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DECEMBER 2007 |
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Statistical Mechanics Seminar |
Topic: |
Random matrices, statistical mechanics and hyperbolic supersymmetry |
Presenter: |
Thomas Spencer, IAS |
Date: |
Wednesday, December 12, 2007, Time: 2:00 p.m., Location: Jadwin 343 |
Abstract: |
We present a statistical mechanics model with a hyperbolic supersymmetry. This model is expected to qualitatively describe properties of random band matrices in N dimensions eg localization and delocalization. The "spins" in this model may be thought of taking values in a Poincare super-disc. In three dimensions we show that this model has a diffusive phase. In one dimension there is only the localized phase. The analysis of this model relies a family of identities coming from SUSY together with estimates of a random walk on a percolation cluster. The surprising relation of this model to linearly reinforced random walk will also be highlighted. No knowledge of SUSY is needed. This is joint work with M. Disertori and M. Zirnbauer. |
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Ergodic Theory and Statistical Mechanics Seminar |
Topic: |
Attractors with Large Invisible Parts |
Presenter: |
Andrei Negut, Princeton University |
Date: |
Thursday, December 13, 2007, Time: 2:00 p.m., Location: Fine Hall 401 |
Abstract: |
Philosophically, an attractor of a dynamical system is a closed subset of the phase space which orbits "approach" as time goes to infinity. Different meanings of the word "approach" produce different versions of attractors: maximal, Milnor, statistical, minimal etc. The question of generic non-coincidence between these various types of attractors has not yet been answered. We will be concerned with a different point of view. Physically, if one looks at the attractor, then one knows where most orbits will go to as time goes to infinity. But it is possible that a large part of the attractor is redundant, in the sense that orbits spend very very little time near it. Thus, it would be more significant to look only at the non-redundant part of the attractor. Concretely, we will present an example of a random dynamical system, given by parameters of "reasonable magnitude" (e.g. 1000). For this dynamical system, roughly half of the attractor is "invisible" in the sense that orbits spend near it only a fraction of 2^{-500} of all time. The number 2^{-500} is equal to zero for all physical or computer experiments, and therefore an observer should not bother with the "invisible" half of the attractor. Moreover, small perturbations of this dynamical system exhibit the same property, and therefore the phenomenon is generic. |
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Discrete Mathematics Seminar |
Topic: |
Embedding a bounded degree tree into a dense graph |
Presenter: |
Endre Szemerédi, Rutgers University |
Date: |
Thursday, December 13, 2007, Time: 2:15 p.m., Location: Fine Hall 224 |
Abstract: |
We will show that if the minimum degree of a graph G is bigger than 1/2 n + logn (where n is the number of vertices of G) then it contains any bounded degree tree on n vertices. It is a joint work with Bela Csaba. |
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Topology Seminar |
Topic: |
Components of Springer fibers and Khovanov's arc algebra |
Presenter: |
Ben Webster, IAS |
Date: |
Thursday, December 13, 2007, Time: 4:30 p.m., Location: Fine Hall 314 |
Abstract: |
Using the structure of certain Springer fibers and their components, I'll describe a geometric construction of an algebra which is painfully close to being isomorphic to the arc algebra defined by Khovanov, but in fact, isn't. I'll then hopefully explain why this is actually a good thing. |
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Differential Geometry and Geometric Analysis Seminar |
Topic: |
On the $\sigma_2$-scalar curvature and its application |
Presenter: |
Yuxin Ge, University Paris 12 |
Date: |
Friday, December 14, 2007, Time: 3:00 p.m., Location: Fine Hall 314 |
Abstract: |
In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. This equation arises from conformal geometry. As application, we prove that, if a compact 3-dimensional manifold $M$ admits a riemannian metric with positive scalar curvature and $\int \sigma_2\ge 0$, then topologically $M$ is a quotient of sphere. |
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Topology Seminar |
Topic: |
Subdirect products of surfaces, homological finiteness, and residually-free groups |
Presenter: |
Martin Bridson, Imperial College |
Date: |
Thursday, December 20, 2007, Time: 4:30 p.m., Location: Fine Hall 314 |
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