Forthcoming Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar

Tue, 22/01/2008
14:45 		Dominic Joyce (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Kuranishi bordism and Kuranishi homology, Part I.
A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $ J $-holomorphic curves in symplectic geometry. Let $ Y $ be an orbifold, and $ R $ a commutative ring. We shall define four topological invariants of $ Y $: two kinds of Kuranishi bordism ring $ KB_*(Y;R) $, and two kinds of Kuranishi homology ring $ KH_*(Y;R) $. Roughly speaking, they are spanned over $ R $ by isomorphism classes $ [X,f] $ with various choices of relations, where $ X $ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $ f:X\rightarrow Y $ is a strong submersion. Our main result is that weak Kuranishi homology is isomorphic to the singular homology of $ Y $. These theories are powerful tools in symplectic geometry for several reasons. Firstly, using them eliminates the issues of virtual cycles and perturbation of moduli spaces, yielding technical simplifications. Secondly, as $ KB_*,KH_*(Y;R) $ are very large, invariants defined in these groups contain more information than invariants in conventional homology. Thirdly, we can define Gromov-Witten type invariants in Kuranishi bordism or homology groups over $ \mathbb Z $, not just $ \mathbb Q $, so they can be used to study the integrality properties of Gromov-Witten invariants. This is the first of two talks. Today we deal with motivation from symplectic geometry, and Kuranishi bordism. Next week's talk discusses Kuranishi homology.
Tue, 29/01/2008
14:45 		Dominic Joyce (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Kuranishi bordism and Kuranishi homology, Part II.
This is the second of two talks, and probably will not be comprehensible unless you came to last week's talk. A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $ J $-holomorphic curves in symplectic geometry. Let $ Y $ be an orbifold, and $ R $ a commutative ring. We define four topological invariants of $ Y $: two kinds of Kuranishi bordism ring $ KB_*(Y;R) $, and two kinds of Kuranishi homology ring $ KH_*(Y;R) $. Roughly speaking, they are spanned over $ R $ by isomorphism classes $ [X,f] $ with various choices of relations, where $ X $ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $ f:X\rightarrow Y $ is a strong submersion. These theories are powerful tools in symplectic geometry. Today we discuss the definition of Kuranishi homology, and the proof that weak Kuranishi homology is isomorphic to the singular homology.
Tue, 12/02/2008
14:45 		Ivan Losev (Belarusian State University and University of Manchester) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Uniqueness property for smooth affine spherical varieties
Let G be a connected reductive algebraic group over an algebraically closed field of characteristic 0. A normal irreducible G-variety X is called spherical if a Borel subgroup of G has an open orbit on X. It was conjectured by F. Knop that two smooth affine spherical G-varieties are equivariantly isomorphic provided their algebras of regular functions are isomorphic as G-modules. Knop proved that this conjecture implies a uniqueness property for multiplicity free Hamiltonian actions of compact groups on compact real manifolds (the Delzant conjecture). In the talk I am going to outline my recent proof of Knop's conjecture (arXiv:math/AG.0612561).
Tue, 26/02/2008
14:45 		Raphael Rouquier (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar
Local non-commutative Calabi-Yaus

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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