https://example.com/tags/notes/ Recent content in Notes on James Colliander Hugo -- gohugo.io en © Copyright 2020 Thu, 21 Mar 2013 00:00:00 +0000 https://example.com/post/2013-03-21-zaher-hani-a-continuum-large-box-limit-for-the-cubic-nonlinear-schr%C3%B6dinger-equation/ Thu, 21 Mar 2013 00:00:00 +0000 https://example.com/post/2013-03-21-zaher-hani-a-continuum-large-box-limit-for-the-cubic-nonlinear-schr%C3%B6dinger-equation/ Analysis & Applied Math Seminar 2013-03-21 Speaker: Zaher Hani Institution: New York University Abstract: Inspired by the general paradigm of weak turbulence theory, we consider the 2D cubic nonlinear Schrödinger equation on a box of size L with periodic boundary conditions. In an appropriate “large box regime” (L very large), we derive a continuum equation on ℝ2 that governs the dynamics of the discrete frequency modes over nonlinear time scales. This equation turns out to satisfy many surprising symmetries and conservation laws, as well as several families of explicit solutions. https://example.com/post/2013-02-11-larry-guth-colloquium-video-unexpected-applications-of-polynomials-in-combinatorics/ Mon, 11 Feb 2013 00:00:00 +0000 https://example.com/post/2013-02-11-larry-guth-colloquium-video-unexpected-applications-of-polynomials-in-combinatorics/ My former colleague Larry Guth (now at MIT) visited us recently and gave a beautiful colloquium talk. The Department has recently deployed a video streaming service so we are able to share Larry’s talk with the world. We look forward to sharing other videos in the future.   Here is the video: Your browser does not support the <video> tag.     Unexpected applications of polynomials in combinatorics by Larry Guth | MIT Time: 16:10 (Wednesday, Jan. https://example.com/post/2012-05-28-g-w-p-of-gross-pitaevskii-equation-on-r4/ Mon, 28 May 2012 00:00:00 +0000 https://example.com/post/2012-05-28-g-w-p-of-gross-pitaevskii-equation-on-r4/ Last week, I had a chance to visit Edinburgh in part to serve as the external examiner on the PhD Thesis (papers) of Tim Candy. Tim is now Dr. Timothy Candy and has an exciting research program to develop as a postdoc at Imperial. It turned out I had lucky timing since my visit overlapped with a visit by Oana Pocovnicu. I had a chance to hear her speak about her recent work on the Gross-Pitaevskii equation. https://example.com/post/2012-03-17-i-a-s-workshop-on-symplectic-dynamics-2-friday/ Sat, 17 Mar 2012 00:00:00 +0000 https://example.com/post/2012-03-17-i-a-s-workshop-on-symplectic-dynamics-2-friday/ IAS School of Mathematics   Workshop web page Friday: 2012-03-16 9:00 - 10:00 James Colliander, University of Toronto, “Big frequency cascades in the cubic nonlinear Schroedinger flow on the 2-torus” abstract 10:15 - 11:15 Marcel Guardia, IAS, “Growth of Sobolev norms for the cubic defocusing nonlinear Schroedinger equation in polynomial time” abstract 11:30 - 12:30 Yann Brenier, University of Nice, “Approximate geodesics on groups of volume preserving diffeomorphisms and adhesion dynamics” abstract James Colliander: Big frequency cascades in the cubic nonlinear Schrödinger flow on the 2-torus (chalk talk) https://example.com/post/2012-03-15-i-a-s-workshop-on-symplectic-dynamics-2-thursday/ Thu, 15 Mar 2012 00:00:00 +0000 https://example.com/post/2012-03-15-i-a-s-workshop-on-symplectic-dynamics-2-thursday/ IAS School of Mathematics   Workshop web page Thursday: 2012-03-15 9:00 - 10:00 Peter Topalov, Northeastern University, “Qualitative features of periodic solutions of KdV” abstract 10:15 - 11:15 Jiansheng Geng, Nanjing University, “Invariant tori for the nonlinear lattice one-dimensional Schroedinger equations with real analytic potential” abstract 11:30 - 12:30 Massimiliano Berti, UNINA, “Quasi periodic solutions of Hamiltonian PDEs” abstract 2:30 - 3:30 Ralph Saxton, University of New Orleans, “The generalized inviscid Proudman Johnson equation” abstract 4:30 - 5:30 Dongho Chae, Sungkyunkwan University, “On the blow-up problem for the Euler equations and the Liousville type results in the fluid equations” abstract Peter Topalov: Qualitative features of periodic solutions of KdV I need to do some detailed setup to expose the ideas I want to describe. https://example.com/post/2012-03-14-i-a-s-workshop-on-symplectic-dynamics-2-wednesday/ Wed, 14 Mar 2012 00:00:00 +0000 https://example.com/post/2012-03-14-i-a-s-workshop-on-symplectic-dynamics-2-wednesday/ IAS School of Mathematics Workshop web page Happy Einstein Birthday! Wednesday: 2012-03-13 9:00 - 10:00 Wilfrid Gangbo, Georgia Institute of Technology, “Lifting absolutely continuous curves from P(Td) to P2(Rd)” abstract 10:15 - 11:15 Jonatan Lenells, Baylor University, “Geometry of diffeomorphism groupos, complete integrability and optimal transport” abstract 11:30 - 12:30 David Ebin, SUNY, “Groups of diffeomorphisms and geodesics on them” abstract 2:30 - 3:30 Susan Friedlander, University of Southern California, “Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even”. https://example.com/post/2012-03-14-princeton-math-colloquium-soliton-resolution-for-the-radial-energy-critical-focusing-nonlinear-wave-equation/ Wed, 14 Mar 2012 00:00:00 +0000 https://example.com/post/2012-03-14-princeton-math-colloquium-soliton-resolution-for-the-radial-energy-critical-focusing-nonlinear-wave-equation/ (Apologies for typos or misquotations…; please help me improve the post. –J. Colliander)   Princeton Math Colloquium 2012-03-14 Presenter: Carlos Kenig University of Chicago Title: A case study for critical non-linear dispersive equations: the energy critical wave equation Abstract: We will discuss recent work on the energy critical wave equation. The issues studied are global existence, scattering, finite time blow-up, universal profiles at blow-up and soliton resolution. This is viewed not as an isolated series of results, but as a way of approaching many similar critical non-linear dispersive equations. https://example.com/post/2012-03-13-i-a-s-workshop-on-symplectic-dynamics-2-monday/ Tue, 13 Mar 2012 00:00:00 +0000 https://example.com/post/2012-03-13-i-a-s-workshop-on-symplectic-dynamics-2-monday/ I’m participating in a workshop at the Institute for Advanced Study on Symplectic Dynamics. This is part of the special concentration this year organized by Helmut Hofer. I’ll try to write notes on the talks I hear. Apologies to the speakers for typos and misquotations…. Comments (pending approval) are open for suggestions and edits. IAS School of Mathematics Workshop web page 10:15 - 11:15 Peter Constantin, Princeton University, “Long time, vanishing viscosity limits” abstract 11:30 - 12:30 Steve Preston, University of Colorado, “The inextensible string as a toy model of fluids” abstract 2:30 - 3:30 Emanuele Caglioti, University of Rome, “Long time behavior of solutions of Vlasov-like equations” abstract 4:30 - 5:30 Roberto Camassa, University of North Carolina, “Large amplitude internal waves and their stability” abstract Peter Constantin: Long time, vanishing viscosity limits (pdf slides) Navier-Stokes Navier-Stokes equations. https://example.com/post/2012-03-13-i-a-s-workshop-on-symplectic-dynamics-2-tuesday/ Tue, 13 Mar 2012 00:00:00 +0000 https://example.com/post/2012-03-13-i-a-s-workshop-on-symplectic-dynamics-2-tuesday/ IAS School of Mathematics   Workshop web page Tuesday: 2012-03-13 9:00 - 10:00 Laszlo Szekelyhidi, University of Leipzig, “The h-principle for the Euler equations” abstract 10:15 - 11:15 Camilo de Lellis, University of Zurich, “The h-principle for the Euler equations Part 2” abstract 11:30 - 12:30 Vladimir Sverak, University of Minnesota, “On the long-time dynamics of some infinite-dimensional Hamiltonian systems” abstract 2:30 - 3:30 Antoine Choffrut, University of Leipzig, “On the local structure of the set of stationary flows to the 2D incompressible Euler equations” abstract 4:30 - 5:30 Thomas Kappeler, University of Zurich, “Symplectic techniques for integrable PDEs” abstract Laszlo Szekelyhidi: The h-principle for the Euler equations (chalk talk; joint work w. https://example.com/post/2012-02-14-academic-trendiness-at-a-a-a-s--vancouver--by-keyword-search-frequency/ Tue, 14 Feb 2012 00:00:00 +0000 https://example.com/post/2012-02-14-academic-trendiness-at-a-a-a-s--vancouver--by-keyword-search-frequency/ The American Association for the Advancement of Science is gathering for their Annual Meeting for 2012. One way to measure academic trendiness is to look at search frequencies of keywords using the search tool for the AAAS conference. “Basic Research” shows up 18 times; “Applied Research” shows up 93 times. “Africa” appears as frequently as “mathematics” and both appear less frequently than “warming” and “online”. “NSERC” appears as frequently as “Wellcome” and nearly 10 times less frequently than “NSF”. https://example.com/post/2011-12-13-2011-winter-c-m-s-meeting-notes/ Tue, 13 Dec 2011 00:00:00 +0000 https://example.com/post/2011-12-13-2011-winter-c-m-s-meeting-notes/ This past weekend, I attended the 2011 Winter meeting of the Canadian Mathematical Society. I have posted below some (often rough) notes on the talks that I attended. Robert McCann (joint work with J. Denzler and H. Koch) ROBERT MCCANN, University of Toronto Higher-order asymptotics of fast diffusion in Euclidean space: a dynamical systems approach. With Denzler and Koch, we quantify the speed of convergence and higher-order asymptotics of fast diffusion dynamics on Rn to the Barenblatt (self similar) solution. https://example.com/post/2011-12-13-2011-winter-c-m-s-town-hall-notes/ Tue, 13 Dec 2011 00:00:00 +0000 https://example.com/post/2011-12-13-2011-winter-c-m-s-town-hall-notes/ The 2011 Winter CMS Meeting took place this past weekend. On Sunday (2011-12-11), there was a CMS Town Hall meeting. The discussion was led by a panel consisting of Long Range Plan Committee Chair Nancy Reid, CMS President Jacques Hurtubise, and CMS Director Johan Rudnick (who was partially obscured by a poinsetta). Attendees to the town hall meeting were fed a free lunch. Long Range Plan Update Nancy gave a status update on the long range plan. https://example.com/post/2011-10-24-addendum-to-arnold-memorial-workshop-khesin-on-pinzari-talk/ Mon, 24 Oct 2011 00:00:00 +0000 https://example.com/post/2011-10-24-addendum-to-arnold-memorial-workshop-khesin-on-pinzari-talk/   The following message is a guest post by Boris Khesin. Boris summarizes the wonderful talk given by Gabriella Pinzari at the workshop. –Jim Colliander Gabriella Pinzari (30min talk) described her joint result with her advisor Luigi Chierchia on a recently found fix for the famous KAM theorem, or rather for its application to the stability of the Solar system. Namely, the original KAM theorem inArnold’s 1963 paper claimed the persistence of the Liouville tori for perturbations of integrable systems under some nondegeneracy assumption - some determinant must be nonzero. https://example.com/post/2011-10-03-edinburgh-arnold-memorial-workshop-notes/ Mon, 03 Oct 2011 00:00:00 +0000 https://example.com/post/2011-10-03-edinburgh-arnold-memorial-workshop-notes/ I am at an interesting workshop in Edinburgh entitled Dynamical systems and classical mechanics: a conference in celebration of Vladimir Arnold 1937 - 2010. Boris Khesin and Serge Tabachnikov have coordinated a Tribute to Vladimir Arnold which will soon appear in consecutive issues of the Notices of the AMS. These tributes were shared at the workshop and are also available here: Arnold1 Arnold2 I will post below my notes from the talks. https://example.com/post/2011-09-08-%C3%AEle-de-berder-workshop-notes/ Thu, 08 Sep 2011 00:00:00 +0000 https://example.com/post/2011-09-08-%C3%AEle-de-berder-workshop-notes/ I am at a workshop on Île de Berder. The post below contains the notes I am taking during the talks. I apologize (especially to the speakers) for misquotations and typos but I hope the notes might be useful. Tuesday 2011-09-06 Rafik Imekraz: Non resonant normal form for perturbed quantum harmonic oscillator Study high sobolev norms of solutions $\psi$ solving a NLS with harmonic trap and a nice smooth bounded potential. https://example.com/post/2011-01-20-edinburgh-meeting-notes-3/ Thu, 20 Jan 2011 00:00:00 +0000 https://example.com/post/2011-01-20-edinburgh-meeting-notes-3/ Galina Perelman: 2 soliton collision in NLS $$ i \partial_t \psi = - \psi_{xx} + F(|\psi|^2) \psi, ~ x \in R $$ where $F(\xi) = -2 \xi + O (\xi^2), ~ \xi \rightarrow 0.$ This family of equations has solitary wave solutions $$ e^{i \theta(x,t) \phi (x - b(t), E)} $$ where $\theta(x,t) = \omega t + \gamma + v \frac{x}{2}, ~b(t) =vt + c$ (all reall parameters). The profile $\phi$ is the associated ground state, which is $C^2$, decays exponentially, is even, … https://example.com/post/2011-01-19-edinburgh-meeting-notes-2/ Wed, 19 Jan 2011 00:00:00 +0000 https://example.com/post/2011-01-19-edinburgh-meeting-notes-2/ Sergei Kuksin (École Polytechnique): Nonlinear Schrödinger Equation We consider Hamiltonian PDE. This is of course very interesting. In physics, there is a class of pdes which is also of interest: Hamiltonian PDE = small damping + small forcing Why is it so important? This class contains a very important equation: Navier-Stokes. $$ \dot{u} + (u\cdot \nabla) u + \nabla p = \epsilon \Delta u + force; ~ \nabla \cdot u = 0. https://example.com/post/2011-01-18-edinburgh-meeting-notes-1/ Tue, 18 Jan 2011 00:00:00 +0000 https://example.com/post/2011-01-18-edinburgh-meeting-notes-1/ These are notes from a meeting entitled Advanced Numerical Studies in Nonlinear PDEs in Edinburgh, Scotland. Walter Craig (McMaster): Water Wave Interactions I’m an analyst but I’m going to talk about numerics and experiments as well as analysis. We will discuss the problem of water waves and then I’ll talk about two specific settings in which the theory has led to good and quite elegant numerics and the numerics have started to answer some questions. https://example.com/post/2010-08-28-notes-on-nonlinear-dispersive-equations-workshop-istanbul/ Sat, 28 Aug 2010 00:00:00 +0000 https://example.com/post/2010-08-28-notes-on-nonlinear-dispersive-equations-workshop-istanbul/ During last week’s NDE meeting in Istanbul, I experimented and took real-time notes in MultiMarkDown during the talks. If any of the speakers wish, I can post links to their presentations provided they send me a copy of their slides. –J. Colliander James Colliander (Toronto) Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2({\mathbb{T}})$ I spoke about recent work with Hiro Oh . Here is alink to my slides.