Conference Schedule (preliminary)
All the talks will take place in Blocker 166.
- 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
- 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
- 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 L² to L² inequality
holds if and only if two L² to weak-L² 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
- 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
is almost periodic and
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
defines a bounded linear operator on the Hardy spaces
H^p, 1≤ p <∞, and on the Korenblum spaces
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,
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.