Analysis Seminar, Spring 2018
During Spring 2018, the Analysis Seminar will meet on Tuesdays 3:30-4:20pm in Keller 401.
If you would like to give a talk, please contact me at email@example.com.
- January 30 : Thomas Hangelbroek (University of Hawaii)
Nonlinear and anisotropic approximation with Gaussians.
In this talk I'll introduce and discuss a pair of nonlinear, multi-scale Gaussian approximation problems. For approximation with most positive definite kernels --
perhaps most obviously for Gaussian radial basis functions -- the selection of a scale by tuning of shape parameters is challenging, yet necessary in the following sense:
for constant scale approximation, high approximation rates are possible but at the cost of conditioning, while for stationary scaling (fixing the scale to fit a grid) the approximation problem can be made very stable but approximation power is lost.
Multi-scale strategies which use kernels at different dilation levels,
have been considered, but this aspect of kernel approximation is relatively new,
and its theoretical understanding lags far behind traditional techniques multi-scale tools
like wavelets and tensor product splines.
The first problem I'll discuss has been considered with Amos Ron.
It treats nonlinear approximation by Gaussians at multiple,
spatially varying scales, and provides correct rates for functions in standard smoothness spaces
for nonlinear approximation.
The second, recently considered with Amos Ron and Wolfgang Erb, treats $N$-term Gaussian
approximation of cartoon class functions with rates comparable to those of curvelets and shearlets.
- February 6 : Malik Younsi (University of Hawaii)
Removable sets, analytic capacity and the Cauchy transform.
A compact subset of the plane is called removable (for bounded analytic functions) if every bounded analytic function on the complement of the set is constant. Is there a geometric characterization of removability? This difficult question was raised by Painleve at the end of the 19th century, and it took more than a hundred years before a reasonable solution was obtained, due to the work of Melnikov, Tolsa, and many others. In this talk, I will give a brief introduction to Painleve's problem and discuss two relevant notions, namely analytic capacity and the Cauchy transform. We shall also see how Tolsa's work naturally leads to another notion of capacity, the Cauchy capacity. Finally, I will present new results on the relationship between the Cauchy capacity and analytic capacity, which shed new light on a question raised by Murai.
- February 13 : Rufus Willett (University of Hawaii)
Almost commuting matrices and Bott periodicity.
The following is an old question, made explicit by Halmos: given epsilon, is there a delta such that whenever u,v are unitary matrices with uv-vu smaller than delta in operator norm, can we find unitary u’,v’ within epsilon of u,v respectively, and with u’,v’ actually commuting? Importantly, delta is asked to exist independently of the size of the matrices. The answer, found by Voiculescu, is no. I’ll try to explain one way (due to Loring) of looking at Voiculescu’s solution, and why the existence of such an example is more-or-less equivalent to the Bott periodicity theorem from topology.
- March 13 : Wayne Smith (University of Hawaii)
Uniform approximation of Bloch functions and the boundedness of the integration operator on $H^\infty$.
I will present a necessary and sufficient condition for the operator of integration to be bounded on $H^\infty$ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions. I will also explain how this problem came up in operator-related
complex function theory on the unit disk. This is joint work with D. Stolyarov and A. Volberg.
- March 20 : Clement Dellaiera (University of Hawaii)
From a notion of dynamical dimension to cutting and pasting algebras.
We will present how some notions of dimension in geometric group theory and dynamical systems can be translated into operator algebraic properties, and how that can be useful for computing some of their invariants.
More precisely: I will try to introduce dynamical asymptotic dimension, introduced by E. Guentner, R. Willett and G. Yu, and explain how it relates to one of my current projects: stability of the Baum-Connes conjecture.
The talk doesn't assume any special background, and is directed to a non-specialized audience.
- April 24 : Malik Younsi (University of Hawaii)
Peano curves in complex analysis.
A Peano curve is a continuous function from the unit interval into the plane whose image contains a nonempty open set. The first example of such a curve was constructed by Peano in 1890, motivated by Cantor's proof of the fact that the unit interval and the unit square have the same cardinality.
One year later, in 1891, Hilbert gave an elegant geometric construction that has now become quite classical and is usually taught at the undergraduate level. Much less known, however, is the fact that Peano curves can be obtained as boundary values of certain power series on the disk, as observed by Salem and Zygmund in 1945. In this talk, I will describe how Peano curves can also be obtained from Cauchy transforms.