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. A major objective is to expose attending graduate students to different areas of analysis. We hope these meetings will help to strengthen the partnership and increase the collaboration between analysts in the midwest.
This year workshop is supported by National Science Foundation grant DMS1545971. It will take place on IU Bloomington campus. The talks will be held at Swain Hall East 105 with coffee breaks set up at Rawles Hall lounge. If you are planning to drive to campus, parking will be free on any campus lot during the weekend. The closest ones are the Atwater parking garage or the big outdoor lots between Atwater St. and Third St. (between Fess St. and Woodlawn).
Accommodation of speakers is arranged by the organizers. All other participants are expected to book their hotel themselves. Nearby hotels include Courtyard by Marriott (15 minute walk from workshop), Hilton Garden Inn (15 minute walk from workshop), Hampton Inn (2 miles from workshop), Holiday Inn (2.2 miles from workshop), Comfort Inn (2.2 miles from workshop), SummerHouse (2.6 miles from workshop). There also will be a conference dinner on Saturday night (cost around $25).
The program and the list of submitted abstracts can be found here. If you have any further questions, please email the organizers at email@example.com. We would appreciate if you could help us to disseminate information about the workshop by printing and posting our poster at your department.
More picture from the workshop can be found here.
Joint work with V. Eiderman, B. Jaye, A. Volberg, X. Tolsa, and Maria Reguera
We shall discuss one possible approach to the DavidSemmes problem asking to describe Borel measures μ in Rn such that the associated sdimensional Riesz transform acts in L2(μ). This approach leads naturally to several easy to state questions from the classical potential theory, the answers to which are still waiting to be found.
The problem by Steklov consists in obtaining sharp bounds on the polynomials orthogonal on the unit circle (or a segment on the real line) with respect to a measure in the given class. We will discuss some recent developments which gave the complete solution to this problem. The connections to sharp estimates on the Hilbert transform in weighted spaces will be explained and some new bounds will be presented.
For Ω a smooth domain in the plane and w∈L1(Ω) a nonnegative weight, we consider the orthonormal polynomials Pn in L2(Ω,w). We prove exterior asymptotics for Pn outside the convex hull of Ω under weak regularity assumptions for the weight. This generalizes previous results by Korovkin, Suetin, Miña-Díaz, and Simanek.
In this talk, we survey equilibrium problems in the presence of external fields, focusing our attention on the case where the conductor is the real axis. In addition, some recent advances and new results are presented, especially concerning the number of cuts (intervals) comprising the support of the equilibrium measure. Some important applications, in particular those related to asymptotics of Orthogonal Polynomials and Random Matrix models, are reviewed.
Joint work with R. Shafikov.
It is wellknown that any complexvalued continuous function on the unit circle can be approximated by rational combinations of exactly one continuous function. We will discuss analogous phenomena for certain function spaces on general manifolds. This problem has interesting connections to complex analysis and symplectic geometry.
We will define local regularity of functions and survey recent results connecting the multifractal analysis to local regularity. These ideas are applied to the study of boundary behavior of harmonic functions in the upper halfspace. We use wavelet characterization and martingale technique to prove the law of the iterated logarithm for oscillation of harmonic functions of controlled growth along vertical lines.
Joint work with Ph. Jaming, Yu. Malinnikova, and K.M. Perfekt.
We prove that if a solution of a discrete timedependent Schrödinger equation with bounded timeindependent real potential decays fast at two distinct times then the solution is trivial. The continuous case was studied by L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega. We consider a semidiscrete equation, where time is continuous and spatial variables are discretized. For the free Shrödinger operator, or operators with compactly supported potential, a sharp analogue of the Hardy uncertainty principle is obtained. The argument is based on the theory of entire functions. The logarithmic convexity of weighted norms is employed for the case of general realvalued bounded potential, following the ideas developed for the continuous case. Our result for the case of a bounded potential is not optimal.
Joint work with R. Buckingham.
We formulate and study a class of initialvalue problems for the sine-Gordon equation in the semiclassical limit. The initial data parametrizes a curve in the phase portrait of the simple pendulum, and near points where the curve crosses the separatrix, a doublescaling limit reveals a universal wave pattern constructed of superluminal kinks located in the spacetime along the real graphs of all of the rational solutions of the inhomogeneous PainlevéII equation. The kinks collide at the real poles, and there the solution is locally described in terms of certain doublekink exact solutions of sineGordon.
This study naturally leads to the question of the largedegree asymptotics of the rational solutions of PainlevéII themselves. In the time remaining we will describe recent results in this direction, including a formula for the boundary of the polefree region, strong asymptotics valid also near poles, a weak limit formula, planar and linear densities of complex and real poles, and specialized asymptotic formulas valid near the boundary of the polefree region.
The universality phenomenon in the context of random polynomials says that asymptotic distribution of (appropriately normalized) zeros of random polynomials should become independent of the choice of distribution of random coefficients as their degree goes to infinity. In this talk, I will present some recent results on universality of limiting zero distribution of multivariate random polynomials. I will also describe generalization of these results to the setting of random holomorphic sections of high powers L⊗n of an ample line bundle L→X over a projective manifold X.
How many gas pumps should one build to ensure that every driver has a pump at a fixed small distance? Where should one place these pumps to minimize the number? It turns out that randomly placed points cover a regular set very well (even though they may be separated very badly). We will discuss this and related results.
The problem of constructing a Feller process on Rd, d≥3, having infinitesimal generator −Δ + b · ∇, with a singular vector field b:Rd→Rd (“a diffusion with drift b”) has been thoroughly studied in the literature, motivated by applications to Mathematical Physics, as well as the search for the maximal (general) class of vector fields b such that the associated process exists. This search culminated in several distinct classes of singular drifts (including Ld+L∞, the best possible result in terms of Lp spaces). I construct a process with b in a wide class of vector fields (measures), containing the classes previously known, and combining, for the first time, critical point singularities and critical hypersurface singularities.
I introduce a new method “of constructing the resolvent”: the starting object is an operatorvalued function, a ‘candidate’ for the resolvent of an operator realization Λp(b) of −Δ + b · ∇ generating a holomorphic C0semigroup in Lp (the key observation here is a link between −Δ + b · ∇ and (λ-Δ)1/2+|b|, as opposed to the classical approach, which compares −Δ + b · ∇ to −Δ + b2 ). The very form of this function provides detailed information about smoothness of the domain D(Λp(b)). Now, this information about D(Λp(b)), combined with the Sobolev embedding theorem, allows us to move the burden of the proof of convergence in the space C∞ of continuous functions on Rd vanishing at ∞ (as needed to construct the transition probability function of the process) to Lp, a space having much weaker topology (locally), hence the gain in the admissible singularities of the drift.
Joint work with Y. Liu, Z. Chen, and Y. Pan.
Motivated by a UCP problem, we investigate the local existence of flat solutions to CauchyRiemann equations. We construct a germ of a smooth ∂closed (0,1) form f, flat at 0∈Cn, such that there is no flat smooth solution to ∂u = f locally. The construction also provides a family of Cauchy Riemann equations whose minimal solutions restricted onto any proper subsets of the domain are no longer minimal with respect to any bounded plurisubharmonic weights.