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DECEMBER 2009 |
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| Department Colloquium |
| Topic: |
Random Matrices: Universality of Local Eigenvalues Statistics |
| Presenter: |
Van Vu, Rutgers University |
| Date: |
Wednesday, December 9, 2009, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
One of the main goals of the theory of random matrices is to establish the limiting distributions of the eigenvalues. In the 1950s, Wigner proved his famous semi-cirle law (subsequently extended by Anord, Pastur and others), which established the global distribution of the eigenvalues of random Hermitian matrices. In the last fifty years or so, the focus of the theory has been on the local distributions, such as the distribution of the gaps between consecutive eigenvalues, the k-point correlations, the local fluctuation of a particular eigenvalue, or the distribution of the least singular value. Many of these problems have connections to other fields of mathematics, such as combinatorics, number theory, statistics and numerical linear algebra.
Most of the local statistics can be computed explicitly for random matrices with gaussian entries (GUE or GOE), thanks to Ginibre's formulae of the joint density of eigenvalues. It has been conjectured that these results can be extended to other models of random matrices. This is generally known as the Universality phenomenon, with several specific conjectures posed by Wigner, Dyson, Mehta etc.
In this talk, we would like to discuss recent progresses concerning the Universality phenomenon, focusing on a recent result (obtained jointly with T. Tao), which asserts that all local statistics of eigenvalues of a random matrix are determined by the first four moments of the entries. This (combining with results of Johansson, Erdos-Ramirez-Schlein-Yau and many others) provides the answer to several old problems. |
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| Discrete Mathematics Seminar |
| Topic: |
Proof of the Bollobas-Catlin-Eldridge conjecture |
| Presenter: |
Gabor Kun, IAS |
| Date: |
Thursday, December 10, 2009, Time: 2:15 p.m., Location: Fine Hall 224 |
| Abstract: |
We say that two graphs G and H pack if G and H can be embedded into the same vertex set such that the images of the edge sets do not overlap. Bollobas and Eldridge, and independently Catlin conjectured that if the graphs G and H on n vertices with maximum degree M(G) and M(H), respectively, satisfy (M(G) + 1)(M(H) + 1) ≤ n + 1 then G and H pack. Aigner and Brandt and, independently, Alon and Fischer proved this in the case M(G),M(H) < 3, Csaba, Shokoufandeh and Szemeredi proved the conjecture if M(G),M(H) < 4. Bollobas, Kostochka and Nakprasit settled the case when one of the graphs is degenerate. Kaul, Kostochka and Yu showed that if M(G)M(H) < 3/5n and the maximal degrees are large enough then G and H pack. We prove the conjecture for graphs with at least 10^8 vertices. |
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| Algebraic Topology Seminar |
| Topic: |
Periodicity and Duality |
| Presenter: |
John Klein, Wayne State University |
| Date: |
Thursday, December 10, 2009, Time: 3:00 p.m., Location: Fine Hall 1203 |
| Abstract: |
This talk will produce periodic families of Poincare duality spaces, giving a partial answer to a problem posed in the proceedings of the 1982 Northwestern homotopy theory conference. The ideas will also relate James Periodicity to the four-fold periodicity of the surgery obstruction groups. |
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| Joint IAS/Princeton University Number Theory Seminar |
| Topic: |
An effective proof of the Oppenheim Conjecture |
| Presenter: |
Elon Lindenstrauss, Princeton University |
| Date: |
Thursday, December 10, 2009, Time: 4:30 p.m., Location: IAS S-101 |
| Abstract: |
In the mid 80's Margulis proved the Oppenheim Conjecture regarding values of indefinite quadratic forms. I will present new work, joint with Margulis, where we quantify this statement by giving bounds on the size of integer vectors for which |Q(x)|<epsilon for an irrational indefinite quadratic form Q in three variables. |
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| Topology Seminar |
| Topic: |
Bundle structures and Algebraic K-theory |
| Presenter: |
John Klein, Wayne State University |
| Date: |
Thursday, December 10, 2009, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
This talk will describe (Waldhausen type) algebraic K-theoretic obstructions to lifting fibrations to fiber bundles having compact smooth/topological manifold fibers.The surprise will be that a lift can often be found in the topological case. Examples will be given realizing the obstructions. |
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| Ergodic Theory and Statistical Mechanics Seminar ***Please note special day |
| Topic: |
Diophantine Properties of Dynamical Systems and IETs |
| Presenter: |
Michael Boshernitzan, Rice University |
| Date: |
Friday, December 11, 2009, Time: 2:00 p.m., Location: Fine Hall 401 |
| Abstract: |
The lecture is based on a recent preprint with the same title, joint with J. Chaika and put recently on arXiv. One of the results is that for ergodic IETs (Interval Exchange Transformations) almost sure $\liminf_{n\to\infty}\limits n|T^nx-y|=0$.
The result is optimal in two ways: \\ (1) the normalizing factor \ $n$ \ cannot be improved, even for rotations;\\ (2) the assumption of ergodicity cannot be replaced by just minimality. |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Uniqueness of constant scalar curvature K\"ahler metrics |
| Presenter: |
Song Sun, Wisconsin |
| Date: |
Friday, December 11, 2009, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
We will show that constant scalar curvature K\"ahler(cscK) metric "adjacent" to a given K\"ahler class is unique up to isomorphism. This generalizes the previous uniqueness theorems of Chen-Tian and Donaldson, where the complex structure is fixed. This is joint work with X-X. Chen. |
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| Special Group Actions Seminar ***Please note special date and time |
| Topic: |
Expansion in SL(d, Z/qZ), q square-free |
| Presenter: |
Peter Varju, Princeton University |
| Date: |
Friday, December 11, 2009, Time: 3:30 p.m., Location: Fine Hall 322 |
| Abstract: |
I discuss the problem whether certain Cayley graphs form an expander family. A family of graphs is called an expander family, if the number of edges needed to be deleted from any of the graphs to make it disconnected is at least a constant multiple of the size of the smallest component we get. Let S be a subset of SL(d, Z) closed for taking inverses. For each square-free integer q consider the graph whose vertex-set is SL(d, Z/qZ) two of which is connected by an edge precisely if we can get one from the other by left multiplication by an element of S. Bourgain, Gamburd and Sarnak proves that if d = 2 and S generates a Zariski dense subgroup of SL2, then these graphs form an expander family. In the talk I outline a modification of their argument which leads to a simpler proof and allows a generalization to d = 3 or to general number fields. |
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| Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Victor Lie, Princeton University |
| Date: |
Monday, December 14, 2009, Time: 4:00 p.m., Location: Fine Hall 110 |
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| Analysis Seminar ***Please note special time |
| Topic: |
Radiation field for Einstein Vacuum equations |
| Presenter: |
Fang Wang, MIT |
| Date: |
Monday, December 14, 2009, Time: 5:00 p.m., Location: Fine Hall 110 |
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| Algebraic Geometry Seminar |
| Topic: |
TBA |
| Presenter: |
Rahul Pandharipande, Princeton University |
| Date: |
Tuesday, December 15, 2009, Time: 4:30 p.m., Location: Fine Hall 322 |
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| Department Colloquium |
| Topic: |
Ideas around Symplectic Field Theory |
| Presenter: |
Helmut Hofer, IAS |
| Date: |
Wednesday, December 16, 2009, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Symplectic Field Theory (SFT) is the study of (pseudo-)holomorphic curves in symplectic cobordisms and contains Gromov-Witten theory and symplectic Floer theory as special cases. The algebraic invariants of SFT are obtained by a simultaneous study of infinitely many interdependent first order elliptic systems which exhibit compactness and transversality issues. A treatment of SFT with classical (nonlinear) Fredholm theory, though possible, would be extremely cumbersome. This lead to the development of a new generalized Fredholm theory in a new class of general spaces called polyfolds. In the talk a certain number of ideas are described which might be also useful in different contexts. |
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| Discrete Mathematics Seminar |
| Topic: |
TBA |
| Presenter: |
Alexandra Ovetsky Fradkin, Princeton University |
| Date: |
Thursday, December 17, 2009, Time: 2:15 p.m., Location: Fine Hall 224 |
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| Topology Seminar |
| Topic: |
Bordered Floer homology and factoring mapping classes |
| Presenter: |
Peter Ozsvath, Columbia University |
| Date: |
Thursday, December 17, 2009, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
I will discuss "bordered Floer homology", an invariant for three-manifolds with parameterized boundary. The theory associates a differential graded algebra to a (parameterized) surface; and a module over that algebra to a three-manifold which bounded by the surface. I will describe this construction, and then focus on computational aspects of this theory, including an algorithm for calculating HF-hat of closed three-manifolds, obtained by factoring mapping classes. This is joint work with Robert Lipshitz and Dylan Thurston. |
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