| FEBRUARY 2012 |
| |
| Department Colloquium |
| Topic: |
Congruent numbers and Heegner points |
| Presenter: |
S. Zhang, Princeton University |
| Date: |
Wednesday, February 8, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
An anonymous Arab manuscript, written before 972, contains a `` problem of congruent numbers": given an integer n, to find a rational square x^2 such that x^2+n and x^2-n are both rational squares. For example 1, 2, 3 are not congruent numbers but 5, 6, 7 are. A modern equivalence of this problem is to find a point with infinite order on the elliptic curve: y^2=x^4-n^2. A special case of the Birch and Swinnerton--Dyer conjecture assets that any positive integer n=5, 6, 7 mod 8 are congruent number while almost all n=1, 2, 3 mod 8 are not congruent. |
| |
| Discrete Mathematics Seminar |
| Topic: |
New classes of tournaments satisfying the Erdos-Hajnal conjecture |
| Presenter: |
Krzysztof Choromanski, Columbia University |
| Date: |
Thursday, February 9, 2012, Time: 2:15 p.m., Location: Fine Hall 224 |
| Abstract: |
The Erdos-Hajnal conjecture states that for every graph H there exists a constant c>0 such that every graph G that does not contain H as an induced subgraph contains a clique or a stable set of size at least |G|^c. The conjecture is still open. However some time ago a version for tournaments was proven to be equivalent to the original. In the tournament version, graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. Both the tournament and the graph versions of the conjecture are known to be true for some small graphs (or tournaments) H, and there are operations that build bigger graphs (or tournaments) for which the conjecture holds. I will show the conjecture for an infinite class of tournaments that is not obtained in the way described above. To the best of our knowledge, this is the first result of this kind. The only five-vertex tournament for which the conjecture was open was the tournament C_5 in which every vertex has outdegree two. I will show a proof that C_5 satisfies the conjecture. Consequently all 5-vertex tournaments satisfy the conjecture. Finally I will talk about the tournaments that satisfy the conjecture in an "almost linear sense" (so-called pseudocelebrities). A construction of all pseudocelebrities will be given. The part of the talk about pseudocelebrities is joint work with Maria Chudnovsky and Paul Seymour. The other part is joint work with Eli Berger and Maria Chudnovsky. |
| |
| Algebraic Topology Seminar |
| Topic: |
Piecewise Laurent polynomials and (operational) equivariant K-theory of toric varieties |
| Presenter: |
Dave Anderson, University of Washington |
| Date: |
Thursday, February 9, 2012, Time: 3:00 p.m., Location: Fine Hall 214 |
| Abstract: |
For a smooth compact toric variety X, results of Bifet-de Concini-Procesi and Brion show that the equivariant cohomology of X is identified with the ring of piecewise polynomials on the associated fan. In 2006, Payne extended this to arbitrary toric varieties, identifying the ring of piecewise polynomials with the operational equivariant Chow cohomology of X. It turns out that a similar story holds for K-theory: when X is smooth and compact, Brion-Vergne and Vezzosi-Vistoli show that the equivariant K-theory of algebraic vector bundles on X can be identified with the ring of "piecewise Laurent polynomials" on the associated fan. On the other hand, the bivariant machinery of Fulton-MacPherson can be applied to construct an "operational" equivariant K-theory for singular toric varieties. In this talk, I will describe ongoing joint work with Sam Payne: for an arbitrary toric variety X, we show that the ring of piecewise Laurent polynomials on the fan is identif ied with the operational equivariant K-theory of X. The proof requires us to develop some foundational aspects of operational K-theory, as well as the usual equivariant K-theory of coherent sheaves. Our point of view leads to the curious result that the abstract operational theory is tractable and computable on varieties where the usual K-theory (of algebraic vector bundles) is completely unknown. |
| |
| Joint IAS and Princeton University Number Theory Seminar |
| Topic: |
Sup-norms, Whittaker periods and hypergeometric sums |
| Presenter: |
Nicolas Templier, Princeton University |
| Date: |
Thursday, Feb. 9, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
We begin with a survey of recent results on the problem of bounding the sup-norm of automorphic forms. If f is a cuspidal automorphic forms on a reductive group G it is classical to study its value distribution and in the particular the maximum of |f(g)| for all g in G. Then we will explain an approach to this problem via Whittaker periods. We establish a new formula for non-archimedean Whittaker functions. The formula involves 2F1 hypergeometric sums and generalizes classical results of Casselman and others. As an application we disprove a folklore conjecture on the sup-norm of GL(2) modular forms. |
| |
| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
h-projective geometry on compact Kähler manifolds |
| Presenter: |
Stefan Rosemann, Uni Jena, Germany |
| Date: |
Friday, February 10, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
The basic geometric structure in h-projective geometry is the family of h-planar curves, associated to a given Kähler metric. Such curves can be seen as generalisations of geodesics on Kähler manifolds. In this context, one problem of interest is the investigation of Kähler manifolds admitting another Kähler metric having the same h-planar curves as the given one. Such a pair of metrics is called h-projectively equivalent. Besides a general introduction to h-projective geometry I want to present a result which was obtained in a joint work with V. S. Matveev: every compact Kähler manifold which admits an h-projective vector field (that is a one-parameter group of transformations mapping the metric to an h-projectively equivalent one) is isomorphic to the complex projective space with Fubini-Study metric provided the h-projective vector field is not a Killing vector field. |
| |
| Analysis Seminar **Please note new time** |
| Topic: |
Description of the blow-up for the semi-linear wave equation |
| Presenter: |
Raphael Cote, Ecole Polytechnique/University of Chicago |
| Date: |
Monday, February 13, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| Abstract: |
We study the the blow-up curve of a (blow-up) solution to the semi-linear wave equation in 1D with power nonlinearity: u_tt - u_xx = |u|^{p-1} u. The blow-up curve is a priori 1-Lispchitz. On this curve, we distinguish (geometrically) characteristic points and non-characteristic points. We describe the blow-up behavior in each case, following a series of papers by Frank Merle and Hatem Zaag: in particular, the set of characteristic points is discrete, the blow-up curve is corner-shaped at every characteristic point, and is $C1$ around any non-characteristic point. We also construct construct a blow-up solution with prescribed characteristic point. |
| |
| PACM Colloquium |
| Topic: |
Computability and Complexity of Julia Sets |
| Presenter: |
Mark Braverman, Princeton University |
| Date: |
Monday, February 13, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
Studying dynamical systems is key to understanding a wide range of phenomena ranging from planetary movement to climate patterns to market dynamics. Various computational and numerical tools have been developed to address specific questions about dynamical systems, such as predicting the weather or planning the trajectory of a satellite. However, the theory of computation behind these problems appears to be very difficult to develop. In fact, little is known about computability of even the most natural problems arising from dynamical systems. In this talk I will survey the recent study of the computational properties of dynamical systems that arise from iterating quadratic polynomials on the complex plane. These give rise to the amazing variety of fractals known as Julia sets, and are closely connected to the Mandelbrot set. Julia sets are perhaps the most drawn objects in Mathematics due to their fascinating fractal structure. The theory behind them is even more fascinating, and the dynamical systems generating them are in many ways archetypal. I will present both positive and negative results on the computability and computational complexity of Julia sets. |
| |
| Algebraic Geometry Seminar |
| Topic: |
TBA |
| Presenter: |
Aise Johan de Jong, Columbia University |
| Date: |
Tuesday, February 14, 2012, Time: 4:30 p.m., Location: Fine Hall 322 |
| |
| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
N. Templier, Princeton University |
| Date: |
Wednesday, February 15, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| |
| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
Binary correlations of the Moebius function |
| Presenter: |
Yakov Sinai, Princeton University |
| Date: |
Thursday, February 16, 2012, Time: 2:00 p.m., Location: Fine Hall 601 |
| |
| Discrete Mathematics Seminar **Please note special time** |
| Topic: |
The Traveling Salesman Problem: A Blueprint for Optimization |
| Presenter: |
Bill Cook, Georgia Tech and Princeton University |
| Date: |
Thursday, February 16, 2012, Time: 2:00 p.m.**, Location: Fine Hall 224 |
| Abstract: |
Given a list of cities along with the cost of travel between each pair of them, the traveling salesman problem is to find the cheapest way to visit them all and return to your starting point. Easy to state, but difficult to solve. In this talk we discuss the problem's history, applications, and computation, laying out a blueprint for future work in discrete optimization and the practical solution of large-scale, possibly intractable, decision models. |
| |
| Algebraic Topology Seminar |
| Topic: |
Cohomological rigidity problems in Toric topology |
| Presenter: |
Suyoung Choi, Ajou University, Korea |
| Date: |
Thursday, February 16, 2012, Time: 3:00 p.m., Location: Fine Hall 214 |
| Abstract |
As is well-known, cohomology ring does not distinguish closed smooth manifolds up to diffeomorphism or homeomorphism in general. However, it does if we restrict our attention to a reasonably small class of objects. For instance, it is known that simply connected closed smooth 4-manifolds are classified up to homeomorphism using their integral cohomology rings. The classification of the toric manifolds up to homeomorphism or diffeomorphism is an interesting open problem. In particular one can ask whether the integral cohomology ring determines the homeomorphism or diffeomorphism type. So far, we do not have any counter example but affirmative partial results. In this talk, we survey results on the topological classifiation of toric manifolds. If time allows, we also discuss about the topological classification of real toric manifolds. |
| |
| Joint IAS and Princeton University Number Theory Seminar |
| Topic: |
TBA |
| Presenter: |
Gopal Prasad, University of Michigan |
| Date: |
Thursday, February 16, 2012, Time: 4:30 p.m., Location: IAS, Room S-101 |
| |
| Topology Seminar |
| Topic: |
Representation theory and homological stability |
| Presenter: |
Tom Church, Stanford University |
| Date: |
Thursday, February 16, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Homological stability is the remarkable phenomenon where for certain sequences X_n of groups or spaces -- for example SL(n,Z), the braid group B_n, or the moduli space M_n of genus n curves -- it turns out that the homology groups H_i(X_n) do not depend on n once n is large enough. But for many natural analogous sequences, from pure braid groups to congruence matrix groups to Torelli groups, homological stability fails horribly. In these cases the rank of H_i(X_n) blows up to infinity, and in the latter two cases almost nothing was known about H_i(X_n); indeed it's possible there is no nice "closed form" for the answers. Representation stability is a notion which takes into account the action of certain symmetries to meaningfully talk about "the stable homology of the pure braid group" or "the stable homology of the Torelli group", even though the homology never stabilizes. In this talk I will explain our broad picture of representation stability and describe a number of connections to other areas of math. In particular, I will consider various sequences of integers a_n arising in topology (e.g. Betti numbers of spaces on configurations of points, of n-pointed curves, of matrices of rank at most n, etc.) and in algebra/combinatorics (e.g. dimensions of spaces of harmonic polynomials, of coinvariant algebras, of free Lie algebras, etc.), and explain how to use representation stability to prove that for each of these sequences (and many more) there is a polynomial P(n) with a_n = P(n) for all n big enough. Joint work with Benson Farb and Jordan Ellenberg. |
| |
| Special Algebraic Topology Seminar **Please note special day** |
| Topic: |
TBA |
| Presenter: |
Ileana Streinu, Smith College |
| Date: |
**Friday, February 17, 2012, Time: 3:00 p.m., Location: Fine Hall 214 |
| |
| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
John Head, Courant Institute |
| Date: |
Friday, February 17, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
| |
| IAS-PU Symplectic Geometry Seminar **Please note location** |
| Topic: |
TBA |
| Presenter: |
Tudor Ratiu, EPFL (Ecole Polytechnique Federale de Lausanne), Switzerland |
| Date: |
Friday, February 17, 2012, Time: 4:30 p.m., Location: IAS, Room S-101 ** |
| |
| Analysis Seminar **Please note new time** |
| Topic: |
Applications of multilinear restriction and restriction estimates |
| Presenter: |
Larry Guth, New York University |
| Date: |
Monday, February 20, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| Abstract: |
The restriction problem is an important open problem in harmonic analysis. The 2-dimensional case was proven in the 1970's. The case of three (or more) dimensions remains open, with interesting partial results. In 2005, Bennett, Carbery, and Tao proved a ``multilinear" restriction estimate. We can think of this work as a near-optimal estimate for a significant chunk of the terms in the original restriction problem. It had major philosophical impact, giving a new perspective on what part of the problem is most difficult. Recently, the multilinear estimate has also had some practical impact. A paper by Bourgain and me uses the multilinear inequality to prove estimates for the original restriction problem, matching and sometimes improving the best estimates previously known. I will briefly review the restriction problem, explain the multilinear restriction theorem, and explain how to apply it to the 3-dimensional restriction problem. |
| |
| PACM Colloquium |
| Topic: |
A Random Walk on Image Patches |
| Presenter: |
Francois Meyer, University of Colorado Boulder |
| Date: |
Monday, February 20, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
Algorithms that analyze patches extracted from time series or images have led to state-of-the art techniques for classification, denoising, and the study of nonlinear dynamics. In the first part of the talk we describe two examples of such algorithms: a novel method to estimate the arrival-times of seismic waves from a seismogram, and a new patch-based method to denoise images. Both approaches combines the following two ingredients: the signals (times series or images) are first lifted into a high-dimensional space using time/space-delay embedding; the resulting phase space is then parametrized using a nonlinear method based on the eigenvectors of the graph Laplacian. Both algorithms outperform existing gold standards. In the second part of the talk we provide a theoretical explanation for the success of algorithms that organize patches according to graph-based metrics. Our approach relies on a detailed analysis of the commute time on prototypical graph models that epitomize the geometry observed in general patch-graphs. |
| |
| Algebraic Geometry Seminar |
| Topic: |
Geometrically characterizing representation type of finite-dimensional algebras |
| Presenter: |
Ryan Kinser, Northeastern University |
| Date: |
Tuesday, February 21, 2012, Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: |
Given a finite-dimensional algebra A, the set of A-modules of a fixed dimension d can be viewed as a variety. This variety carries a group action whose orbits correspond to isomorphism classes of A-modules. A natural problem is to characterize various properties of an algebra A in terms of its module varieties. For example, if A is assumed to have global dimension one, then it is not difficult to show that A has finitely many indecomposable modules (up to isomorphism) if and only if all of its module varieties have a dense orbit, which is also if and only if all weight spaces of semi-invariants in the coordinate rings of its module varieties have dimension one. Our goal is to generalize these statements (with modification) to higher global dimension. After explaining the background, we present counterexamples to the naive generalizations, along with plausible modifications and cases where these modifications are correct. (Joint work with Calin Chindris, Piotr Dowbor, and Jerzy Weyman) |
| |
| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
E. Lindenstrauss, Hebrew University of Jerusalem |
| Date: |
Wednesday, February 22, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| |
| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
Effect of emergent distinguishability of particles in a non-equilibrium chaotic system |
| Presenter: |
Itzhak Fouxon, Tel Aviv University |
| Date: |
Thursday, February 23, 2012, Time: 2:00 p.m., Location: Fine Hall 601 |
| Abstract: |
We consider the behavior of classical particles which evolution consists of free motion interrupted by binary collisions. The fluid of hard balls and the dilute gas with arbitrary short-range interactions are treated, where the total number of particles is moderate (say, five particles). We assume that the decay of correlations, characteristic for chaotic systems, holds (it can be considered proved for hard balls). We show that the numbers of collisions of a given particle with other particles grow effectively as a biased random walk. This is used to prove that over indefinitely long periods of time each particle has preferences: it systematically collides more with certain particles and less with others. Thus certain particles are effectively attracted and certain others are repelled, making the particles effectively distinguishable. The effect is of statistical origin and it reminds of entropic forces. |
| |
| Princeton-Rider Workshop on Homotopy Theory and Toric Spaces (sponsored by Algebraic Topology Seminar) |
| Topic: |
Polyhedral products, quasi-toric manifolds, and twisted cohomology |
| Presenter: |
Alex Suciu, Northeastern University |
| Date: |
Thursday, February 23, 2012, Time: 3:00 p.m., Location: Fine Hall 214 |
| Organizers: |
Bill Browder <browder@math.princeton.edu; Martin Bendersky <mbenders@hunter.cuny.edu;
Tony Bahri <bahri@rider.edu |
| Abstract: |
I will discuss the cohomology with coefficients in rank one local systems for various polyhedral products, including real Davis-Januszkiewicz spaces and toric complexes. As one application (joint work with Alvise Trevisan), I will show how to determine the Betti numbers and the cup products of real, quasi-toric manifolds. As another application (joint work with Graham Denham and Sergey Yuzvinsky), I will explain why Cohen-Macaulay toric complexes enjoy a certain "abelian duality" property. |
| |
| Joint IAS and Princeton University Number Theory Seminar |
| Topic: |
TBA |
| Presenter: |
David Geraghty, Princeton University |
| Date: |
Thursday, Feb. 23, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| |
| Princeton-Rider Workshop on Homotopy Theory and Toric Spaces (sponsored by Algebraic Topology Seminar) **Please note time and location** |
| Topic: |
TBA |
| Presenter: |
Santiago Lopez de Medrano, UNAM, Mexico |
| Date: |
Thursday, February 23, 2012, Time: 4:30 p.m., Location: Fine Hall 314 ** |
| Organizers: |
Bill Browder <browder@math.princeton.edu; Martin Bendersky <mbenders@hunter.cuny.edu;
Tony Bahri <bahri@rider.edu |
| |
| Princeton-Rider Workshop on Homotopy Theory and Toric Spaces (sponsored by Algebraic Topology Seminar) **Please note Times/Location** |
| Topic: |
TBA |
| Organizers:: |
Bill Browder <browder@math.princeton.edu; Martin Bendersky <mbenders@hunter.cuny.edu;
Tony Bahri <bahri@rider.edu> |
| Date: |
Friday, February 24, 2012, Times: See below, Location: Rider University, Sweigart Hall, Room 115** |
| Times: |
10:00 COFFEE
10:30-11:30 Fred Cohen, (University of Rochester)
12:00-1:00 Tara Holm, (Cornell University)
LUNCH
3:00-4:00 Sam Gitler, (CINVESTAV), Mexico)
4:30-5:30 Suyoung Choi, (Ajou University, Korea)
6:30 DINNER |
| Abstract: |
Suyoung Choi: PDF |
| |
| Princeton-Rider Workshop on Homotopy Theory and Toric Spaces (sponsored by Algebraic Topology Seminar) **Please note Times/Location** |
| Topic: |
See "times" below |
| Organizers:: |
Bill Browder <browder@math.princeton.edu; Martin Bendersky <mbenders@hunter.cuny.edu;
Tony Bahri <bahri@rider.edu> |
| Date: |
Saturday, February 25, 2012, Times: See below, Location: Princeton University - Room 314 ** |
| Times: |
10:00 COFFEE
10:30-11:30 Benjamin Matschke, (IAS, Princeton)
12:00-1:00 Peter Landweber, (Rutgers University); Spanning sets of distinct point in spheres |
| |
| Analysis Seminar **Please note new time** |
| Topic: |
Well-posedness and finite-time blowup for the Zakharov system on two-dimensional torus |
| Presenter: |
Nobu Kishimoto, Kyoto University |
| Date: |
Monday, February 27, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| Abstract: |
We consider the Zakharov system on two-dimensional torus. First, we show the local well-posedness of the Cauchy problem in the energy space by a standard iteration argument using the X^{s,b} norms. Our result does not depend on the period of torus. Conservation laws and a sharp Gagliardo-Nirenberg inequality imply an a priori bound of solutions, which enables us to extend the local-in-time solution to a global one if its L2 norm is less than that of the ground state solution of the cubic NLS on R2. We then show that the L2 norm of the ground state is actually the threshold for global solvability, namely, that there exists a finite-time blow-up solution to the Zakharov system on 2d torus with the L2 norm greater than but arbitrarily close to that of the ground state. This is joint work with Masaya Maeda (Tohoku University, Japan). |
| |
| PACM Colloquium |
| Topic: |
Dimension reduction, coarse-graining and data assimilation in high-dimensional dynamical systems |
| Presenter: |
Eric Vanden-Eijnden, Courant NYU |
| Date: |
Monday, February 27, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
Modern computing technologies, such as massively parallel simulation, special-purpose high-performance computers, and high-performance GPUs permit to simulate complex high-dimensional dynamical systems and generate time-series in amounts too large to be grasped by traditional "look and see" analyses. This calls for robust and automated methods to extract the essential structural and dynamical properties from these data in a manner that is little or not depending on human subjectivity. To this end, a decade of work has led to the development of analysis techniques which rely on the partitioning of the conformation space into discrete substates and reduce the dynamics to transitions between these states. A particular successful class of methods of this type are Markov state models (MSMs), in which the transitions between the states in the partition are assumed to be memoryless jumps. The accuracy of these models crucially depends on the choice of these states. In this talk, I will discuss systematic strategies that permit to identify these states and quantify the error of the resulting approximation. These methods will be illustrated on examples arising from molecular dynamics simulations of biomolecules. |
| |
| Algebraic Geometry Seminar |
| Topic: |
TBA |
| Presenter: |
Mina Teicher, Bar-Ilan University, IAS |
| Date: |
Tuesday, February 28, 2012, Time: 4:30 p.m., Location: Fine Hall 322 |
| |
| Ergodic Theory and Statistical Mechanics Seminar **Please note special date, time and location** |
| Topic: |
On a Hidden Symmetry of Simple Harmonic Oscillators |
| Presenter: |
Sergei Suslov, Arizona State University |
| Date: |
Wednesday, February 29, 2012, Time: 3:00 p.m., Location: Fine Hall 110 ** |
| Abstract: |
Since the original 1926 Schroedinger's paper, there was a misconception that the "simple" harmonic oscillator can be solved only by the separation of variables, which results in a traditional "static" electron density distribution. It is not entirely accurate and a nontrivial oscillator hidden symmetry group, found by Niederer in 1973, provides "dynamic solutions". The phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications to quantum optics are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented. In addition, these "dynamic harmonic states" possess the nontrivial Berry phase, which may be use for their identification. The corresponding phase is evaluated in terms of elementary functions. In view of importance of the simple harmonic oscillators in numerous applications, these results will be interesting to everybody who is going to study and/or teach quantum mechanics --- It may help better understand a general concept of quantum motion. |
| |
| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
J. Szeftel, Ecole Normale Superieure |
| Date: |
Wednesday, February 29, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| |
| MARCH 2012 |
| |
| Algebraic Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Nitu Kitchloo, Johns Hopkins University |
| Date: |
Thursday, March 1, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
| |
| Joint IAS and Princeton University Number Theory Seminar |
| Topic: |
TBA |
| Presenter: |
Dennis Gaitsgory, Harvard University |
| Date: |
Thursday, March 1, 2012, Time: 4:30 p.m., Location: IAS, Room S-101 |
| |
| Analysis Seminar |
| Topic: |
Hardy spaces with variable exponents and generalized Campanato spaces |
| Presenter: |
Yoshihiro Sawano, Kyoto University |
| Date: |
Monday, March 5, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| Abstract: |
Hardy spaces play an important role not only in harmonic analysis but also in partial differential equations because singular integral operators are bounded on Hardy spaces. The Hardy space H1, which substitute for L1, and the Hardy spaces H^p with 0 < p < 1, are different in that the latter contains non-regular distributions. Although it will turn out to be an equivalent expression of L^p, for 1 < p < \infty, we can define the Hardy space H^p. To have a unified understanding of these situations, we consider and de?ne Hardy spaces with variable exponents on R^n. We will connect harmonic analysis with function spaces with variable exponents. We then obtain the atomic decomposition and the molecular decomposition. With these decomposition proved, we investigate the Littlewood?Paley characterization. Also, we specify the dual spaces of Hardy spaces with variable exponents. They will turn out to be Campanato spaces with variable growth conditions. We shall allude to local Hardy spaces with variable exponents as well. |
| |
| PACM Colloquium |
| Topic: |
Topological Landscape of Networks |
| Presenter: |
Yuan Yao, Peking University |
| Date: |
Monday, March 5, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
We will discuss how one can endow a network with a landscape in a very simple and natural way. Critical point analysis is introduced for functions defined on networks. The concept of local minima/maxima and saddle points of different indices are defined, by extending the notion of gradient flows and minimum energy path to the network setting. Persistent homology is used to design efficient numerical algorithms for performing such analysis. Applications to some examples of social and biological networks (LAO protein binding network) are demonstrated. These examples show that the critical nodes play important roles in the structure and dynamics of such networks. This is a joint work with Weinan E and Jianfeng Lu. |
| |
| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
J. Kahn, Brown University |
| Date: |
Wednesday, March 7, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| |
| Algebraic Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Po Hu, Wayne State University |
| Date: |
Thursday, March 8, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
| |
| Joint IAS and Princeton University Number Theory Seminar |
| Topic: |
TBA |
| Presenter: |
Mark Kisin, Harvard University |
| Date: |
Thursday, March 8, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| |
| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Dan Lee, CUNY |
| Date: |
Friday, March 9, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
| |
| Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Alex Iosevish, University of Rochester |
| Date: |
Monday, March 12, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| |
| Algebraic Geometry Seminar |
| Topic: |
TBA |
| Presenter: |
Karl Schwede, Penn State University |
| Date: |
Tuesday, March 13, 2012, Time: 4:30 p.m., Location: Fine Hall 322 |
| |
| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
C. Kenig, University of Chicago |
| Date: |
Wednesday, March 14, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| |
| Discrete Mathematics Seminar |
| Topic: |
TBA |
| Presenter: |
Ben Clark, Victoria University of Wellington, New Zealand |
| Date: |
Thursday, March 15, 2012, Time: 2:15 p.m., Location: Fine Hall 224 |
| |
| Joint IAS and Princeton University Number Theory Seminar **Please note location** |
| Topic: |
TBA |
| Presenter: |
Fernando Villegas, University of Texas at Austin |
| Date: |
Thursday, March 15, 2012, Time: 4:30 p.m., Location: IAS, West Bldg. Lecture Hall ** |
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| Algebraic Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Igor Kriz, University of Michigan |
| Date: |
Thursday, March 22, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
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| Analysis Seminar |
| Topic: |
Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping |
| Presenter: |
Hans Cristianson, University of North Carolina at Chapel Hill |
| Date: |
Monday, March 26, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| Abstract: |
It is well known that on $\reals^n$, the Schrödinger propagator is unitary on $L2$ based spaces, but that locally in space and on average in time there is a $1/2$ derivative smoothing effect. We consider a family of manifolds with trapped geodesics which are degenerately hyperbolic and prove a sharp local smoothing estimate with loss depending on the type of trapping. Further, we construct a microlocal parametrix extended polynomially beyond Ehrenfest time, and as a consequence, we obtain Strichartz estimates with near-sharp loss depending only on the dimension of the trapping. This is partly joint work with J. Wunsch (Northwestern) |
| |
| PACM Colloquium |
| Topic: |
Geometry and Topology in Dimension Reduction |
| Presenter: |
Sayan Mukherjee, Duke University |
| Date: |
Monday, March 26, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
In the first part of the talk we describe how learning the gradient of a regression function can be used for supervised dimension reduction (SDR). We provide an algorithm for learning gradients in high-dimensional data, provide theoretical guarantees for the algorithm, and provide a statistical interpretation. Comparisons to other methods on real and simulated data are presented. In the second part of the talk we present preliminary results on using the Laplacian on forms for dimension reduction. This involves understanding higher-order versions of the isoperimetric inequality for both manfifolds and abstract simplicial complexes. |
| |
| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
G. Margulis, Yale University |
| Date: |
Wednesday, March 28, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Mattias Jonsson, University of Michigan |
| Date: |
Friday, March 30, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
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| APRIL 2012 |
| |
| Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Brian Street, University of Wisconsin, Madison |
| Date: |
Monday, April 2, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| |
| Discrete Mathematics Seminar |
| Topic: |
TBA |
| Presenter: |
Jeff Kahn, Rutgers University |
| Date: |
Thursday, April 5, 2012, Time: 2:15 p.m., Location: Fine Hall 224 |
| |
| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Meijun Zhu, University of Oklahoma |
| Date: |
Friday, April 6, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
| |
| Algebraic Geometry Seminar |
| Topic: |
TBA |
| Presenter: |
Daniel Erman, Stanford University |
| Date: |
Tuesday, April 10, 2012, Time: 4:30 p.m., Location: Fine Hall 322 |
| |
| Algebraic Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Paul Baum, Penn State University |
| Date: |
Thursday, April 12, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
| |
| Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Michael Lacey, Georgia Institute of Technology |
| Date: |
Monday, April 16, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| |
| Analysis Seminar |
| Topic: |
The Cauchy problem for the Benjamin-Ono equation in L^2 revisited (Joint work with Luc Molinet) |
| Presenter: |
Didier Pilod, Universidade Federal do Rio de Janeiro / University of Chicago |
| Date: |
Monday, April 30, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| Abstract: |
The Benjamin-Ono equation models the unidirectional evolution of weakly nonlinear dispersive internal long waves at the interface of a two-layer system, one being in finitely deep. The Cauchy problem associated to this equation presents interesting mathematical difficulties and has been extensively studied in the recent years. In a recent work (2007), Ionescu and Kenig proved well-posedness for real-valued initial data in L^2(R). In this talk, we will give another proof of Ionescu and Kenig's result, which moreover provides stronger uniqueness results. In particular, we prove unconditional well-posedness in H^s(R), for s > 1/4 . Note that our approach also permits to simplify the proof of the global well-posedness in L^2(T) by Molinet (2008) and yields unconditional well-posedness in H^{1/2}(T). Finally, it is worthwhile to mention that our technique of proof also apply for a higher-order Benjamin-Ono equation. We prove that the associated Cauchy problem is globally well-posed in the energy space H^1(R). |
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| MAY 2012 |
| |
| Algebraic Geometry Seminar |
| Topic: |
TBA |
| Presenter: |
Bhargav Bhatt, University of Michigan |
| Date: |
Tuesday, May 1, 2012, Time: 4:30 p.m., Location: Fine Hall 322 |
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