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| PACM Colloquium | |
| Topic: | Modeling textures with total variation minimization and oscillating patterns in image processing |
| Presenter: | Luminata Vese, University of California, Los Angeles |
| Date: | Monday, March 3, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: | This talk is devoted to the decomposition of a given (possibly textured) image $f$ into a sum of two components $u+v$, where $u$ is a function of bounded variation (a cartoon approximation of $f$), while $v$ is an oscillating function, representing texture or noise. To model the oscillatory component $v$, we investigate the use of some spaces defined by duality, instead of the standard $L^2$ norm. These new techniques for image decomposition and texture modeling follow some recent ideas of Y. Meyer. The obtained algorithms are very simple, making use of differential equations and are easily solved in practice. Finally, I will present various numerical results on real textured images, showing the obtained decomposition $u+v$. I will also illustrate how the proposed methods can be used for image restoration, texture discrimination and texture segmentation. This is joint work with S. Osher and A. Sole. |
| Algebraic Geometry Seminar | |
| Topic: | Volume of the Space of Real Cubic Surfaces |
| Presenter: | James Carlson, University of Utah |
| Date: | Tuesday, March 4, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: |
We show
that the moduli space of real cubic surfaces has, in a natural way, the
structure of real hyperbolic orbifold of dimension four. We discuss the
structure of this space, its fundamental group, and we compute its exact
hyperbolic volume. As a result we can, for instance, show that real cubics
with twenty-seven real lines comprise less than two percent of the full
space.
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| Department Colloquium | |
| Topic: | Proof of the strong perfect graph conjecture |
| Presenter: | Paul Seymour, Princeton University |
| Date: | Wednesday, March 5, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
In
1961, Claude Berge proposed the conjecture that, in every graph with no
odd hole or odd antihole, the number of colours needed to properly colour
the graph equals the size of the largest complete subgraph. (An "odd hole"
means an induced subgraph which is an odd cycle of length >= 5, and an
"odd antihole" is the same in the complement graph.) This became one of
the most well-known and popular open problems in graph theory. In joint
work with Maria Chudnovsky, Neil Robertson and Robin Thomas, we proved
the conjecture last summer.Most
previous approaches to the conjecture were based on studying properties
of a minimal counterexample, but our approach was different.We
proved that every graph with no odd hole or antihole either falls into
one of five well-understood classes, or admits a useful decomposition;
and Berge's conjecture is a consequence.The
proof is lengthy (150 pages), and this talk will just be a survey of some
of the background and ideas involved.
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| Ergodic Theory and Statistical Analysis Seminar | |
| Topic: | Parabolic Anderson Problems |
| Presenter: | Michael Cranston, University of Rochester |
| Date: | Thursday, March 6, 2003, Time: 2:00 p.m., Location: Fine Hall 214 |
| Abstract: | We consider the almost sure behavior of solutions of the equation\partial u(t,x)/\partial t =\kappa \Delta u(t,x) + dW_x(t) u(t,x) , u(0,x)=u_0(x). This equation arises in a variety of situtions and was the subject of the memoir of Carmona and Molchanov. We consider the cases where x could be in the integer lattice, Z^d, or in R^d. In the integer lattice case we take W_x to be a field of independent Levy processes. We prove existence of lim 1/t log u(t,x)=\lambda(\kappa) a.s. when u_0 is a bounded,nonzero, nonnegative function and give asymptotics for the Lyapunov exponent, \lambda(\kappa), as \kappa goes down to zero. In the case of R^d, we take W_x to be a C^2 correlated field of Brownian motions. Then we show again that lim 1/t log u(t,x)=\lambda(\kappa) a.s. when u_0 is a bounded nonnegative function and give asymptotics for the Lyapunov exponent, \lambda(\kappa), as\kappa goes down to zero. The asymptotic behavior of \lambda(\kappa) is quite different in the two cases. |
| Joint Princeton University/IAS/Rutgers University Number Theory Seminar | |
| Topic: | Combinatorial measure theory related to the Kakeya set conjecture |
| Presenter: | Jean Bourgain, Institute for Advanced Study |
| Date: | Thursday, March 6, 2003, Time: 4:15 p.m., Location: IAS SH-101 |
| Topology Seminar | |
| Topic: | Legendrian knots and cables |
| Presenter: | John Etnyre, University of Pennsylvania |
| Date: | Thursday, March 6, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Geometric Analysis Seminar | |
| Topic: | The differential geometry of ordinary differential equations: relations to the conformal structures of Fefferman |
| Presenter: | George Sparling, University of Pittsburgh |
| Date: | Friday, March 7, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: | Two strands of thought due primarily to Elie Cartan will be combined with ideas of Charles Fefferman to give new descriptions of the geometry of second and third order ordinary differential equations. Properties of infinite differential ideals associated naturally to ordinary differential equations will be analyzed. This is joint work with Pawel Nurowski of the University of Warsaw. |
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| Analysis Seminar | |
| Topic: | The F. and M. Riesz theorem for complex vector fields |
| Presenter: | Shiferaw Berhanu, Temple University |
| Date: | Monday, March 10, 2003, Time: 4:00 p.m., Location: Fine Hall 314 |
| Abstract: | The classical F. and M. Riesz theorem for holomorphic functions states that if a holomorphic function defined on a smooth subdomain of the plane has a boundary value that is a measure, then the measure is absolutely continuous with respect to Lebesgue measure. We will discuss generalizations of this result to solutions of first order complex vector fields |
| PACM Colloquium | |
| Topic: | New challenges for hydrodynamics: microfluidics, imaging science, and mobile sensors |
| Presenter: | Andrea Bertozzi, Duke University |
| Date: | Monday, March 10, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: | This talk will showcase three
new research areas involving mathematical fluid dynamics. Microfluidics
is a rapidly growing field being driven by new technological applications
in the medical, materials, and chemical sciences. Surface tension effects
(Marangoni stresses) are important on these scales. We consider the basic
physics of surface tension gradients (used to move liquids) in conjunction
with body forces on fluids and show that the ensuing dynamics can yield
multiple shock structures involving undercompressive waves.
In the field of imaging science, Image inpainting involves filling in part of an image or video using information from the surrounding area. We introduce a class of automated methods for digital inpainting using ideas from classical fluid dynamics. The main idea is to think of the image intensity as a `stream function' for a two-dimensional incompressible flow. The method is directly based on the Navier-Stokes equations for fluid dynamics, which has the immediate advantage of well-developed theoretical and numerical results. An emerging area of mobile sensor control is the design of algorithms for multiple unmanned vehicles. Taking ideas from mathematical biology, we consider swarming algorithms for fluid-like motion based on simple rules for self-propulsion and local interaction. Applications range from mine detection algorithms to perimeter patrol and gradient searching. |
| Algebraic Geometry Seminar | |
| Topic: | Odd theta characteristics: embedding M_g and A_g |
| Presenter: | Samuel Grushevsky, Princeton University |
| Date: | Tuesday, March 11,2003 Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: | Starting from the classical problem of recovering a plane curve from its bitangents, we discuss, following the work of Caporaso and Sernesi, recovering a canonical curve of genus $g$ in $P^g$ from the hyperplanes tangent to it in $g-1$ points. We then present a generalization of this question to the moduli of abelian varieties, construct an embedding of (some level covers of) $M_g$ and $A_g$ into a Grassmanian, and discuss some applications. This is joint work with Riccardo Salvati Manni. |
| Department Colloquium | |
| Topic: | Homology manifolds |
| Presenter: | Frank Quinn, Virginia Polytechnic Institute and State University |
| Date: | Wednesday, March 12, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: | We describe the development, complete with misadventures, of homology manifolds from Poincare to the present day. A new construction giving low-dimensional examples will be sketched, and the status of the major open problems will be discussed. |
| Ergodic Theory and Statistical Analysis Seminar | |
| Topic: | Ergodic and stochastic properties of Hénon and Hénon-like maps |
| Presenter: | Michael Benedicks, Royal Institute of Technology, Stockholm |
| Date: | Thursday, March 13, 2003, Time: 2:00 p.m., Location: Fine Hall 214 |
| Abstract: | We will indicate the fairly recent progress on the understanding of ergodic and statistical properties of the Henon maps with "strange attractors" constructed by L. Carleson and M. Benedicks around 1990 and the corresponding Henon-like map studied by Mora and Viana. We will discuss such topics as the existence of SRB-measures, decay of correlation, the metric basin property (there are no holes in the metric basin of the attractor) and stability under random perturbations. This is exclusively joint work with either L-S Young or M. Viana. |
| Topology Seminar | |
| Topic: | Mu transpososome: a knot-theoretic look at an experiment by Pathania, Jayaram, and Harshey |
| Presenter: | John Luecke, University of Texas at Austin |
| Date: | Thursday, March 13, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: | The Mu transpososome is a bringing-together of three separate segments of a DNA molecule. We describe a recent experiment by Pathania, Jayaram, Harshey (Cell, Vol. 109, 425-436) which discovers the architecture of this complex (thinking of it as a 3-string tangle) by coupling it with the recombination enzyme CRE and observing the resulting knotted products which arise. We will give a knot-theoretic analysis of the results of this experiment. We show that there are other possible architectures, and then classify these architectures in terms of certain kinds of knotted graphs. Using this description, we shot that the PJH solution is the only rational (3-string) tangle solution for the architecture, and furthermore show that any other solution must have at least 11 crossings (where the PJH solution has only 6). This is joint work with Isabel Darcy and Mariel Vasquez. |
| Joint Non-Linear Analysis Seminar | |
| Topic: | Causal Structure of rough solutions to the Einstein equations |
| Presenter: | Sergiu Klainerman, Princeton University |
| Date: | Thursday, March 13, Time: 4:00 p.m., Location: Fine Hall 214 |
| Geometric Analysis Seminar | |
| Topic: | Dehn surgery and Einstein metrics in higher dimensions |
| Presenter: | Michael Anderson, SUNY at Stony Brook |
| Date: | Friday, March 14, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: | We will describe a construction of a large class of Einstein metrics of negative scalar curvature on compact n-manifolds, for any n > 2. These metrics are obtained by performing Dehn surgery on toral ends of a complete non-compact hyperbolic n-manifold, exactly as in Thurston's cusp closing theorem in dimension 3. (The construction gives a new proof of Thurston's theorem). A key ingredient is the use of "twisted" toral black hole metrics discussed in connection with the AdS/CFT correspondence. |
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| SPRING BREAK | |
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| PACM Colloquium *** Please note special time | |
| Topic: | TBA |
| Presenter: | Steven Strogatz, Cornell University |
| Date: | Monday, March 24, 2003, Time: 12:30 p.m., Location: Fine Hall 214 |
| PACM Colloquium | |
| Topic: | New high-order, high-frequency methods in computational electromagnetism |
| Presenter: | Oscar Bruno, California Institute of Technology |
| Date: | Monday, March 24, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: | We present a new set of algorithms and methodologies for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, fast Fourier transforms and highly accurate high-frequency methods, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers --- even in cases in which the scatterers contain geometric singularities such as corners and edges. In all cases the solvers exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our algorithms can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects by direct solution of integral equations --- a goal, otherwise achievable today only by supercomputing. A new class of high-order surface representation methods will be discussed, which allows for accurate high-order description of surfaces from a given CAD representation. A class of high-order high-frequency methods which we developed recently, finally, are efficient where our direct methods become costly, thus leading to a general and accurate computational methodology which is applicable and accurate for the whole range of frequencies in the electromagnetic spectrum. |
| Department Colloquium | |
| Topic: | TBA |
| Presenter: | Madhu Sudan, MIT |
| Date: | Wednesday, March 26, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Topology Seminar | |
| Topic: | Hyperbolic Manifolds with Convex Boundary |
| Presenter: | Jean-Marc Schlenker, Université Paul Sabatier |
| Date: | Thursday, March 27, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Let
M be a compact 3-manifold with boundary, which admits a convex co-compact
hyperbolic metric. One can describe the hyperbolic metrics on M for which
the boundary is smooth and strictly convex.
Theorem A: the induced metrics have curvature K>-1, and each is obtained for a unique hyperbolic metric on M. Theorem B: the third fundamental forms of the boundary have curvature K<1, and their closed geodesics which are contractible in M have length L>2\pi. Each is obtained for a unique hyperbolic metric on M. Theorem B has analogs when the boundary is supposed to look locally like an ideal or a hyperideal polyhedron. As a consequence, we find an extension of the Koebe circle packing theorem when the sphere is replaced by the boundary of M. |
| Joint Princeton / IAS / Rutgers Non-linear Analysis Seminar | |
| Topic: | Topological Singularity in some Non-linear PDE Problems |
| Presenter: | Fang-Hua Lin, Courant Institute |
| Date: | Thursday, March 27, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: |
Many
interesting natural phenomena contain some sort of singular behavior and are
often manifested through energy concentrations. Singularities of
solutions of Partial Differential Equations which describe these phenomena
are, therefore, an important part of facets. One can divide these
singularities into two basic categories: topological and
non-topological. There are many examples of non-topological
singularities such as spikes in the reaction-diffusion systems, concentrated
vorticities in the Euler or the Navier-Stokes equations. Singularities
in these examples may or may not carry quantified amounts of energy. On
the other hand, the topological singularities often not only carry a definite
topological information but also a quantified amount of energy. Because
of this, they are often more stable energetically and dynamically. The
purpose of this lecture is to describe some recent works on analysis of
topological singularity in some variational and evolution problems.
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| Geometric Analysis Seminar | |
| Topic: | TBA |
| Presenter: | Fanghua Lin, Courant Institute, New York University |
| Date: | Friday, March 28, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
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| PACM Colloquium | |
| Topic: | TBA |
| Presenter: | Anna-Karin Tornberg, Courant Institute |
| Date: | Monday, March 31, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Algebraic Geometry Seminar | |
| Topic: | TBA |
| Presenter: | Gavril Farkas, University of Michigan, Ann Arbor |
| Date: | Tuesday, April 1, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Department Colloquium | |
| Topic: | TBA |
| Presenter: | Stanislav Smirnov, Royal Institute of Technology, Stockholm |
| Date: | Wednesday, April 2, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Ergodic Theory and Statistical Analysis Seminar | |
| Topic: | Ergodic properties of boundary actions |
| Presenter: | Tatiana Nagnibeda, Royal Institute of Technology, Stockholm |
| Date: | Thursday, April 3, 2003, Time: 2:00 p.m., Location: Fine Hall 214 |
| Abstract: | We shall discuss ergodic properties of the action of a subgroup H of a free group F on the Poisson boundary of the simple random walk on F. The action is ergodic if and only if the quotient F/H admit no non-constant bounded harmonic function. Methods from combinatorial group theory allow us to identify the conservative and the dissipative part of the action. We also present necessary and sufficient conditions of conservativity of the action in terms of geometry of the quotient. This is a joint work with R. Grigorhcuk and V. Kaimanovich. |
| Topology Seminar | |
| Topic: | TBA |
| Presenter: | John Morgan, Columbia University |
| Date: | Thursday, April 3, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Geometric Analysis Seminar | |
| Topic: | TBA |
| Presenter: | Changyou Wang, University of Kentucky |
| Date: | Friday, April 4, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
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| PACM Colloquium | |
| Topic: |
Interval analysis and set-membership techniques in estimation |
| Presenter: | Isabelle Braems, MAE, Princeton University |
| Date: | Monday, April 7, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: | Interval analysis has been developed more than four decades ago to control numerical round-off errors in computers, in a rigorous way. It has then reached many other fields (assisted proof demonstrations, numerical simulation, estimation…) and applications (biology, chemical engineering, economics, computer vision, robotics…) where guaranteed computations are essential. In this talk we shall focus on parameter and state estimation problem. We will emphasize how interval analysis permits to estimate in a guaranteed way a reliable enclosure of all the global minima in optimization problems, or of all the acceptable solutions in the bounded-error context. This talk will first briefly present (or recall) the bases of interval analysis. Several applications -including non-identifiable kinetic parameteridentification, reliable characterization of a thermal set-up, and robot localization- will illustrate the performance of this approach. |
| Algebraic Geometry Seminar | |
| Topic: | TBA |
| Presenter: | Nikos Tziolas, Max Planck Institute |
| Date: | Tuesday, April 8, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Department Colloquium | |
| Topic: | TBA |
| Presenter: | Percy Deift, New York University |
| Date: | Wednesday, April 9, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Geometric Analysis Seminar | |
| Topic: | TBA |
| Presenter: | Yu Yuan, University of Washington |
| Date: | Friday, April 11, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
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| PACM Colloquium | |
| Topic: | TBA |
| Presenter: | Russel Caflisch, University of California at Los Angeles |
| Date: | Monday, April 14, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Topology Seminar | |
| Topic: | TBA |
| Presenter: | Alejandro Adem, University of Wisconsen |
| Date: | Thursday, April 17, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Geometric Analysis Seminar | |
| Topic: | TBA |
| Presenter: | Aobing Li, Rutgers University |
| Date: | Friday, April 18, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
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| PACM Colloquium | |
| Topic: | TBA |
| Presenter: | Carlos Castillo-Chavez, Cornell University |
| Date: | Monday, April 21, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Algebraic Geometry Seminar | |
| Topic: | TBA |
| Presenter: | Jason Starr, MIT |
| Date: | Tuesday, April 22, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Department Colloquium | |
| Topic: | TBA |
| Presenter: | Russel Lyons, Indiana |
| Date: | Wednesday, April 23, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Geometric Analysis Seminar | |
| Topic: | TBA |
| Presenter: | Xiaodong Wang, MIT |
| Date: | Friday, April 25, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
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| Algebraic Geometry Seminar | |
| Topic: | TBA |
| Presenter: | T. Terasoma, Institute for Advanced Study |
| Date: | Tuesday, April 29, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Department Colloquium | |
| Topic: | TBA |
| Presenter: | S.R.Srinivasa Varadhan, New York University |
| Date: | Wednesday, April 30, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |