## Seminars and Colloquia by Series

Wednesday, April 13, 2011 - 14:00 , Location: Skiles 005 , Mishko Mitkovski , School of Mathematics, Georgia Tech , Organizer: Brett Wick
It is well known that, via the Bargmann transform, the completeness problems for both Gabor systems in signal processing and coherent states in quantum mechanics are equivalent to the uniqueness set problem in the Bargmann-Fock space. We introduce an analog of the Beurling-Malliavin density to try to characterize these uniqueness sets and show that all sets with such density strictly less than one cannot be uniqueness sets. This is joint work with Brett Wick.
Wednesday, April 6, 2011 - 14:00 , Location: Skiles 005 , Karl Deckers , Georgia Tech , Organizer: Brett Wick
Consider a positive bounded Borel measure \mu with infinite supporton an interval [a,b], where -oo <= a < b <= +oo, and assume we have m distinctnodes fixed in advance anywhere on [a,b]. We then study the existence andconstruction of n-th rational Gauss-type quadrature formulas (0 <= m <= 2)that approximate int_{[a,b]} f d\mu. These are quadrature formulas with npositive weights and n distinct nodes in [a,b], so that the quadratureformula is exact in a (2n - m)-dimensional space of rational functions witharbitrary complex poles fixed in advance outside [a,b].
Wednesday, March 30, 2011 - 14:00 , Location: Skiles 005 , Sergey Denissov , University of Wisconsin-Madison , , Organizer:
We consider the 1d wave equation and prove the propagation of the wave provided that the potential is square summable on the  half-line. This result is sharp.
Wednesday, March 16, 2011 - 14:00 , Location: Skiles 005 , Betsy Stovall , UCLA , Organizer: Michael Lacey
We will discuss a proof that finite energy solutions to the defocusing cubicKlein Gordon equation scatter, and will discuss a related result in thefocusing case.  (Don't worry, we will also explain what it means for asolution to a PDE to scatter.)  This is joint work with Rowan Killip andMonica Visan.
Thursday, March 10, 2011 - 15:00 , Location: Skiles 006 , Ka-Sing Lau , Hong Kong Chinese University , Organizer:
There is a large literature to study the behavior of the image curves f(\partial {\mathbb D}) of analytic functions f on the unit disc {\mathbb D}. Our interest is on the class of analytic functions f for which the image curves f(\partial {\mathbb D}) form infinitely many (fractal) loops. We formulated this as the Cantor boundary behavior (CBB). We develop a general theory of this property in connection with the analytic topology, the distribution of the zeros of f'(z) and the mean growth rate of f'(z) near the boundary. Among the many examples, we showed that the lacunary series such as the complex Weierstrass functions have the CBB, also the Cauchy transform F(z) of the canonical Hausdorff measure on the Sierspinski gasket, which is the original motivation of this investigation raised by Strichartz.
Wednesday, March 9, 2011 - 14:00 , Location: Skiles 005 , Michael Loss , School of Mathematics, Georgia Tech , Organizer: Jeff Geronimo
This talk is about a random Schroedinger operator describing the dynamics of an electron in a randomly deformed lattice. The periodic displacement configurations which minimize the bottom of the spectrum are characterized. This leads to an amusing problem about minimizing eigenvalues of a Neumann Schroedinger operator with respect to the position of the potential. While this conﬁguration is essentially unique for dimension greater than one, there are inﬁnitely many different minimizing conﬁgurations in the one-dimensional case. This is joint work with Jeff Baker, Frederic Klopp, Shu Nakamura and Guenter Stolz.
Wednesday, March 2, 2011 - 14:00 , Location: Skiles 005 , Camil Muscalu , Cornell , Organizer: Michael Lacey
Calderon's algebra can be thought of as a world whichincludes singular integral operators and operators of multiplicationwith functions which grow at most linearly (more precisely, whose firstderivatives are bounded).The goal of the talk is to address and discuss in detail the followingnatural question: "Can one meaningfully extend it to include operatorsof multiplication with functions having polynomial growth as well ?".
Wednesday, February 16, 2011 - 14:00 , Location: Skiles 006 , Tim Ferguson , University of Michigan , Organizer: Michael Lacey
I will discuss linear extremal problems in the Bergman spaces $A^p$ ofthe unit disc and a theorem of Ryabykh about regularity of thesolutions to these problems.  I will also discuss extensions I havefound of Ryabykh's theorem in the case where $p$ is an even integer.The proofs of these extensions involve Littlewood-Paley theory and abasic characterization of extremal functions.
Wednesday, February 2, 2011 - 14:00 , Location: Skiles 006 , Yen Do , Georgia Tech , Organizer: Brett Wick
I will survey recent results about the convergence of the Wash-Fourier series near L1. Joint work with Michael Lacey.
Wednesday, January 26, 2011 - 14:00 , Location: Skiles 005 , Prof. Dany Leviatan , Tel Aviv University , Organizer: Doron Lubinsky
Let C[-1, 1] be the space of continuous functions on [-1, 1], and denote by \Delta^2 the set of convex functions f \in C[-1, 1]. Also, let E_n(f) and En^{(2)}_n(f) denote the degrees of best unconstrained and convex approximation of f \in \Delta^2 by algebraic polynomials of degree < n, respectively. Clearly, E_n(f) \le E^{(2)}_n (f), and Lorentz and Zeller proved that the opposite inequality E^{(2)}_n(f) \le CE_n(f) is invalid even with the constant C = C(f) which depends on the function f \in \Delta^2. We prove, for every \alpha > 0 and function f \in \Delta^2, that sup{n^\alpha E^{(2)}_n(f) : n \ge 1} \le c(\alpha)sup{n^\alpha E_n(f): n \ge 1}, where c(\alpha) is a constant depending only on \alpha. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (-1,1) is also investigated. It turns out that there are substantial differences between the cases s \le 1 and s \ge 2.