes we have the following.
Theorem 2 I. s(P ) = 1 iff P = ∆n (i.e. m
− n = 1).
II. If s(P ) = 2, then 2 6 m − n 6 2 + n2 , and either P = I × ∆n , or any
two facets intersect. Moreover, any m − n − 2 facets intersect. Furthermore,
39
• if m − n = 2, then P = ∆i × ∆j ;
• if m − n = 3, then s(P ) = 2 iff N (P ) > 7;
• if n 6 5, then s(P ) = 2 iff m − n = 2, i.e. P = ∆i × ∆j .
Let C n (m)∆ be a polytope polar to a cyclic polytope.
[ n ]+1
Proposition 3 We have s(C n (m)∆ ) = 2 iff 2 6 m − n < 2 + 2 3 . In
particular, for any k > 2 there are simple polytopes P with m − n = k and
s(P ) = 2. The first nontrivial example appears for n = 6, when m − n = 3
and s(P ) = 2.
The work was partially supported by the Russian President grant MK 600.2014.1.
[1] N.Yu. Erokhovets, Theory of the Buchstaber invariant of simplicial complexes and convex polytopes, accepted to Proceedings of the Steklov Institute
of Mathematics, Vol 268, 2014.
[2] N. Erokhovets, Criterion for the Buchstaber invariant of simplicial complexes to be equal to two, arXiv:1212.3970v1.
[3] N. Erokhovets, Buchstaber Invariant of Simple Polytopes, arXiv:0908.3407.
Name : Nikolay Erokhovets
Address : Lomonosov Moscow State University, Moscow, Russia
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 16:00 – 16:30. Room 107
40
On well-definedness of the local index
Hajime Fujita
In our joint work with M.Furuta and T.Yoshida, we gave a formulation of
an analytic index theory for non-compact Riemannian manifolds. The index
theory gives us an localization formula of index and enables us to understand
several equalities, such as [Q, R] = 0 for torus action and RR = BS for
Lagrangian fibration, in a geometric way. Our index theory uses an open
covering of an end of the manifold, a family of torus bundles and Diractype operators along fibers on the open covering as a boundary condition.
The resulting index, which we call the local index, a priori depends on the
covering. In this talk I will talk about the cobordism invariance of the local
index, and as an application I will show that the index does not depend on
the open covering in a suitable category.
Name : Hajime Fujita
Address : Department of Mathematical and Physical Science, Japan Women’s
University, 2-8-1 Mejirodai, Bunkyo-ku Tokyo, 112-8681 Japan
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 11:40 – 12:10. Room 108
41
McKay corespondence in quasitoric orbifolds
Saibal Ganguli
McKay correspondence relates orbifold cohomology with the cohomology
of a crepant resolution. This is a phenomenon in algebraic geometry. It was
proved for toric orbifolds by Batyrev and Dais in the nineties. In this talk we
present a similar correspondence for omnioriented quasitoric orbifolds. The
interesting feature is how we deal with the absence of an algebraic or analytic
structure. In a suitable sense, our correspondence is a generalization of the
algebraic one.
Name : Saibal Ganguli
Address : Institute of Mathematical Sciences, Chennai, India
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 11:40 – 12:10. Room 107
42
Spin toric manifolds associated to graphs
Miho Hatanaka
We describe a necessary and sufficient condition for a toric manifold to
admit a spin structure. This implies that a toric manifold admits a spin
structure if and only if its real part is orientable. It is known that a Delzant
polytope can be constructed from a simple graph, so that one can associate
a toric manifold to a simple graph. We characterize simple graphs whose
associated toric manifolds admit spin structures.
Name : Miho Hatanaka
Address : Osaka City University
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 16:00 – 16:30. Room 107
43
The equivariant cohomology rings of Peterson varieties in all Lie types
Tatsuya Horiguchi
We will give an efficient presentation of the S 1 - equivariant cohomology
ring of Peterson varieties in all Lie types as a quotient of a polynomial ring
by an ideal J generated by quadratic polynomials, in the spirit of the Borel
presentation of the cohomology of the flag variety. Our description of the
ideal J uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant
cohomology rings of Peterson varieties for all Lie types, as obtained in the
work of Drellich. Our result generalizes a previous theorem of FukukawaHarada-Masuda, which was only for Lie type A.
Name : Tatsuya Horiguchi
Address : Osaka City University
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 11:00 – 11:30. Room 108
44
Toric varieties and finite domination of chain
complexes
Thomas Huettemann
Let R be a ring with unit. A chain complex C of R-modules is called
R-finitely dominated if it is homotopy equivalent to a bounded complex of
finitely generated projective R-modules. This notion has been considered in
various areas of mathematics, for example algebraic topology (finite domination of spaces), group theory (groups of type FP), and algebraic geometry
(under the name of “perfect complexes").
If C is a bounded complex of finitely generated free modules over a
Laurent polynomial ring R[x, 1/x], there is a homological criterion: C is
R-finitely dominated if and only if the homology of C with coefficients in
the two rings of formal Laurent power series in x resp. 1/x vanishes.
It is non-trivial to extend this criterion to Laurent polynomial rings in
several indeterminates. One such extension involves arguments that are best
understood from the point of view of toric geometry. A complex of finitely
generated free modules over a Laurent polynomial ring in n indeterminates
is a complex of trivial vector bundles on an n-dimensional algebraic torus.
Every suitable compactification of the torus yields a homological criterion
for finite domination: The original complex is required to be homologically
trivial “infinitesimally near the divisors at infinity", that is, near the complement of the torus in its compactification. Armed with the theory of toric
varieties one can give a hands-on, combinatorial formulation of the criterion:
Every n-dimensional lattice polytope yields a new characterisation of finite
domination.
In the talk I will explain the algebro-geometric background of homological finiteness criteria, focusing on ideas rather than technical details. Time
permitting I will also comment on a rather different set of tools (homotopy
commutative diagrams of chain complexes, generalised mapping tori, truncated product totalisation of multi-complexes) that is required to establish
the equivalence of finite domination with vanishing Novikov homology.
Name : Thomas Huettemann
Address : Queen’s University, Belfast
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 11:40 – 12:10. Room 107
45
Symplectic capacities from Hamiltonian circle actions
Taekgyu Hwang
Symplectic capacities are symplectic invariants related to embeddings
of symplectic manifolds. Under certain conditions on Hamiltonian circle
actions on symplectic manifolds, we compute two symplectic capacities, the
Gromov width and the Hofer-Zehnder capacity, in terms of moment map.
This is joint with Dong Youp Suh.
Name : Taekgyu Hwang
Address : KIAS, Korea
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 16:40 – 17:10. Room 108
46
Complexity one GKM graph with symmetries and an obstruction to be a torus graph
Shintaro Kuroki
An m-valent GKM graph is an m-valent graph whose edges are labeled
by elements of H 2 (BT n ), where n is less than or equal to m. A GKM graph
is often induced from a nice 2m-dimensional manifold with n-dimensional
torus action, called a GKM manifold, and its GKM graph contains many
topological information of a GKM manifold. For instance, a toric manifold
M is a GKM manifold and its induced GKM graph is called a torus graph,
and we can compute the equivariant cohomology of M by using its torus
graph. In this talk, we study the case when m = n + 1, called a complexity one, and give a classification of complexity one GKM graphs with Weyl
group actions (in particular, for n = 2, 3). Moreover, we also discuss when
they are induced from torus graphs (i.e., the case when m = n) by introducing an obstruction (which is different from the obstruction defined by
Takuma in his unpublished paper).
Name : Shintaro Kuroki
Address : The University of Tokyo
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 16:40 – 17:10. Room 107
47
Grossberg-Karshon twisted cubes and hesitant walk avoidance
Eunjeong Lee
Let G be a complex semisimple simply connected linear algebraic group.
Let λ be a dominant weight for G and I = (i1 , i2 , . . . , in ) a word decomposition for an element w = si1 si2 · · · sin of the Weyl group of G, where the si
are the simple reflections. In the 1990s, Grossberg and Karshon introduced
a virtual lattice polytope associated to λ and I, which they called a twisted
cube, whose lattice points encode (counted with sign according to a density
function) characters of representations of G. In recent work, M. Harada
and J. Yang prove that the Grossberg-Karshon twisted cube is untwisted
(so the support of the density function is a closed convex polytope) precisely when a certain torus-invariant divisor on a toric variety, constructed
from the data of λ and I, is basepoint-free. In this talk, we translate this
toric-geometric condition to the combinatorics of I and λ. More precisely,
we introduce the notion of hesitant λ-walks and then prove that the associated Grossberg-Karshon twisted cube is untwisted precisely when I is
hesitant-λ-walk-avoiding. This is joint work with M. Harada.
Name : Eunjeong Lee
Address : Department of Mathematical Sciences, KAIST, 291 Daehak-ro
Yuseong-gu, Daejeon 305-701, South Korea
e-mail : [email protected]
Talk schedule : August 9 (Sat), 2014. 11:00 – 11:30. Room 108
48
On equivariant Pieri rule of isotropic Grassmannians
Changzheng Li
In this talk, we will discuss the equivariant Pieri rules for the torusequivariant cohomology of Grassmannians of classical Lie types. We will
introduce a first manifestly positive formula for Grassmannians of Lie types
B, C and D beyond the equivariant Chevalley formula. This is my joint
work with Vijay Ravikumar.
Name : Changzheng Li
Address : Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), Todai Institutes for Advanced Study, The University
of Tokyo, 5-1-5 Kashiwa-no-Ha,Kashiwa City, Chiba 277-8583, Japan
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 11:40 – 12:10. Room 108
49
Combinatorics of simple polytopes, their
Stanley-Reisner rings and moment-angle manifolds
Ivan Limonchenko
Toric topology gives us a deep relation between combinatorial properties
of a simplicial complex K, algebraic invariants of its Stanley–Reisner ring
k[K] over an integral domain k and topological properties of the corresponding moment-angle complex ZK .
In my talk I will show this for boundary complexes of some simplicial polytopes, namely for neighbourly ones, stellar subdivisions of a simplex and
their generalizations.
The author was supported by the Russian Science Foundation (grant no.
14-11-00414).
Name : Ivan Limonchenko
Address : Faculty of Geometry and Topology, Department of Mathematics
and Mechanics, Moscow State University, Leninskiye Gory, Moscow 119992,
Russia
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 16:40 – 17:10. Room 107
50
Equivariant Giambelli formula for isotropic
flag varieties
Tomoo Matsumura
The Giambelli problem in Schubert calculus is to find a closed formula for
a Schubert class in terms of the special classes that generate the cohomology
as a ring. In this talk, I will explain the equivariant Giambelli formula for
the Grassmann of non-maximal isotropic subspaces in a symplectic vector
space. The formula expresses an equivariant Schubert class as a sum of
Pfaffians. To prove the formula, we use the left divided difference operators,
that are essential in the theory of double Schubert polynomials. This is a
joint work with T. Ikeda.
Name : Tomoo Matsumura
Address : KAIST, Korea
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 16:40 – 17:10. Room 108
51
Torsions of cohomology of real toric manifolds
Hanchul Park
Unlike toric manifolds and quasitoric manifolds, it is difficult to describe
integral cohomology rings of real toric manifolds and small covers. In this
talk, we compute cohomology rings of real toric objects for coefficient ring
Q or Zq , where q is an odd integer. As an application, for any given odd
integer q > 1, we construct a real toric manifold whose cohomology ring has
a q-torsion. This is a joint work with Suyoung Choi (Ajou Univ.).
Name : Hanchul Park
Address : KIAS, Korea
e-mail : [email protected]
Talk schedule : August 11 (Mon), 2014. 11:40 – 12:10. Room 107
52
Betti numbers of toric origami manifolds
Seonjeong Park
The notion of a toric origami manifold was introduced by Cannas da
Silva- Guillemin-Pires by weakening the notion of a symplectic toric manifold. If an orientable toric origami manifold has a fixed point, it can be a
torus manifold with a locally standard torus action. In this talk, we will
compute the Betti numbers of toric origami manifolds. This is a joint work
with Anton Ayzenberg, Mikiya Masuda, and Haozhi Zeng.
Name : Seonjeong Park
Address : Division of Mathematical Models, National Institute for Mathematical Sciences, 463-1 Jeonmin-dong, Yuseong-gu, Daejeon 305-811, Korea
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 16:40 – 17:10. Room 107
53
Hamiltonian loops on symplectic blow ups
Andrés Pedroza
We present a criteria for when a Hamiltonian diffeomorphism on a symplectic manifold can be lifted to the symplectic blow up at one point. Based
on this we give necessary conditions under which a Hamiltonian loop on the
symplectic blow up, induced from a Hamiltonian loop on the base manifold,
is not homotopic to zero.
Name : Andrés Pedroza
Address : Universidad de Colima
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 16:00 – 16:30. Room 108
54
Explicit triangulation of complex projective spaces
Soumen Sarkar
In 1983, Banchoff and Kuhnel constructed a minimal triangulation of
with 9 vertices. CP 3 was first triangulated by Bagchi and Datta in
2012 with 18 vertices. Known 2lower bound on number of vertices of a triangulation of CP n is 1 + (n+1)
for n ≥ 3. We give explicit construction of
2
CP 2
n+1
some triangulations of complex projective space CP n with (4 3 −1) vertices
for all n. No explicit triangulation of CP n is known for n ≥ 4.
Name : Soumen Sarkar
Address : University of Regina
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 11:00 – 11:30. Room 107
55
Orbifold and non-orbifold linear symplectic
quotients
Christopher Seaton
Let T be a torus and V a unitary T -representation. Choosing the homogeneous quadratic moment map, zero is a singular value, and the corresponding symplectic reduced space yields a symplectic stratified space.
For T nontrivial, the resulting space has infinite isotropy groups and hence
is never an orbifold. In some cases, however, the reduced space is symplectomorphic to an orbifold. These symplectomorphisms are constructed
using isomorphisms between the graded rings of regular functions on these
(semi-algebraic) spaces; we refer to such maps as graded regular symplectomorphisms.
We will present results related to the question of which T -representations
yield a reduced space that is graded regularly symplectomorphic to an orbifold. This includes a complete answer when T is the circle as well as progress
in other cases. In addition, we will present results indicating similarities between the Hilbert series of invariants on symplectic T -quotients and orbifolds
in general.
Name : Christopher Seaton
Address : Department of Mathematics and Computer Science, Rhodes College, 901-843-3721
e-mail : [email protected]
Talk schedule : August 11 (Mon), 2014. 11:40 – 12:10. Room 108
56
Schubert polynomials and pipe dreams
Evgeny Smirnov
Schubert polynomials were introduced by A.Lascoux and M.-P.Schuetzenberger
as a tool for studying the cohomology ring of a full flag variety. They
naturally generalize well-known Schur polynomials. In 1996 S.Fomin and
An.Kirillov provided a realization for Schubert polynomials using combinatorial objects usually referred to as pipe dreams, or rc-graphs. Geometrically,
pipe dreams correspond to irreducible components of the images of Schubert
varieties under a certain toric degeneration of a flag variety.
It turns out that these pipe dreams have many interesting combinatorial
properties which relate them, among others, to plane partitions (=threedimensional Young diagrams) and Stasheff associahedra. Some of them were
already observed in 1990s by Fomin and Kirillov, Billey et al.; some other
properties are new. In my talk I will give an overview of these properties.
Time permitting, I will also explain how pipe dreams are related to our work
with V.Kiritchenko and V.Timorin on the realization of Schubert calculus
on full flag varieties via Gelfand-Zetlin polytopes.
Name : Evgeny Smirnov
Address : Higher School of Economics, Moscow
e-mail : [email protected]
Talk schedule : August 7 (Thu), 2014. 16:00 – 16:30. Room 108
57
The integral cohomology ring of toric orbifolds and weighted projective towers
Jongbaek Song
We call a toric orbifold the toric verities associated to simplicial fans.
It is well-known by Danilov and Jurkiewicz that the cohomology ring with
Z or Q-coefficients of non-singular toric variety is isomorphic to StanleyReisner ring of underlying simplicial complex modulo linear relations. Also,
under the Q-coefficients, cohomology ring of toric orbifold is isomorphic to
Q[Σ] modulo linear relations. In this talk, we shall discuss the cohomology
ring of toric orbifold with integer coefficients and introduce an interesting
class of toric orbifolds, weighted projective towers. Finally, we introduce the
formula of its integral cohomology ring. This is a joint work with A.Bahri,
N.Ray, and S.Sarkar.
Name : Jongbaek Song
Address : KAIST
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 11:00 – 11:30. Room 108
58
Examples of toric manifolds whose orbit
spaces by the compact torus are not simple
polytopes
Yusuke Suyama
It is known that if a toric manifold is projective or has complex dimension
n ≤ 3, then its orbit space by the restricted action of the compact torus is
a simple polytope. We show that there are infinitely many toric manifolds
whose orbit spaces by the compact torus are not simple polytopes for any
complex dimension n ≥ 4. This implies that there are infinitely many toric
manifolds which are not quasitoric manifolds.
Name : Yusuke Suyama
Address : Osaka city university
e-mail : [email protected]
Talk schedule : August 9 (Sat), 2014. 11:00 – 11:30. Room 107
59
Okounkov bodies, toric degenerations, and
Bott-Samelson varieties
Jihyeon Jessie Yang
Let X be a complex projective variety of dimension n equipped with
a very ample line bundle L and a choice of valuation ν on its homogeneous coordinate ring R = R(L). Given this data, we can associate to
(X; R; ν) a convex body of (real) dimension n, called the Okounkov body
∆ = ∆(X; R; ν). In many cases ∆ is in fact a rational polytope; indeed,
in the case when X is a nonsingular projective toric variety, the ring R
and valuation ν may be chosen so that ∆ is the Newton polytope of X. It
has been proved (Anderson, Kaveh) that, in many cases of interest (such as
those arising in representation theory and Schubert calculus), the Okounkov
body gives rise to a toric degeneration of X; in particular, this construction
simultaneously generalize many toric degenerations given in the literature
(e.g. Alexeev-Brion, Caldero, Kogan-Miller). However, Okounkov bodies
(and the associated toric degenerations) depend in general on the valuation
ν in a subtle way which is not well-understood. In this talk we report on
work in progress related to these ideas. Specifically, for a toric degeneration
of a Bott-Samelson variety to a toric variety constructed by Pasquier (based
on work by Grossberg and Karshon), we ask: does this toric degeneration
arise as a special case of Anderson’s general construction?
Name : Jihyeon Jessie Yang
Address : McMaster university
e-mail : [email protected]
Talk schedule : August 8 (Fri), 2014. 16:40 – 17:10. Room 108
60
Torus fibrations and localization of index
Takahiko Yoshida
We report a recent progress of the joint work with H. Fujita and M.
Furuta on a localization of index of a Dirac-type operators on possibly noncompact Riemannian manifolds. We also describe some applications to the
torus actions.
We make use of a structure of torus fibration on the end, but the mechanism of the localization does not rely on any group action. In the case of
Lagrangian fibration, we show that the index is described as a sum of the
contributions from Bohr-Sommerfeld fibers and singular fibers.
To show the localization we introduce a deformation of a Dirac-type operator for a manifold equipped with a fiber bundle structure which satisfies
a kind of acyclic condition. The deformation allows an interpretation as an
adiabatic limit or an infinite dimensional analogue of Witten deformation.
Joint work with Hajime Fujita and Mikio Furuta.
Name : Takahiko Yoshida
Address : Department of Mathematics, School of Science and Technology,
Meiji University
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 11:00 – 11:30. Room 108
61
The cohomology algebra of polyhedral product spaces
Qibing Zheng
We compute the cohomology algebra of polyhedral product spaces over
a field and in certain cases, over a ring. This is done by first constructing
a chain complex homotopic equivalent to the singular chain complex of the
polyhedral product space, and then by constructing a coproduct on the chain
complex such that the dual homomorphism induces the cup product of the
cohomology algebra of the polyhedral product space.
Name : Qibing Zheng
Address : Nankai University
e-mail : [email protected]
Talk schedule : August 10 (Sun), 2014. 11:40 – 12:10. Room 107
62
Memo
64
65
66
67