• Vanishing cycles and their algebraic computation, Claude Sabbah (Ecole Polytechnique).
In these lectures, we review recent results concerning the algebraic computation of vanishing cycles of an algebraic function on a complex quasi-projective variety. The first lecture presents various constructions of the complex of vanishing cycles and its fundamental properties. In the second lecture, we consider the case of a projective function and we explain an algebraic formula of Barannikov and Kontsevich for computing the dimension of each vanishing hypercohomology space. Lastly, in the third lecture, we relax the assumption of projectivity of the function and we focus on the algebraic computation of the monodromy.
LECTURE NOTES *link will be available beginning Friday 5-24-13
• Donaldson-Thomas invariants and the motivic Milnor fiber, Balázs Szendrői (Oxford).Moduli of sheaves on varieties and their cohomology and counting invariants; sheaf counting on Calabi-Yau manifolds: Donaldson-Thomas theory and the Behrend function; motivic and cohomological vanishing cycles and the corresponding variants of Donaldson-Thomas theory; examples from local geometries.
• Monodromy Conjecture, François Loeser (UPMC).
Lecture 1 - Milnor fiber, vanishing cycles, monodromy
Lecture 2 - Motivic Igusa zeta function, motivic Milnor fiber, monodromy conjecture
Lecture 3 - The trace of the monodromy
• Motivic integration, Tommaso de Fernex (Utah).1. Jet schemes and arc spaces of smooth varieties 2. Grothendieck ring of varieties and the motivic ring 3. Motivic integration on smooth varieties 4. Change-of-variable formula for birational maps between smooth varierties 5. Application to Batyrev conjecture on the Hodge numbers of CY's 6. (Time Permitting) Divisorial valuations via contact loci in arc spaces
• Nash Conjecture, Javier Fernández de Bobadilla (Madrid).The proof of the Nash problem for surfaces and its background, higher dimensional counterexamples, and possible higher dimensional formulations.
Please feel free to contact us at cmnd@nd.edu with any questions.