Abstracts

I will review a few examples of mechanical systems and explain the relevance of symplectic geometry and torus actions. I will then give examples of topological techniques (e.g. Morse theory) used to investigate these actions, and of classical results (e.g. convexity or classification theorems)

Properties of the stable structure of generalized moment-angle complexes as well as applications are given. This lecture is based on joint work with Anthony Bahri, Martin Bendersky and Sam Gitler.

Toric topology provides interesting examples in equivariant cohomology and brave new commutative algebra. It is hoped that some of them will be discussed in the talk.

tbc

I will discuss some properties of the number of solutions of a generic system of n algebraic equations on an affine n-dimensional algebraic variety X. These properties are analogous to the properties of mixed volumes of n-dimensional convex bodies (e.g. some of the well-known geometric inequalities). Our main construction is to associate a convex body (usually a polytope) to a finite dimensional linear subspace L of regular functions such that its volume is responsible for the number of solutions of a generic system in L. For the proofs we use only Hilbert theory on the degree of an algebraic projective variety and the well-known isoperimetric inequality in the plane. Kushnirenko--Bernstein theorem for the number of solutions of a generic system of equations with given Newton polyhedra, Hodge Index Theorem for an algebraic surface, Alexandrov--Fenchel inequality and some other properties of mixed volumes of convex bodies are straight forward corollaries from our results.

Suppose X is equipped with a reductive group action and take a finite dimensional spaces L of regular functions on X which is invariant under the group action. In this case, under some extra assumptions, one can make all the above constructions much more precise and extend the Bernstein theorem to some spherical varieties.

My talk is based on a joint work with Kiumars Kaveh.

The classification of compact toric manifolds as varieties has been completed in some cases by classifying associated fans. However not much is known for their topological classification. A naive question is Problem. Are two compact toric manifolds homeomorphic (or diffeomorphic) if their cohomology rings with Z coefficients are isomorphic as graded rings? As is well-known, there are many closed smooth manifolds which are not homeomorphic but have isomorphic cohomology rings. So the problem above seems unlikely but no counterexample is known. One can ask the same question for compact real toric manifolds (with Z/2 coefficients). I will report some partial affirmative solutions to the problem above and discuss related problems.

Moment angle complexes and manifolds are one of the most important objects of study in toric topology. The moment-angle complex Z_K is a T^m-space associated to a finite simplicial complex K on m vertices; this construction is functorial with respect to simplicial maps. The first definition of Z_K as an identification space was given in a 1991 paper of Davis and Januszkiewicz; it goes back to Vinberg universal space construction for Coxeter groups of 1970's. If K is a sphere triangulation, then Z_K is a (closed) manifold; in particular, a moment-angle manifold is assigned to every simplicial (or simple) convex polytope.

Equivalent constructions of moment-angle complexes and manifolds appear in different disguises in several seemingly unrelated fields; these include homotopy fibres of simple cellular inclusions in homotopy theory, coordinate subspace arrangement complements in algebraic geometry, level sets for toric moment maps in symplectic geometry, and, most recently, complete intersections of real quadrics in C^m. The latter interpretation provides a new series of complex non-K\"ahler manifolds, generalising those of Hopf and Calabi--Eckmann, and creates a new bridge between toric topology and complex analysis.

We shall give an overview of known results on cohomological and homotopical properties of moment-angle complexes and manifolds, including the relationships with the cohomology of Stanley--Reisner face rings and recent applications to cobordisms of quasitoric manifolds. We shall also propose a list of open problems.

Let M be a closed, smooth manifold with nontrivial fundamental group. Cohomology with local system for abelian representation of fundamental group can be calculated be maens of spectral sequence with Massay type differentials, as Novikov showed. We generelized this spectral sequence for nonabelian case and studied some conditions of convergence.

A well-know construction from toric geometry due to D. Cox (1995) realizes an arbitrary toric variety without non-constant invertible functions as a categorical quotient of an open subset of a vector space by a quasitorus action. More generally, with any normali algebraic variety X with a free finitely generated divisor class group one may associate a multigraded factorial ring R(X) that is called the total coordinate ring or the Cox ring. If R(X) is finitely generated, then X is realized as the categorical quotient of an open subset U of an affine factorial variety Z=Spec(R(X)) by the Neron-Severi torus H. We give a new proof of factoriality of R(X) based on the notion of graded factoriality, i.e. a uniqueness of factorization into irreducible elements for the multiplicative semigroup of homogeneous elements of R(X). If the divisor class group of X has torsion, the ring R(X) is also factorially graded, but factoriality may be lost. We demonstrate this effect by calculating the Cox ring of a homogeneous space of an affine algebraic group. Applications of the Cox's construction to algebraic geometry, combinatorics and toric topology are widely known. In this talk we discuss further applications to the theory of algebraic transformation groups. Namely, assume that a connected reductive group G acts on the variety X. Passing from G to its finite covering, one lifts the G-action on X to HxG-action on Z with U being an invariant subset. This leads to the equivariant Cox's construction and allows to reduce some problems to the case of an affine factorial variety. We illustrate this approach with two problems. Firstly, we give a combinatorial description of open equivariant embeddings of a given homogeneous space G/H with small boundary. The last condition means that the complement of the image of G/H contains no divisors. Secondly, we deal with a central problem of Geometric Invariant Theory (GIT): for a given G-variety X describe all maximal open invariant subsets admitting a good G-quotient.

The talk is based on a joint project with J. Hausen (Tuebingen, Germany).

To any simplicial complex K on m vertices there corresponds so-called partial product functor which associates to m based topological spaces X_1,...,X_m a certain subspace W_K(X_1,...,X_m) in the cartesian product of the X_i's. For some particular simplicial complexes these partial products coincide with classical constructions like wedges, fat wedges or products. This functor plays an important role in toric topology. We study the rational homology algebras of loop spaces of partial products. It turns out that the structure of these algebras is determined by the homology of certain labelled configuration spaces. This leads to a presentation of the loop homology algebras in terms of higher operations. The explicit presentation will be given for some special classes of complexes. In connection with this result I will discuss stable splittings of these loops spaces, formulas for Poincare series, and applications to toric topology.

In 1940s N. Steenrod posed the following problem called the problem on realization of cycles. For a given integral homology class z of a topological space X, is there an oriented manifold N and a continuous mapping f:N\to X such that f_*[N]=z? A famous theorem due to R. Thom claims that every integral homology class can be realized with some multiplicity by an image of a smooth oriented manifold. Our goal is to point out a class \mathcal{M}_n of smooth oriented n-dimensional manifolds such that \mathcal{M}_n suffices for realization with some multiplicity of every integral homology class of every topological space X. The main object of our construction is the manifold M^n of isospectral symmetric tridiagonal real (n+1)\times (n+1) matrices. For \mathcal{M_n} we take the class of all finite-fold coverings of M^n. Our main result can be formulated in the following way. Theorem. Suppose z\in H_n(X;\mathbb{Z}). Then there exist a finite-fold covering p:\widehat{M}^n\to M^n and a continuous mapping f:\widehat{M}^n\to X such that f_*[\widehat{M}^n]=qz for some nonzero integer q. In 1984 C. Tomei proved that the manifold M^n is aspherical. Hence the above theorem implies that every integral homology class of an arcwise connected toplogical space can be realized with some multiplicity by an image of an aspherical manifold.

This is a preliminary report on work in progress with Graeme Wilkin. Let G be a compact Lie group. The well-known Kirwan surjectivity theorem in equivariant symplectic geometry states that the G-equivariant rational cohomology of a Hamiltonian G-space (M,\omega) surjects onto the ordinary rational cohomology of the symplectic quotient of M by G. This surjective ring homomorphism ("the Kirwan map") has been a key tool in computations of the topology of symplectic quotients. I will discuss our recent progress on the analogous hyperk\"ahler question, namely: if (M,\omega_1, \omega_2, \omega_3) is a hyperk\"ahler hyperhamiltonian G-space, then does the G-equivariant cohomology of M surject onto the ordinary rational cohomology of the hyperk\"ahler quotient of M by G? We restrict to the case of Nakajima quiver varieties and develop a Morse theory for the hyperk\"ahler moment map analogous to the case of the moduli space of Higgs bundles. In particular, we show that the Harder-Narasimhan stratification of spaces of representations of quivers coincide with the Morse-theoretic stratification associated to the norm-square of the real moment map. Our approach also provides insight into the topology of specific examples of small-rank quiver varieties, including hyperpolygon spaces and some ADHM quivers.

I will discuss symplectic techniques that can be used to compute stringy invariants of orbifolds. The main examples include symplectic toric orbifolds, where we will see that the orbifold K-theory is torsion free. This talk is based on joint works with Goldin, Harada, Kimura, and Knutson.

Buchstaber conjectured that the cohomology of Landweber-Novikov algebra is generated by two elements with respect to non-trivial Massey products. Fukhs, Feigin and Retakh found realisations of all basic cocycles by means of trivial Massey products. Later, Buchstaber's PhD student Artelnykh found some non-trivial realisations but not for all cohomology. We will discuss a recent progress in the direction of the whole answer.

I will present an overview of current research on equivariant vector bundles on toric varieties, their Chern classes, positivity properties, and cohomology. This includes joint work with Milena Hering and Mircea Mustata.