Conference Schedule (preliminary)

Monday
August 6
Tuesday
August 7
Wednesday
August 8
Thursday
August 9
Friday
August 10
8:30--9 Coffee Coffee Coffee Coffee Coffee
9--10 Helge Krueger Helge Krueger Helge Krueger Svetlana Jitomirskaya Svetlana Jitomirskaya
10:10--11:10 Stephen Gustafson Stephen Gustafson Stephen Gustafson Brett Wick Brett Wick
11:10--11:30 Coffee Coffee Coffee Coffee Coffee
11:30--12:20 Mike Jury Maxim Zinchenko Brett Wick
  (11:30--12:30)
Alfonso Montes-Rodriguez Svetlana Jitomirskaya
  (11:30--12:30)
12:20--2 Lunch Lunch Lunch Lunch Lunch
2--2:50 Anna Skripka Paul Mueller Michael Lacey Franciszek Szafraniec Jingguo Lai (2:25 -- 2:50)
2:55--3:20 Mishko Mitkovski Stephen Fulling Keshav Raj Acharya Yen Do Sohail Ahmed Dayo
3:25--3:50 Milivoje Lukic Kabe Moen Injo Hur Greg Knese Lupita Morales
3:50--4:10Coffee Coffee Coffee Coffee Tea
4:10--4:35 Kazuo Yamazaki talk 7 August Krueger Daniel Seco
6:30--... Dinner
at Madden's

All the talks will take place in Blocker 166.


Abstracts

Keshav Raj Acharya: Remling's Theorem on Canonical System
I will present the Remling's Theorem on Canonical System and sketch the proof of the theorem.
Stephen Fulling: An inverse problem in one-dimensional scattering theory
Consider a one-dimensional Schrödinger operator with potential V(z) that monotonically approaches 0 at -∞ and +∞ at +∞. Its spectral decomposition is characterized by the scattering phase shift, δ(p), for momenta p > 0. What technical conditions must I place on δ to assure that a V of the desired type exists? And given δ, what is V?
Stephen Gustafson: Regularity and singularities of Schrödinger maps
Geometric evolution equations such as the harmonic map heat-flow and (more recently) its dispersive cousins the wave and Schrödinger map equations, have recently received a great deal of attention in the PDE literature, not least because they are both geometrically natural, and physically relevant (ferromagnetism, liquid crystals, general relativity, etc.). In two space dimensions, these equations are all 'energy critical', and the basic question of whether solutions form singularities or remain globally smooth is a delicate and interesting one. In these 3 lectures, we will introduce these equations, their structure and properties, and the global well-posedness question, and then give a brief survey of some recent results on this question in the context of symmetric maps, with an emphasis on the Schrödinger case. Along the way we touch on some key recent developments in dispersive PDE, employing variational and harmonic analysis.
Injo Hur: Approximation results for 1D reflectionless Schrödinger operator
We will see that a reflectionless Schrödinger operator can be approximated by periodic Schrödinger operators.
Svetlana Jitomirskaya: Analytic quasiperiodic matrix cocycles: questions of continuity and applications
As, beginning with the famous Hofstadter's butterfly, all numerical studies of spectral and dynamical quantities related to quasiperiodic operators are actually performed for their rational frequency approximants, the questions of continuity upon such approximation are of fundamental importance. The fact that continuity issues may be delicate is illustrated by the recently discovered discontinuity of the Lyapunov exponent for non-analytic potentials. We will discuss the ideas behind several continuity results, focusing on harmonic analysis connections, and, as time permits, the applications to Avila's global theory of one-frequency Schrodinger operators and to genericity of dominated cocycles.
Michael Jury: Aleksandrov-Clark measures in one and several variables
This talk will review some of the theory of Aleksandrov-Clark measures, particularly connections with backward shift operators and rank-one perturbations. We then discuss some generalizations of these results to a multivariable setting
Greg Knese: Uchiyama's lemma and the John-Nirenberg inequality
Using integral formulas based on Green's theorem and in particular a lemma of Uchiyama, we give simple proofs of comparisons of different BMO norms without using the John-Nirenberg inequality while we also give a simple proof of the strong John-Nirenberg inequality. Along the way we prove the inclusions of BMOA in the dual of H^1 and BMO in the dual of real H^1
August Krueger: Some nonlinear bound states on a 1D half-lattice
We will discuss some recent work on the existence/uniqueness of real bound states for some nonlinear Schrödinger-like operators on a lattice. We will focus on a particular choice of kinetic operator with variable coefficients, polynomial nonlinearities, and an absence of external potential. We will include a short summary of the linear theory of Jacobi operators.
Helge Krueger: The skew-shift: Localization and Beyond
The basic plan is: Monday: The Green's function: Basic properties, recurrence of the skew-shift, and Cartan's lemma. Tuesday: Semi-algebraic sets & Localization. Wednesday: The spectrum contains intervals.
The presentation will largely follow:
J. Bourgain, Estimates on Green's functions, localization and the quantum kicked rotor model, Ann. of Math. (2) 156 (1) (2002) 249-294.
H. Krueger, The spectrum of skew-shift Schrödinger operators contains intervals. J. Funct. Anal. 262 (2012), no. 3, 773-810.
Time permitting, I will discuss recent results on the microscopic distribution of eigenvalues.
Michael Lacey: On the two weight inequality for the Hilbert transform
The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the to inequality holds if and only if two to weak- inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions. Joint work with Eric Sawyer, Chun-Yun Shen, and Ignacio Uriate-Tuero.
Jingguo Lai: Bellman Function for L^p Carleson Embedding Theorem
We will see an elegant proof of the L^p Carleson Embedding Theorem via Bellman Function which also claims the constant we get is sharp.
Milivoje Lukic: Schrödinger operators with decaying oscillatory potentials
We consider Schrödinger operators on the half-line with decaying oscillatory potentials. Our class includes potentials of the form V(x)=W(x)γ(x), where W(x) is almost periodic and γ(x) is a decaying function of bounded variation. We discuss several results which guarantee preservation of absolutely continuous spectrum and give restrictions on possible singular spectrum.
Kabe Moen: Abstract sampling theory
We examine extensions of the well known Shannon sampling theorem. We obtain new sampling functions in a general setting. We will use several analytic tools, including the Zak and Laurent transforms, as well as algebraic tools such as convolution idempotents. This is a joint work with Hrvoje Sikic, Guido Weiss, and Edward Wilson.
Alfonso Montes-Rodriguez: The Eigenfunctions of the Hilbert Matrix
For each non-integer complex number λ, the Hilbert matrix
H_λ=(1/(n+m+λ))_{n,m≥ 0}
defines a bounded linear operator on the Hardy spaces H^p, 1≤ p <∞, and on the Korenblum spaces A^{-τ}, τ>0. In this work, we determine the point spectrum with multiplicities of the Hilbert matrix acting on these spaces. This extends to complex λ. results by Hill and Rosenblum for real λ. We also provide a closed formula for the eigenfunctions. They are in fact closely related to the associated Legendre functions of the first kind. The results will be achieved through the analysis of certain differential operators in the commutator of the Hilbert matrix. This is a joint work with Alexandru Aleman (Lund University) and Andreea Sarafoleanu (Sibiu University).
Lupita Morales: Some aspects about the Riemann-Lebesgue Lemma
The Riemann-Lebesgue Lemma is related with the convergence of integrals of product functions this way f(x)g_n(x), where f is an integrable function and each g_n is its multiplier. In this study we present two results which are associated whith that lemma for functions in the completion of Henstock-Kurzweil space. Moreover, for finite intervals case, we show an analogous result to Riemann-Lebesgue Lemma and we get some consequences, as the Riemann-Lebesgue property for Dirichlet kernel, or nth partial sum behavior of the Fourier series on such completion. For Dirichletnite intervals case, we generalize the Riemann-Lebesgue Lemma for bounded variation functions such that their limits at infinity are zero. To end, we present functions which are of bounded variation that vanish at infinity and they are Henstock-Kurzweil integrable but are not Lebesgue integrable.
Paul Mueller: Compensated Compactness, separately convex Functions, Interpolatory Estimates
Motivated by a problem arising with the Calculus of Varitions we present Interpolatory estimates between Riesz Transforms and directional Haar projections. We discuss extensions thereof to Wavelet projections and to the vector valued case.
The talk is based on joint work with J. Lee and S. Müller : Compensated compactness, separately convex functions and interpolatory estimates between Riesz transforms and Haar projections. Comm. Partial Differential Equations 36 (2011), no. 4, 547--601.
Daniel Seco: Cyclicity and extremal polynomials
We study cyclic functions in the Dirichlet space (and other Hilbert spaces of analytic functions) and the properties of the optimal polynomials p_n in terms of the norm of p_n f - 1.
Anna Skripka: Spectral shift for contractions
The talk will discuss perturbation of functions of contractions and existence of (higher order) spectral shift functions for pairs of contractions. Jointly with D. Potapov and F. Sukochev we give an affirmative answer to a question on existence of a spectral shift function for an arbitrary pair of contractions with non-trace class difference V∈S^n, n≥2 posed by F. Gesztesy, A. Pushnitski, and B. Simon in [Zh. Mat. Fiz. Anal. Geom., 4 (2008), no. 1].
Franciszek Szafraniec: Naimark extensions for indeterminacy in the moment problem
Brett Wick: The Corona Theorem for Besov-Sobolev Spaces on the unit Ball
We discuss the ideas behind the absence of a corona in the multiplier algebra of the Drury-Arveson space. Included are the Toeplitz corona theorem, the Koszul complex, Charpentier's solution operators to d-bar equations, and the interplay with complex tangential vector fields. These ideas extend to other Besov-Sobolev spaces of holomorphic functions on the ball having varying degrees of smoothness, as well as to vector-valued settings.
Kazuo Yamazaki: On the regularity criteria of a surface quasi-geostrophic equation
We obtain regularity criteria for a quasi-geostrophic equation that depends more on one direction than the others. In particular, we show that in the critical case, the global regularity depends only on a partial derivative rather than a gradient of the solution.
Maxim Zinchenko: Spectral theory for finite gap Jacobi matrices
This talk is a review of major recent results in the spectral theory for Jacobi matrices with an emphasis on the single interval and finite gap cases.