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| DECEMBER 2010 |
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| Statistical Mechanics Seminar |
| Topic: |
Nonequilibrium: Thermostats, BBGKY Hierarchy, Fourier's Equation |
| Presenter: |
Giovanni Gallavotti, Rutgers University |
| Date: |
Wednesday, December 8, 2010, Time: 2:00 p.m., Location: Jadwin Hall 343 |
| Abstract: |
Review of rigorous results on thermostats. Families of exact formal solutions of the BBGKY hierarchy for hard sphere systems with free boundary conditions at collisions and Fourier equation emergence, to first order in the temperature difference, after boundary conditions are imposed. Formal means that the solutions are given by series with well defined terms but whose convergence is not discussed. |
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| Symplectic Geometry Seminar |
| Topic: |
Aspects of stringy global quotients, de Rham, singularities and gerbes |
| Presenter: |
Ralph Kaufmann, Perdue University |
| Date: |
Thursday, December 9, 2010, Time: 1:30 p.m., Location: Fine Hall 601 |
| Abstract: |
We discuss stringy functors from the pull back point of view of Jarvis-K-Kimura and the push forward point of view of our orbifold Milnor ring constructions. We show how these approaches merge to give a de Rham theory and apply back to singularity theory. If time allows, we also give results on global gerbe twists and the Drinfel'd Double. |
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| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
On the distribution of gaps for saddle connection directions |
| Presenter: |
Jayadev Athreya, University of Illinois Urbana-Champaign |
| Date: |
Thursday, December 9, 2010, Time: 2:00 p.m., Location: Fine Hall 401 |
| Abstract: |
In joint work with J. Chaika, we prove results on the distribution of gaps of angles between saddle connections on flat surfaces. Our techniques draw on the work of Marklof-Strombergsson on the periodic Lorentz gas and that of Eskin-Masur on flat surfaces. We describe some applications to billiards in polygons. |
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| Discrete Mathematics Seminar |
| Topic: |
Proving the Lovász-Plummer conjecture |
| Presenter: |
Andrew King, Columbia University |
| Date: |
Thursday, December 9, 2010, Time: 2:15 p.m., Location: Fine Hall 224 |
| Abstract: |
In the 1970s, Lovász and Plummer conjectured that every cubic bridgeless graph has exponentially many perfect matchings with respect to the number of vertices. The conjecture was proven by Voorhoeve for bipartite graphs and by Chudnovsky and Seymour for planar graphs. In this talk I will describe our proof of the general case, which uses elements of both aforementioned partial results as well as Edmonds' characterization of the perfect matching polytope. This is joint work with Louis Esperet, Frantisek Kardos, Daniel Kral, and Sergey Norin. |
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| Algebraic Topology Seminar |
| Topic: |
Cohomology of graph products of infinite groups with group ring coefficients |
| Presenter: |
Mike Davis, Ohio State University |
| Date: |
Thursday, December 9, 2010, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
I will explain a computation of the cohomology of any graph product of infinite groups in terms of the factor groups. For example, this gives a calculation for right-angled Artin groups, which are, by definition, graph products of copies of the infinite cyclic group. The method of proof is a simple spectral sequence argument which I don't think has been used previously. |
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| Princeton University and IAS Number Theory Seminar |
| Topic: |
Parahoric subgroups and supercuspidal representations of p-adic groups |
| Presenter: |
Benedict Gross, Harvard University |
| Date: |
Thursday, December 9, 2010, Time: 4:30 p.m., Location: IAS S-101 |
| Abstract: |
This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this induced representation can (in certain situations) have finite length. I will describe the parahorics and characters for which this occurs, and what the Langlands parameters of the corresponding irreducible summands must be. |
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| Topology Seminar |
| Topic: |
Gromov's knot distortion |
| Presenter: |
John Pardon, Princeton University |
| Date: |
Thursday, December 9, 2010, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Gromov defined the distortion of an embedding of S^1 into R^3 and asked whether every knot could be embedded with distortion less than 100. There are (many) wild embeddings of S^1 into R^3 with finite distortion, and this is one reason why bounding the distortion of a given knot class is hard. I will show how to give a nontrivial lower bound on the distortion of torus knots. I will also mention some natural conjectures about the distortion, for example that the distortion of the (2,p)-torus knots is unbounded. |
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| Special Seminar |
| Topic: |
Geometric problems arising in theoretical computer science |
| Presenter: |
Zeev Dvir, Princeton University |
| Date: |
Friday, December 10, 2010, Time: 2:30 p.m., Location: Fine Hall 314 |
| Abstract: |
In this talk I will demonstrate two cases where basic geometric questions, regarding intersections of lines, come up in theoretical computer science. In the first part of the talk I will discuss the finite field kakeya problem. This is a problem regarding intersections of lines in different directions which originated in real analysis and arose independently in computer science in relation to explicit constructions of certain pseudo-random graphs. In the second part of the talk I will discuss certain robust generalizations of the Sylvester-Gallai theorem. In these type of problems, combinatorial information about intersections of lines is transformed into dimension bounds. These type of questions come up in computer science when studying properties of special families of error-correcting codes. In both of these cases tools (mostly algebraic) and intuitions from theoretical computer science have proven to be quite useful in making progress. |
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| Differential Geometry and Geometric Analysis Seminar ***Please note special time |
| Topic: |
Metric flips with Calabi symmetry |
| Presenter: |
Yuan Yuan, Johns Hopkins |
| Date: |
Friday, December 10, 2010, Time: 3:30 p.m., Location: Fine Hall 314 |
| Abstract: |
I will discuss the metric behavior of the Kahler-Ricci flow on $\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus (m+1)})$, assuming that the initial metric satisfies the symmetry defined by Calabi. I will describe the Gromov-Hausdorff limit of the flow as time approaches the singular time and how the Kahler-Ricci flow can be continued. This is a joint work with Jian Song. |
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| Differential Geometry and Geometric Analysis Seminar ***Please note special time |
| Topic: |
Initial time singularities for mean curvature flow |
| Presenter: |
Tom Ilmanen, ETH Zürich |
| Date: |
Friday, December 10, 2010, Time: 4:30 p.m., Location: Fine Hall 314 |
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| Special Group Actions Seminar |
| Topic: |
Deformation of compact quotients of homogeneous spaces |
| Presenter: |
Fanny Kassel, University of Chicago |
| Date: |
Monday, December 13, 2010, Time: 12:00 p.m., Location: McDonnell 201A |
| Abstract: |
Many mathematicians have worked on the problem of determining which homogeneous spaces G/H admit proper and cocompact actions by discrete groups Gamma. This question is highly nontrivial when H is noncompact, and still far from being solved. I will consider homogeneous spaces G/H that do admit such actions and examine the deformation of the compact quotients Gamma\G/H. I will prove that for most known examples with G and H reductive, the proper discontinuity of the action is preserved under any small deformation of Gamma in G. For G/H=SO(2,2)/SO(1,2), this is related to the existence of Thurston's asymmetric distance on Teichmuller space. I will also address similar questions in the setting of p-adic homogeneous spaces. |
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| PACM Colloquium |
| Topic: |
Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles |
| Presenter: |
Sal Torquato, Princeton University |
| Date: |
Monday, December 13, 2010, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: |
I reformulate the covering and quantizer problems, well-known problems in discrete geometry, as the determination of the ground states of interacting particles in d-dimensional Euclidean space that generally involve single-body, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the "void" nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplifies the deep interplay between geometry and physics, allow one now to employ optimization techniques to analyze and solve these energy minimization problems. The covering and quantizer problems have relevance in numerous applications, including wireless communication network layouts, the search of high-dimensional data parameter spaces, stereotactic radiation therapy, data compression, digital communications, meshing of space for numerical analysis, and coding and cryptography, among other examples. The connections between the covering and quantizer problems and the sphere-packing and number-variance problems (related to problems in number theory) are discussed. I also show that disordered saturated sphere packings provide relatively thin (economical) coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions. I derive improved upper bounds on the quantizer error using sphere-packing solutions, which are generally substantially sharper than an existing upper bound in low to moderately large dimensions. I demonstrate that disordered saturated sphere packings yield relatively good quantizers. Finally, I remark on possible applications of the results to the detection of gravitational waves. |
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| Discrete Mathematics Seminar ***Please note special date and time |
| Topic: |
Concrete mathematical incompleteness |
| Presenter: |
Harvey Friedman, Columbia University |
| Date: |
Tuesday, December 14, 2010, Time: 10:00 a.m., Location: Fine Hall 224 |
| Abstract: |
An unprovable theorem is a theorem about basic mathematical objects that can only be proved using more than the usual axioms for mathematics (ZFC = Zermelo Frankel set theory with the Axiom of Choice) - and that has been proved using standard extensions of ZFC generally adopted in the mathematical logic community. The highlight of the talk is the presentation of a new unprovable theorem concerning the structure of maximal cliques in certain graphs on Cartesian powers of the rational numbers. We first review some previous examples of unprovable theorems. 1-5 are unprovable in the weaker sense that any proof demonstrably requires some use of logical principles transcendental to the problem statement. These previous contexts include 1. Patterns in finite sequences from a finite alphabet. 2. Pointwise continuous embeddings between countable sets of reals (or, more concretely, rationals). 3. Relations between f(n_1,...,n_k) and f(n_2,...,n_k+1). 4. Homeomorphic embeddings between finite trees. 5. Borel sets in the plane and graphs of one dimensional Borel functions. 6. Boolean relations between sets of integers and their images under integer functions. |
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| Algebraic Geometry Seminar |
| Topic: |
Restriction varieties and geometric branching rules |
| Presenter: |
Izzet Coskun, UIC |
| Date: |
Tuesday, December 14, 2010, Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: |
In representation theory, a branching rule describes the decomposition of the restriction of an irreducible representation to a subgroup. Let $i: F' \rightarrow F$ be the inclusion of a homogeneous variety in another homogeneous variety. The geometric analogue of the branching problem asks to calculate the induced map in cohomology in terms of the Schubert bases of $F$ and $F'$. In this talk, I will give a positive, geometric rule for computing the branching coefficients for the inclusion of an orthogonal flag variety in a Type-A flag variety. The geometric rule has many applications including to the restrictions of representations of $SL(n)$ to $SO(n)$, to the study of the moduli spaces of rank 2 vector bundles on hyperelliptic curves and to presentations of the cohomology ring of orthogonal flag varieties. |
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| Mathematical Physics Seminar |
| Topic: |
Geometric methods for nonlinear quantum many-body systems |
| Presenter: |
Mathieu Lewin, Universite de Cergy-Pointoise |
| Date: |
Tuesday, December 14, 2010, Time: 4:30 p.m., Location: Jadwin 343 |
| Abstract: |
Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this talk I will present a formalism which also allows to study nonlinear systems. I will in particular define a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. As an application I prove the existence of multi-polaron systems in the Pekar- Tomasevich approximation, in a certain regime for the coupling constant. |
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| Special Seminar |
| Topic: |
Erdos distinct distance problem in the plane |
| Presenter: |
Larry Guth, IAS and U. Toronto |
| Date: |
Wednesday, December 15, 2010, Time: 4:00 p.m., Location: Fine Hall 314 |
| Abstract: |
Erdos conjectured that N points in the plane determine at least c N (log N)^{-1/2} different distances. Recently Nets Katz and I came close to proving the conjecture, showing that the number of distinct distances is at least c N (log N)^{-1}.
Elekes and Sharir made a connection between the distinct distance problem and incidence geometry - the study of intersection patterns of points and lines. There has been a lot of progress in this area over the last few years starting with Dvir's solution of the Kakeya problem in finite fields using the polynomial method. The new wrinkle in our proof is a way to mix polynomial methods with topological methods. |
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| Discrete Mathematics Seminar |
| Topic: |
TBA |
| Presenter: |
Menachem Kojman, Ben Gurion University and IAS |
| Date: |
Thursday, December 16, 2010, Time: 2:15 p.m., Location: Fine Hall 224 |
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| Joint Princeton University and IAS Number Theory Seminar |
| Topic: |
Weyl's sums for roots of quadratic congruences |
| Presenter: |
Henryk Iwaniec, Rutgers University |
| Date: |
Thursday, December 16, 2010, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
It is known that the roots of congruences for a fixed irreducible quadratic polynomial are equidistributed. This statement translates to getting cancellation in the corresponding sum of Weyl's sums. In a recent work by W. Duke, J. Friedlander and H. Iwaniec we succeeded to get cancellation (so also the equidistribution) in very short sums of Weyl's sums relatively to the discriminant. The spectral theory of metaplectic automorphic forms is the basic tool, of which some special aspects will be the subject of this talk. Numerous applications of the result will be also discussed. |
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| Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Andy Cotton-Clay, Harvard University |
| Date: |
Thursday, December 16, 2010, Time: 4:30 p.m., Location: Fine Hall 314 |
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