The goal of MWAA is to facilitate interactions between mathematicians from the midwestern United States working in approximation theory, mathematical physics, potential theory, complex analysis by bringing them together in an informal way for a two day workshop. Anyone interested in participating should email any of the organizers. Speakers may email abstracts to Adam Coffman.
This year the workshop is held in conjunction with the IPFW annual Minisymposium on Analysis. It will take place on IPFW campus with the talks being given on the second floor of Kettler Hall. Those of the participant that say at the university housing should check in at the Club House first. You can find the Club House, Housing, and Kettler Hall on this map.
The workshop is in part supported by IPFW Department of Mathematical Sciences, IPFW Office of Research, Engagement, and Sponsored Programs, and Simons Foundation Collaboration Grant.
When attempting to generalize results from orthogonal polynomials on the unit circle to more general settings, a natural case to consider is that when the measure of orthogonality is concentrated on a region whose boundary is defined by the level set of a polynomial. In this talk, we will explain why this is the case and explore some recent results on this topic. One of our main results is a set of conditions on the entries of the Bergman Shift matrix that is equivalent to the measure being concentrated (in an appropriate sense) near the boundary of a polynomial lemniscate.[slides]
Joint work with P. Boyvalenkov, P. Dragnev, E. Saff, M. Stoyanova
Based upon the works of DelsarteGoethalsSeidel, Levenshtein, Yudin, and CohnKumar we derive universal lower bounds for the potential energy of spherical codes, that are optimal (in the framework of the standard linear programming approach) over a certain class of polynomial potentials whose degrees are upper bounded via a familiar formula for spherical designs. We classify when improvements are possible employing polynomials of higher degree. Our bounds are universal in the sense of Cohn and Kumar; i.e., they apply whenever the potential is given by an absolutely monotone function of the inner product between pairs of points.[slides]
We describe some simple sufficient geometric conditions on a compact set E of the plane under which the normalized counting measures of the zeros of any asymptotically extremal sequence of polynomials necessarily converges in the weakstar topology to the equilibrium measure for E. The question of existence of “electrostatic skeletons” for compact sets E arises naturally in the context of such asymptotic problems.[slides]
Lunch will be provided by the workshop
In this talk we will examine asymptotic properties of a family of polynomials that naturally arises in CR geometry. In particular we will show how these polynomials are intimately related to Chebyshev polynomials.
Joint work with X. Huang
In this talk, we discuss CR transversality of holomorphic maps between CR hypersurfaces. Let Mℓ be a smooth Levinondegenerate hypersurface of signature ℓ in Cn with n≥3, and write HℓN for the standard hyperquadric of the same signature in CN with N-n< (n-1)/2. Let F be a holomorphic map sending Mℓ into HℓN. Assume F does not send a neighborhood of Mℓ in Cn into HℓN. We show that F is necessarily CR transverse to Mℓ at any point.[slides]
It was once hoped that whenever a compact set in complex Euclidean space has a nontrivial polynomially convex hull, there must be analytic structure in the hull. This hope was dashed by a counterexample given by Stolzenberg in 1963. I will present recent joint work with Samuelsson Kalm and Wold showing that every smooth manifold of dimension at least three can be smoothly embedded in some complex Euclidean space so as to have hull without analytic structure and present current work with Stout extending this to two dimensional manifolds. (It is well known that a smoothly embedded one dimensional manifold never has hull without analytic structure.)[slides]
Ordinary potential theory is concerned with (pluri)subharmonic functions in the complex plane (or on higher dimensional real or complex manifolds). These functions can also be thought of as defining hermitian metrics on line bundles. Noncommutativity enters when one passes to holomorphic vector bundles with fibers of dimension > 1 and hermitian metrics on them. Such hermitian metrics locally can be represented by self adjoint matrix functions, and taking the curvature of the metric is analogous to applying the Laplacian to a scalar valued function. In the talk I will discuss properties of positively/negatively curved metrics, i.e. matrix functions, that generalize properties of (pluri)sub and superharmonic functions.
Joint work with K. Liechty
We present an exact solution to the large N limit of the sixvertex model with partial domain wall boundary conditions in the ferroelectric phase. The solution consists of two steps. In the first step we derive a formula for the partition function involving the determinant of a matrix of mixed Vandermonde/Hankel type. This determinant can be expressed in terms of a system of discrete orthogonal polynomials, which can then be evaluated asymptotically by comparison with the Meixner polynomials.
Joint work with T. Bothner, P. Deift, and I. Krasovsky
We study the determinant det(I-γKs), 0 < γ < 1, of the Fredholm operator Ks acting on the interval (-1,1) with kernel Ks(λ,μ)= sin(s(λ - μ))/π(λ-μ). This determinant represents one of the fundamental distributions functions of the random matrix theory. We evaluate, in terms of elliptic thetafunctions, the double scaling limit of det(I-γKs) as s→∞ and γ→1, in the region cs-ε≤-(1/2s)ln(1-γ)≤1-δ, for any fixed 0 < δ< 1. This problem was first considered by Dyson in 1995.
For Y a subset of the complex plane, a β ensemble is a sequence of probability measures Probn,β,Q on Yn for n=1,2,... depending on a positive real parameter β and a realvalued continuous function Q on Y. We consider the associated sequence of probability measures on Y where the probability of a subset W of Y is given by the probability that at least one coordinate of Yn belongs to W. With appropriate restrictions on Y, Q we prove a large deviation principle for this sequence of probability measures. This extends a result of BorotGuionnet to subsets of the complex plane and to β ensembles defined with measures using a BernsteinMarkov condition.
We construct a smooth function f that is flat at the origin, and is such that ∂u= f has no flat solutions.
Joint work with Beneteau, Kosinski, Liaw, Seco, and Sola
A vector is cyclic for an operator or family of commuting operators if the closed invariant subspace it generates is the whole Hilbert space. A famous result of Smirnov and Beurling says that the cyclic vectors for the shift operator on the Hardy space on the disk are exactly the outer functions. Generalizing this result to more dimensions and in particular to polydisks is well-motivated by the fact that characterizing cyclic vectors for the Hardy space on the infinite polydisk is closely related to Nyman's dilation completeness problem, which is known to be equivalent to the Riemann hypothesis. In this talk we confine ourselves to two variables and we completely characterize the cyclic *polynomials* for the shift operators on a range of Hilbert spaces of analytic functions on the bidisk which include the Hardy space and the Dirichlet space. The answer depends on the size and nature of the zero set of the polynomials on the distinguished boundary of the bidisk.[slides]
Joint work with A. Draux, V.A. Kalyagin and D.N. Tulyakov
The classical A. Markov inequality establishes a relation between the maximum modulus or the L∞([-1,1]) norm of a polynomial Qn and of its derivative: ||Q'n||≤ Mn n2||Qn||, where the constant Mn=1 is sharp. The limiting behavior of the sharp constants Mn for this inequality, considered in the space L2([-1,1],w(α,β)) with respect to the classical Jacobi weight w(α,β)(x):=(1-x)α(x+1)β, is studied. We prove that, under a technical condition |α - β| < 4, the limit is limn → ∞ Mn = 1/(2 jν) where jν is the smallest zero of the Bessel function Jν(x) and 2ν= min(α,β) - 1. Recently V. Totik, basing on our result, has removed this technical condition on the parameters of the Jacobi weights.[slides]
Joint work with A. Aptekarev and G. LópezLagomasino
Pollaczek multiple orthogonal polynomials are type II HermitePadé polynomials orthogonal with respect to two simple measures supported on the positive semiaxis. These measures form a socalled Nikishin pair, with the feature that one of its generators is purely discrete. It is known that the largedegree asymptotics of such polynomials is governed by the solution of a vector equilibrium problem, which was previously computed by V. Sorokin. For the strong asymptotics we use the RiemannHilbert characterization of the HermitePadé polynomials and the corresponding nonlinear steepest descent method. We discuss some of the main ingredients of this analysis and the asymptotic results obtained by this method.[slides] [slides-keynote]