Abstracts - Fall 2009 

Speaker: Dan Dugger 

Title:  Motivic stable homotopy groups

Abstract:  After the foundational work of Morel and Voevodsky, it became possible to define motivic stable homotopy groups of spheres over any ground field (or even ground scheme).  These generalize the classical stable homotopy groups of spheres from algebraic topology, but they are a finer invariant which contain more "algebraic" information, even when the field is the complex numbers.  I will give a gentle (I hope) introduction to these objects, and a survey about what is known about them.

Speaker: Tom Graber 

Title:  Relative Gromov-Witten invariants and Local Gromov-Witten invariants

Abstract:  I will describe joint work with B. Hassett which gives a very simple explanation and generalization of a fact first conjectured by Takahashi and proven by Gathmann - the virtual number of maps from A^1 to the complement of a smooth plane cubic is the same (up to a simple factor) as the virtual number of maps from P^1 to the canonical bundle over P^2.

Speaker: Mihai Putinar 

Title: A duality approach to Positivstellensatze in real algebraic geometry

Abstract:  Positivstellensatze in real algebraic geometry were predicted to exist more than half a century time ago by Tarski.  Their precise formulation was found only much later, sometimes indirectly and most of the time accidentally. A general framework to derive all such positivity certificates was developed only in the last decade, as a result of a classical duality scheme, having moment problems (of positive measures, or C*-algebra states) at the center.  The transition to positivity in non-commutative *-algebras is then very natural. Applications to polynomial optimization will be discussed.

Speaker: Ravi Vakil 

Title: The ring of invariants of n points on the projective line

Abstract: The GIT quotient of a small number of points on the projective line has long been known to have beautiful geometry. For example, the case of six points is intimately connected to the outer automorphism of S_6. We extend this picture to an arbitrary number of points, completely describing the equations of the moduli space. (In some sense there is only one equation.) The case of eight points is particularly entertaining. This is joint work with Ben Howard, John Millson, and Andrew Snowden.

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