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Mathematics Colloquium
2008 - 2009

Wednesday, November 5, 2008
4:15 p.m.  //  151 Sloan
Monica Visan (University of Chicago)

Nonlinear Schrodinger equations at critical regularity.

Abstract: We introduce the nonlinear Schrodinger equation (NLS) and define criticality.
We then survey the history of the two most studied cases of critical NLS, namely, the mass-critical NLS and the energy-critical NLS. This includes recent joint work with Rowan Killip, Terry Tao and Xiaoyi Zhang.

 

Thursday, November 6, 2008
4:15 p.m.  //  151 Sloan
Rowan Killip (UCLA)

Some operators with random-matrix eigenvalue statistics

Abstract: Several deterministic (i.e. manifestly non-random) sequences have been observed numerically to exhibit the same local behaviour as the eigenvalues of a random matrix. One example is characteristic frequencies of certain oddly shaped drums.
In this talk I will describe some individual operators (of less intrinsic interest) which have the same local eigenvalue statistics. Much of the talk will be devoted to the journey leading to this result, including the suprising role of numerical linear algebra.

 

 

Tuesday, November 11, 2008
4:15 p.m.  //  151 Sloan
Sourav Chatterjee (Berkeley)

A rigorous theory of chaos in disordered systems

Abstract: Disordered systems are an important class of models in statistical mechanics, having the defining characteristic that the energy landscape is a fixed realization of a random field. Examples include various models of glasses and polymers. They also arise in other subjects, like fitness models in evolutionary biology. The ground state of a disordered system is the state with minimum energy. The system is said to be chaotic if a small perturbation of the energy landscape causes a drastic shift of the ground state. In this talk I will present a rigorous theory of chaos in disordered systems that confirms long-standing physics intuition about connections between chaos, anomalous fluctuations of the ground state energy, and the
existence of multiple valleys in the energy landscape. Combining these results with mathematical tools like hypercontractivity, I will present a proof of the existence of chaos in directed polymers. This is the first rigorous proof of chaos in any nontrivial disordered
system. Applications to other models like spin glasses, fitness models, and general Gaussian fields will also be discussed.
 

 

Tuesday, November 18, 2008
4:15 p.m.  //  151 Sloan
Yi Ni (
MIT)

Floer homology and fibered 3-manifolds

Abstract: There are several Floer homologies of 3-manifolds defined using gauge theory and symplectic geometry. Three of them are Instanton Floer homology, Monopole Floer homology and Heegaard Floer homology. It turns out that each of these three homologies determines whether a 3-manifold is a surface bundle over the circle. I will give a survey on this topic,
which will cover the works of Ghiggini, Ni, Juhasz, Kronheimer and Mrowka.
 

 


Tuesday, November 25, 2008

4:15 p.m.  //  151 Sloan
Adrian Ioana (
Caltech and CMI)

Rigidity in orbit equivalence and von Neumann algebras.

Abstract: The first examples of von Neumann algebras came from actions of countable groups on probability spaces. As it turns out, their study (up to
isomorphism) is closely related to the study of countable equivalence relations (up to orbit equivalence). Recently, this connection proved to be very useful, leading to remarkable rigidity results in both orbit equivalence and von Neumann algebras theory. In this talk, I will survey some of these results including: existence of II_1 factors without symmetries, new cocycle superrigidity results, and existence of non-orbit equivalent actions for arbitrary non-amenable groups.
 

 

Tuesday, February 17, 2009
4:15 p.m.  //  151 Sloan
Zeev Dvir (Institute for Advanced Study)

The finite field Kakeya problem  


Abstract: The finite field Kakeya problem deals with finding lower bounds on the size of sets in a vector space over a finite field that contain a line in every direction. This problem was introduced by Wolff in 1999 and is connected to several problems in analysis, combinatorics and theoretical computer science.

In this talk I will survey recent progress on this problem [Dvir 08, Saraf Sudan 08, Dvir Kopparty Saraf Sudan 09] which give nearly optimal bounds on the size of Kakeya sets.

 

 


 

 



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