11.0012:00

Stephen Theriault (Aberdeen)
Homotopy types of polyhedral products for shifted
complexes
We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a
shifted simplicial complex on n vertices, X_1, ... , X_n are spaces and
CX_i is the cone on X_i, then the polyhedral product determined by K and
the pairs (CX_i,X_i) is homotopy equivalent to a wedge of suspensions of
smashes of the X_i's. This is joint work with Jelena Grbic, and it
generalises our earlier work in the case when each X_i is a loop space.
Connections are made to toric topology, combinatorics, and classical
homotopy theory.

2:003:00

David Quinn (Belfast)
Generalized finite domination via projective toric schemes
A finite f.g. free R[T,T^1]module chain complex C is
said to be _Rfinitely dominated_ if it is
Rmodule chain equivalent to a finite chain complex of f.g. projective
Rmodules. Using purely algebraic methods, Ranicki gives a condition for
finite domination when T is a single indeterminate. We describe how this
condition may be recovered using toric methods, and how these methods can
be used to obtain conditions for finite domination when T is a set of
finitely many indeterminates.
This talk is based on joint work with Thomas H\"uttemann.

4:005:00 
Taras Panov (Moscow State University)
Geometric structures on momentangle manifolds
Momentangle complexes are spaces acted on by a torus and parametrised by
finite simplicial complexes. They are central objects in toric topology,
and currently are gaining much interest
in the homotopy theory. Due to their combinatorial origins, momentangle
complexes also find applications in combinatorial geometry and commutative
algebra. Momentangle complexes corresponding to simplicial subdivision of
spheres are topological manifolds, and those
corresponding to simplicial polytopes admit smooth realisations as
intersection of real quadrics in C^m.
After an introductory part describing the general properties of
momentangle complexes we shall concentrate on the complexanalytic and
Lagrangian aspects of the theory.
We show that the momentangle manifolds corresponding to complete
simplicial fans admit nonKaehler complexanalytic structures. This
generalises the known construction of complexanalytic structures on
polytopal momentangle manifolds, coming from identifying them as LVMmanifolds.
We proceed by describing the Dolbeault cohomology and certain Hodge
numbers of momentangle manifolds by applying the Borel spectral sequence
to holomorphic principal bundles over toric varieties.
A new wide family of minimal Lagrangian submanifolds N in C^m or CP^m can
be constructed from intersections of real quadrics. These submanifolds
have the following topological properties: every N embeds in the
corresponding momentangle manifold Z, and every N is the total space of
two different fibrations, one over the torus T^{mn} with fibre a real
momentangle manifold R, and another over a small cover with fibre a torus.
These properties are used to produce new examples of Lagrangian
submanifolds with quite complicated topology.
Different parts of this talk are based on joint works with Victor
Buchstaber, Andrei Mironov and Yuri Ustinovsky.
