Department of Mathematics

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On the Cauchy problem of the one-dimensional cubic NLS in the low regularity setting

by Tadahiro Oh | Princeton University
Time: 11:00 — 12:00  (Monday, May. 28, 2012)
Location: BA2135, Bahen Center, 40 St George St
Abstract:
In this talk, we discuss some recent developments on the Cauchy problem on the one-dimensional cubic NLS both on R and T.  We first go over the classical well-posedness theory: energy method (by Sobolev embedding), space-time estimates (Strichartz estimates, Bourgain's Fourier restriction norm method).  In view of "ill-posedness" results below L^2, there is no satisfactory well-posedness theory below L^2.  On R, Christ-Colliander-Tao and Koch-Tataru proved existence of solutions below L^2.  It turns out, however, that the periodic cubic NLS is genuinely ill-posed below L^2.  We introduce the Wick ordered cubic NLS (which is equivalent to the usual cubic NLS in L^2) and discuss probabilistic well-posedness results (with Jim Colliander) and an existence result (with Zihua Guo) below L^2.  In particular, for the second result, we use the so-called short-time Fourier restriction norm method (along with the I-method and normal form reduction) to establish an a priori estimate.  As a byproduct, we also prove that non-existence of a weak solution to the usual cubic NLS below L^2.  If time permits, we will discuss how one can use an infinite iteration of normal form reductions (which is the same as the infinite iteration of the I-method) to prove uniqueness and continuous dependence below L^2(T).

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DEPARTMENTAL PHD THESIS EXAM -- Maximal-in-time behavior of deterministic and stochastic dispersive partial differential equations

by Geordie Richards  (www) | University of Toronto
Time: 14:00  (Monday, May. 28, 2012)
Location: BA6183, Bahen Center, 40 St George St
Abstract:
This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem?

The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions.

The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported in H^s(T) for s<1/2, and the local-in-time solution to the quartic KdV cannot be obtained below H^{1/2}(T) using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}(T). Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}(T).

The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.

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DEPARTMENTAL PHD THESIS EXAM - Stably non-stable C*-algebras with no bounded trace

by Henning Petzka  (www) | University of Toronto
Time: 11:10  (Monday, May. 28, 2012)
Location: BA6183, Bahen Center, 40 St George St
Abstract:
A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. My thesis deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stably finiteness.

For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra A that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over A are finite, while A has no bounded quasitrace. The second example is a non-unital, simple C*-algebra B that is stably non-stable, i.e. no matrix algebra over B is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra B has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.

Everyone welcome. Refreshments will be served in the Math Lounge before the exam.

Dates in this series

· Friday, Jan. 15, 2010: Learning Seminar on Teichmuller Theory and Dynamics (Sylvain Bonnot )

· Monday, Feb. 22, 2010: Departmental PhD Thesis Exam -- The Gromov Width of Coadjoint Orbits of Compact Lie Groups (Masrour Zoghi)

· Tuesday, Feb. 23, 2010: An introduction to derived manifolds (David Spivak)

· Wednesday, Feb. 24, 2010: The construction of derived manifolds (David Spivak)

· Wednesday, Feb. 24, 2010: Departmental PhD Thesis Exam -- Scalar and Vector Coherent State Representations of Compact and Non-Compact Symplectic Groups in a Unitary Basis (Lindsey Shorser)

· Thursday, Feb. 25, 2010: The cobordism theory of derived manifolds (David Spivak)

· Thursday, Feb. 25, 2010: FIELDS INSTITUTE OPERATOR ALGEBRA SEMINAR -- An introduction to Thompson's group F (Uffe Haagerup)

· Friday, Mar. 05, 2010: UNDERGRADUATE MATH UNION TALK -- Mathematical & Computational Challenges in the Biomedical Sciences (Siv Sivaloganathan)

· Friday, Mar. 12, 2010: MASTER'S THESIS PRESENTATION -- On moments of class numbers of real quadratic fields (Alexander Dahl)

· Tuesday, Mar. 16, 2010: FIELDS INSTITUTE OPERATOR ALGEBRA SEMINAR -- TFT and topological K-theory (Snigdhayan Mahanta)

· Wednesday, Mar. 17, 2010: Special Informal Seminar (Jacob Lurie)

· Monday, Mar. 22, 2010: Departmental PhD Thesis Exam -- Ramsey property of posets and related structures (Miodrag Sokic)

· Wednesday, Mar. 24, 2010: UNDERGRADUATE MATH UNION TALK -- The Mathematics of Arbitrage (Matheus Grasselli)

· Tuesday, Apr. 06, 2010: FIELDS INSTITUTE OPERATOR ALGEBRA SEMINAR -- Gerbes and generalized Kahler geometry (Marco Gualtieri)

· Tuesday, Apr. 13, 2010: Departmental PhD Thesis Exam -- The Multivariable Alexander Polynomial on Tangles (Jana Archibald)

· Wednesday, Apr. 21, 2010: Departmental PhD Thesis Exam -- Geometric Theory of Parshin Residues (Mikhail Mazin)

· Monday, Apr. 26, 2010: Departmental PhD Thesis Exam -- Singularity formation in nonlinear heat and mean curvature flow equations (Wenbin Kong)

· Friday, Apr. 30, 2010: Departmental PhD Thesis Exam -- Grassmann dynamics (Mario Morfin Ramirez)

· Tuesday, May. 04, 2010: Departmental PhD Thesis Exam -- Classical Lie Algebra Weight Systems of Arrow Diagrams (Louis Leung)

· Tuesday, May. 04, 2010: SDE Summer Reading Group -- Organizational Meeting ()

· Wednesday, May. 05, 2010: Departmental PhD Thesis Exam -- L-functions in number theory (Yichao Zhang)

· Thursday, May. 06, 2010: Random Groups Summer Reading Group -- Organizational Meeting ()

· Monday, May. 10, 2010: Departmental PhD Thesis Exam -- Standing Ring Blowup Solutions for the Cubic Nonlinear Schrodinger Equation (PhD Candidate: Ian Zwiers )

· Friday, May. 14, 2010: One Dimensional Dynamics Reading Group -- Organizational Meeting (Pablo Carrasco)

· Tuesday, May. 18, 2010: W-knots and the Alexander polynomial (Dror Bar-Natan)

· Thursday, May. 20, 2010: Undergraduate Math Union Summer Seminar Series -- Seminar Series Debut: Research Project in Number Theory, Equidistribution of Roots (Mitsuru Wilson)

· Wednesday, May. 26, 2010: SPECIAL COLLOQUIUM -- Introduction to Resolution of Singularities (Jaroslaw Wlodarczyk)

· Wednesday, May. 26, 2010: Resolution of Singularities (Jaroslaw Wlodarczyk)

· Monday, Jun. 07, 2010: Departmental PhD Thesis Exam -- Dynamical Foliations (Tanya Firsova)

· Monday, Jun. 21, 2010: Departmental PhD Thesis Exam -- Non-Abelian Localization and U(1) Chern-Simons Theory (Brendan McLellan)

· Monday, Jun. 28, 2010: Departmental PhD Thesis Exam -- Compact Dynamical Foliations (Pablo Carrasco)

· Tuesday, Jun. 29, 2010: Departmental PhD Thesis Exam -- Contributions towards a fine structure theory of Aronszajn orderings (Carlos Martinez Ranero)

· Tuesday, Jun. 29, 2010: Departmental PhD Thesis Exam -- Non-chainable continua and Lelek's problem (Logan Hoehn)

· Thursday, Jul. 08, 2010: A few remarks on Mahler's conjecture on convex symmetric bodies  (Fedor Nazarov)

· Monday, Jul. 19, 2010: Departmental PhD Thesis Exam -- Abrikosov Lattice Solutions of the Ginzburg-Landau Equations of Superconductivity (Tim Tzaneteas)

· Tuesday, Aug. 17, 2010: Reidemeister distance of knot diagrams and complexity of relations in the braid group* (Arkadius Kalka)

· Wednesday, Sep. 29, 2010: On the determinantal representations of plane curves (Dmitry Kerner)

· Friday, Oct. 01, 2010: Communicating Mathematics (Ivars Peterson)

· Wednesday, Oct. 06, 2010: On the local determinantal representations of plane curves (Dmitry Kerner)

· Thursday, Oct. 14, 2010: Departmental PhD Thesis Exam -- On the JLO Character and Loop Quantum Gravity (Chung Lun Alan Lai)

· Friday, Oct. 15, 2010: WORKING GROUP IN HAMILTONIAN SYSTEMS -- Pentagram map (Fedor Soloviev)

· Tuesday, Oct. 19, 2010: Optimal Transportation Minicourse: Geometry, Regularity and Applications (Part 1 of 2) (Robert McCann)

· Wednesday, Oct. 20, 2010: Departmental PhD Thesis Exam -- De Rham Theory and Semialgebraic Geometry (Leonid Shartser)

· Thursday, Oct. 21, 2010: Optimal Transportation Minicourse: Geometry, Regularity and Applications (Part 2 of 2) (Robert McCann)

· Tuesday, Oct. 26, 2010: GRADUATE MINI-COURSE: An introduction to infinity harmonic functions (Part 1A) (Scott Armstrong)

· Tuesday, Oct. 26, 2010: GRADUATE MINI-COURSE: An introduction to infinity harmonic functions (Part 1B) (Charles Smart )

· Thursday, Oct. 28, 2010: GRADUATE MINI-COURSE: An introduction to infinity harmonic functions (Part 1C) (Scott Armstrong)

· Thursday, Oct. 28, 2010: GRADUATE MINI-COURSE: An introduction to infinity harmonic functions (Part 1D) (Charles Smart )

· Wednesday, Nov. 10, 2010: Departmental PhD Thesis Exam -- Hrushovski and Ramsey Properties of Classes of Finite Inner Product Structures, Finite Euclidean Metric Spaces, and Boron Trees (Jakub Jasinski)

· Tuesday, Jan. 18, 2011: SPECIAL GRADUATE STUDENT SEMINAR -- Motivic Integration (Nir Avni)

· Wednesday, Jan. 19, 2011: On tropical homology theories and/or other details for the talk the day before (Grisha Mikhalkin)

· Friday, Jan. 21, 2011: SPECIAL GRADUATE STUDENT SEMINAR -- Quantum ergodicity and quantum maps (Dubi Kelmer )

· Monday, Jan. 24, 2011: On local behavior of holomorphic functions along complex submanifolds of $C^N$ (Alexander Brudnyi)

· Wednesday, Mar. 16, 2011: Joint Fields Institute Colloquium in Applied Mathematics and Math Department Colloquium -- Breakdown of Smoothness in the Muskat Problem (Charles Fefferman)

· Monday, Apr. 04, 2011: DEPARTMENTAL PHD THESIS EXAM - The Left Regular Representation of a Semigroup,  (Barry Rowe)

· Thursday, Apr. 07, 2011: DEPARTMENTAL PHD THESIS EXAM -- Structural results on optimal transportation plans ,  (Brendan Pass)

· Thursday, Apr. 21, 2011: DEPARTMENTAL PHD THESIS EXAM -- Loewner Theory in Several Complex Variables and Related Problems,  (Mircea Voda)

· Thursday, May. 19, 2011: PUBLIC LECTURE - Structural results on optimal transportation plans (Brendan Pass)

· Monday, May. 30, 2011: DEPARTMENTAL PHD THESIS EXAM -- The Cuntz Semigroup of C(X,A),  (Aaron Tikuisis)

· Friday, Jun. 17, 2011: DEPARTMENTAL PHD THESIS EXAM - Low Regularity Stability for Subcritical Generalized Korteweg-de Vries Equations (Brian Pigott)

· Friday, Jun. 24, 2011: DEPARTMENTAL PHD THESIS EXAM -- Toric Varieties Associated with Moduli Spaces (James Uren)

· Monday, Jul. 18, 2011: DEPARTMENTAL PHD THESIS EXAM - On van der Waals forces (Ioannis Anapolitanos)

· Wednesday, Jul. 27, 2011: DEPARTMENTAL PHD THESIS EXAM -- Finite Rank Perturbations of Random Matrices and Their Continuum Limits (Alexander Bloemendal)

· Monday, Aug. 15, 2011: DEPARTMENTAL PHD THESIS EXAM - A Universal Finite Type Invariant of Knotted Trivalent Graphs (Zsuzsanna Dancso)

· Friday, Aug. 19, 2011: Master's Thesis Presentation - On the Logarithmic Calculus and Sidorenko’s Conjecture (Janet Li)

· Wednesday, Sep. 07, 2011: DEPARTMENTAL PHD THESIS EXAM -- Lie 2-Algebras as Homotopy Algebras over a Quadratic Operad (Travis Squires)

· Monday, Sep. 12, 2011: DEPARTMENTAL PHD THESIS EXAM -- Resolution of singularities of pairs preserving semi-simple normal crossings,  (Franklin Vera Pacheco)

· Monday, Oct. 24, 2011: DEPARTMENTAL PHD THESIS EXAM - Geometric analysis on solutions of some differential inequalities and within restricted classes of holomorphic functions (Damir Kinzebulatov)

· Friday, Oct. 28, 2011: DEPARTMENTAL PHD THESIS EXAM - On the local and global classification of generalized complex structures (Michael Bailey)

· Tuesday, Nov. 01, 2011: SPECIAL SEMINAR -- Taylor projectors defined on the spacer of holomorphic functions on manifolds (Sh. Izumi)

· Monday, Nov. 07, 2011: DEPARTMENTAL PHD THESIS EXAM -- The Pure Virtual Braid Group is Quadratic (Stephen Peter Lee)

· Tuesday, Nov. 15, 2011: Proving that Artinian implies Noetherian without proving that Artinian implies finite length (Chris Conidis)

· Friday, Nov. 18, 2011: SEVERAL COMPLEX VARIABLE SEMINAR -- Mappings of 2-nondegenerate hypersurfaces in dimension three (Ilya Kossovskiy)

· Friday, Nov. 25, 2011: Algorithm Design Using Spectral Graph Theory (Richard Peng)

· Wednesday, Feb. 08, 2012: DEPARTMENTAL PHD THESIS EXAM - L-functions and Number Theory (Jaehyun Cho)

· Monday, Feb. 27, 2012: DEPARTMENTAL PHD THESIS EXAM - Solving Nested Recursions with Trees (Abraham Isgur)

· Monday, Mar. 05, 2012: DEPARTMENTAL PHD THESIS EXAM - $\mathcal{LA}$-Courant Algebroids and their Applications (David Li-Bland)

· Monday, Apr. 02, 2012: DEPARTMENTAL PHD THESIS EXAM - Topological Methods in Galois Theory,  (Yuri Burda)

· Tuesday, Apr. 03, 2012: DEPARTMENTAL PHD THESIS EXAM - Bases for Invariant Spaces and Geometric Representation Theory (Bruce Fontaine)

· Wednesday, Apr. 18, 2012: The icosahedron: stem cell of the solid state? (Very preliminary "report") (Marjorie Wikler Senechal)

· Wednesday, Apr. 25, 2012: DEPARTMENTAL PHD THESIS EXAM - Tropical Severi Varieties and Applications (Jihyeon Jessie Yang)

· Monday, Apr. 30, 2012: DEPARTMENTAL PHD THESIS EXAM - Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets (Artem Dudko)

· Monday, May. 14, 2012: DEPARTMENTAL PHD THESIS EXAM - Transfer relations in essentially tame local Langlands correspondence (Kam-Fai Tam)

· Friday, May. 18, 2012: DEPARTMENTAL PHD THESIS EXAM - Flat Virtual Pure Tangles (Karene Chu)

· Monday, May. 28, 2012: On the Cauchy problem of the one-dimensional cubic NLS in the low regularity setting (Tadahiro Oh)

· Monday, May. 28, 2012: DEPARTMENTAL PHD THESIS EXAM -- Maximal-in-time behavior of deterministic and stochastic dispersive partial differential equations (Geordie Richards)

· Monday, May. 28, 2012: DEPARTMENTAL PHD THESIS EXAM - Stably non-stable C*-algebras with no bounded trace (Henning Petzka)

· Tuesday, May. 29, 2012: DEPARTMENTAL PHD THESIS EXAM - Covering Arrays with Row Limit (Nevena Francetic)

· Monday, Jun. 11, 2012: DEPARTMENTAL PHD THESIS EXAM - On $C^1$ rigidity for maps with a break point (Elio Mazzeo)

· Friday, Jun. 15, 2012: DEPARTMENTAL PHD THESIS EXAM - Diffeologies, Differential Spaces, and Symplectic Geometry (Jordan Watts)

· Thursday, Jun. 28, 2012: DEPARTMENTAL PHD THESIS EXAM - Optimal Transport Approach to Non-linear Evolution Equations  (Ehsan Kamalinejad )

· Wednesday, Dec. 05, 2012: DEPARTMENTAL PHD THESIS EXAM: The self-power map and its image modulo a prime (Catalina V. Anghel)

· Wednesday, Dec. 05, 2012: DEPARTMENTAL PHD THESIS EXAM: The distribution of values of logarithmic derivatives of real L-functions (Mariam Mourtada)

· Thursday, Dec. 06, 2012: SPECIAL PROBABILITY SEMINAR -- Couplings of probability spaces and related issues ( Balázs Szegedy)

· Tuesday, Jan. 22, 2013: DEPARTMENTAL PHD THESIS EXAM: The (co)isoperimetric problem in (random) polyhedra  (Dominic Dotterrer)