Forthcoming Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar

Tue, 06/05/2008
15:45 		Alastair King (University of Bath) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Exceptional curves on del Pezzo surfaces
I plan to discuss some aspects the mysterious relationship between the symmetries of toroidal compactifications of M-theory and helices on del Pezzo surfaces.
Tue, 20/05/2008
15:45 		Thomas Nevins (UIUC) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Mirabolic Langlands duality and the Quantum Calogero-Moser system II
The geometric Langlands program aims at a "spectral decomposition" of certain derived categories, in analogy with the spectral decomposition of function spaces provided by the Fourier transform. I'll explain such a geometrically-defined spectral decomposition of categories for a particular geometry that arises naturally in connection with integrable systems (more precisely, the quantum Calogero-Moser system) and representation theory (of Cherednik algebras). The category in this case comes from the moduli space of vector bundles on a curve equipped with a choice of “mirabolic” structure at a point. The spectral decomposition in this setting may be understood as a case of “tamely ramified geometric Langlands”. In the talk, I won't assume any prior familiarity with the geometric Langlands program, integrable systems or Cherednik algebras.
Tue, 03/06/2008
14:15 		Yinan Song (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L1
Generalized Donaldson-Thomas invariants. I. An invariant counting pairs.
This is the first of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. We shall define invariants "counting" semistable coherent sheaves on a Calabi-Yau 3-fold. Our invariants are invariant under deformations of the complex structure of the underlying Calabi-Yau 3-fold, and have known transformation law under change of stability condition. This first seminar constructs an auxiliary invariant "counting" stable pairs (s,E), where E is a Gieseker semistable coherent sheaf with fixed Hilbert polynomial and s : O(-n) –> E for n >> 0 is a morphism of sheaves, and (s,E) satisfies a stability condition. Using Behrend-Fantechi's approach to obstruction theories and virtual classes we prove this auxiliary invariant is unchanged under deformation of the underlying Calabi-Yau 3-fold.
Tue, 03/06/2008
15:45 		Dominic Joyce (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Generalized Donaldson-Thomas invariants. II. Invariants and transformation laws.
This is the second of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. (Still work in progress.) Behrend showed that conventional Donaldson-Thomas invariants can be written as the Euler characteristic of the moduli space of semistable sheaves weighted by a "microlocal obstruction function" \mu. In previous work, the speaker defined Donaldson-Thomas type invariants "counting" coherent sheaves on a Calabi-Yau 3-fold using Euler characteristics of sheaf moduli spaces, and more generally, of moduli spaces of "configurations" of sheaves. However, these invariants are not deformation-invariant. We now combine these ideas, and insert Behrend's microlocal obstruction \mu into the speaker's previous definition to get new generalized Donaldson-Thomas invariants. Microlocal functions \mu have a multiplicative property implying that the new invariants transform according to the same multiplicative transformation law as the previous invariants under change of stability condition. Then we show that the invariants counting pairs in the previous seminar are sums of products of the new generalized Donaldson-Thomas invariants. Since the pair invariants are deformation invariant, we can deduce by induction on rank that the new generalized Donaldson-Thomas invariants are unchanged under deformations of the underlying Calabi-Yau 3-fold.

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

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