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| DECEMBER 2010 |
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| Statistical Mechanics Seminar |
| Topic: |
Hidden Symmetries at the Percolation Point in Two Dimensions |
| Presenter: |
Peter Kleban, University of Maine |
| Date: |
Wednesday, December 1, 2010, Time: 2:00 p.m., Location: Jadwin Hall 343 |
| Abstract: |
Percolation is perhaps the simplest non-trivial model in statistical mechanics, but has remained under active study for more than forty years. In 2-D it exhibits a second-order phase transition, at which a number of interesting and little-understood symmetries manifest themselves. We discuss three of these: (a) the horizontal crossing probability, which reveals a triangular symmetry, (b) an exact factorization of certain correlation functions, and (c) a generalization of this factorization that shows a mysterious independence of one coordinate. We demonstrate (c) via the explicit calculation of a certain six-point correlation function. Both (b) and (c) generalize to a variety of other two-dimensional critical points. The main tool employed is conformal field theory. |
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| Department Colloquium |
| Topic: |
Natural maps old and new |
| Presenter: |
Gerard Besson, Grenoble |
| Date: |
Wednesday, December 1, 2010, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
In 1995, G. Courtois, S. Gallot and myself constructed a family of maps with very good properties regarding volume elements between certain manifolds. We used it to give an alternative proof of Mostow's rigidity for rank one closed symmetric spaces as well as a rigidity result for their geodesic flow, conjectured by A. Katok. Various modifications of the original construction have been made since yielding new results in different settings. We shall describe the basic construction, the modifications, some applications and open questions. |
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| Symplectic Geometry Seminar |
| Topic: |
A conjecture of Arnold |
| Presenter: |
Heather Macbeth, Princeton University |
| Date: |
Thursday, December 2, 2010, Time: 1:30 p.m., Location: Fine Hall 601 |
| Abstract: |
The "chord conjecture" of Vladimir Arnold is a contact-geometry analogue of his well-known Lagrangian intersections conjecture in symplectic geometry. It proposes that, for each Legendrian submanifold of a contact form on a compact manifold, there should be a integral curve of the Reeb vector field which crosses the Legendrian submanifold at least twice. I will present the 2001 paper http://www.jstor.org/stable/3062116 of Klaus Mohnke, which proves this conjecture for a class of compact contact manifolds including spheres. |
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| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
Two results on rigidity of commutative actions by toral automorphisms |
| Presenter: |
Zhiren Wang
, Princeton University |
| Date: |
Thursday, December 2, 2010, Time: 2:00 p.m., Location: Fine Hall 401 |
| Abstract: |
In 1983 Berend proved rigidity of higher-rank commutative actions by toral automorphisms under some hyperbolicity and irreduciblity assumptions. We will present two rigidity results that respectively extend Berend's theorem to certain non-hyperbolic and reducible cases. We will also discuss some counterexamples of non-homogeneous orbit closures. This is joint work with Elon Lindenstrauss. |
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| Discrete Mathematics Seminar |
| Topic: |
Higher-order Fourier analysis of F_p^n and the complexity of systems of linear forms |
| Presenter: |
Shachar Lovett, Tel-Aviv University and IAS |
| Date: |
Thursday, December 2, 2010, Time: 2:15 p.m., Location: Fine Hall 224 |
| Abstract: |
We study the density of small linear structures (e.g. arithmetic progressions) in subsets A of the group F_p^n. It is possible to express these densities as certain analytic averages involving 1_A, the indicator function of A. In the higher-order Fourier analytic approach, the function 1_A is decomposed as a sum f_1+f_2 where f_1 is structured in the sense that it has a simple higher-order Fourier expansion, and f_2 is pseudorandom in the sense that the kth Gowers uniformity norm of f_2, dentoted \|f_2\|_{U^k}, is small for a proper value of k.
For a given linear structure, we find the smallest degree of uniformity k such that assuming that \|f_2\|_{U^k} is sufficiently small, it is possible to discard f_2 and replace 1_A with f_1, affecting the corresponding analytic average only negligibly. Previously, Gowers and Wolf solved this problem for the case where f_1 is a constant function. Furthermore, our result extends to analytic averages that involve more than one subset of F_p^n, and resolves an open problem posed by Gowers and Wolf.
Joint work with Hamed Hatami. |
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| Algebraic Topology Seminar |
| Topic: |
The EHP sequence and the Goodwillie tower |
| Presenter: |
Mark Behrens, MIT |
| Date: |
Thursday, December 2, 2010, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
The EHP sequence and the Goodwillie tower of the identity give two different spectral sequences for computing the unstable homotopy groups of spheres. I will explain how the two can be mixed, so that each provides information about the differentials in the other. To demonstrate the effectiveness of the techniques presented, the methods will be applied to recompute the 2-primary Toda range (first 20 unstable stems). |
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| Princeton University and IAS Number Theory Seminar |
| Topic: |
The Iwasawa Main Conjectures for Modular Forms |
| Presenter: |
Christopher Skinner, Princeton University and IAS |
| Date: |
Thursday, December 2, 2010, Time: 4:30 p.m., Location: Fine Hall 214 |
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| Topology Seminar |
| Topic: |
Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities. |
| Presenter: |
Sa'ar Hersonsky, University of Georgia |
| Date: |
Thursday, December 2, 2010, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Consider a planar, bounded, $m$-connected region $\Omega$, and let $\partial\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\partial\Omega$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$ singular flat surface tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$.
The subject has an interesting history that started with Dehn (1903). References may be found here: http://www.math.uga.edu/~saarh/Papers/Papers1.htm (#18 & #19). |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Differentiable rigidity with Ricci bounded below |
| Presenter: |
Gilles Courtois, École/ Polytechnique/ |
| Date: |
Friday, December 3, 2010, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
We consider a closed hyperbolic manifold $(N,h)$ of dimension $n\geq 3$ and a manifold $(M,g)$ with a degre one map $f:M \to N$. We will show that if $Ricci_g \geq -(n-1)$ and $Vol (M,g) \leq (1+\epsilon) Vol (N,h)$, then the manifolds $M$ and $N$ are diffeomorphic. The proof relies on Cheeger-Colding theory of limits of Riemannian manifolds under lower Ricci curvature bound. |
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| Analysis Seminar |
| Topic: |
Periodic DNLS: weighted Wiener measures, gauge transformation and almost global well-posedness |
| Presenter: |
Gigliola Staffilani, MIT |
| Date: |
Monday, December 6, 2010, Time: 4:00 p.m., Location: Fine Hall 314 |
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| PACM Colloquium |
| Topic: |
Diffusions Interacting Through Their Ranks, and the Stability of Large Equity Markets |
| Presenter: |
Ioannis Karatzas, Columbia University |
| Date: |
Monday, December 6, 2010, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: |
We introduce and study ergodic multidimensional di usion processes interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods. The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset. Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study. The methodologies used in this study touch upon the question of triple points for systems of interacting di usions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no e ect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of intersection local times. The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics, such as the Sherrington-Kirkpatrick model for spin-glasses. |
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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: |
TBA |
| Presenter: |
Andrew King, Columbia University |
| Date: |
Thursday, December 9, 2010, Time: 2:15 p.m., Location: Fine Hall 224 |
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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: |
TBA |
| 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: 4:00 p.m., Location: Fine Hall 224 |
| 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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| 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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| 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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