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

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