Past Algebraic and Symplectic Geometry Seminar

12 February 2008
14:45
Abstract
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).
  • Algebraic and Symplectic Geometry Seminar
29 January 2008
14:45
Dominic Joyce
Abstract
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.
  • Algebraic and Symplectic Geometry Seminar

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