The goal of MWAA is to facilitate interactions between mathematicians from the midwestern United States working in approximation theory, mathematical physics, potential theory, and 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. We would appreciate if you could help us to disseminate information about the workshop by printing and posting **our poster** at your department.

This year workshop is supported in part by National Science Foundation and Simons Foundation Collaboration Grant. It will take place on IUPUI campus. The keynote address will take place in the Science building, room LD010 (refreshments will be available in LD 259). The talks will be held in the Technology building, room ET202, with coffee breaks set up in LD259.

If you are planning to drive to campus, please, look for visitors parking (you will have to pay, but we shall reimburse you later). The closest visitor parking to Technology and Science buildings are North Street Parking Garage and Gateway Parking Garage. For more visitor parking, please, consult this **map**. Please, consider carpooling.

Accommodation of speakers is arranged by the organizers. All other participants are expected to book their hotel themselves. Participants are encouraged to stay at Candlewood Suites, where a block of rooms is reserved for MWAA participants. Reservations can be made at 317-536-7700. The rooms reserved include studios and 2 bedroom suites. Please, consider sharing a 2 bedroom suite as this year workshop funding is very constrained.

The program and the list of submitted abstracts can be found **here**.

IUPUI

University of Toronto

University of Toledo

Indiana University

Central Michigan University

American Mathematical Society

University of Kentucky

Texas Tech University

IUPUI

University of Chicago

Michigan State University

University of Indiana

IPFW

Purdue University

Indiana University

Central Michigan University

University of Michigan

PUCMM

Michigan State University

Indiana University

IUPUI

Oklahoma State University

Michigan State University

Texas Tech University

Indiana University

IUPUI

Mathematical Reviews

University of Toledo

IUPUI

**3:00—3:30pm**

**3:30—4:30pm**

*Parts of this talk are joint work with D. Calegari and A. Walker, and parts of this talk are joint work with X. Buff and A. Epstein*

In William Thurston's last paper, * Entropy in Dimension One*, there is a spectacular image associated to entropy values of quadratic polynomials on the very first page. This set displays some amazing fractal structure which can be (somewhat) understood when viewed as a subset of parameter space for a particular family of iterated function systems (IFS). In this talk, we compare this with the parameter space discussion of the family \( z\mapsto z^2+c \), and investigate the associated connectedness locus in parameter space for the IFS. If time permits, we study another subset arising in the dynamical realm which displays similar structure; this set is also associated to postcritically finite quadratic polynomials.

**9:00—9:40am**

**09:40—10:20am**

This talk summarizes some recent results in the weighted theory of commutators of multiplication with Calderón–Zygmund operators \(T\). The prototypical such operator is the Hilbert transform in dimension \(1\), and the Riesz transforms in higher dimensions. We are looking at operators of the form \([b, T]f = b Tf - T(bf)\). It is a famous theorem of Coifman, Rochberg and Weiss that this operator is bounded in \(L^p(\mathbb{R}^n)\) if \(b\) is in \(BMO(\mathbb{R}^n)\) (and conversely, if the commutator with the Riesz/Hilbert transforms is bounded, then \(b\) is in BMO). It is also known that this same result holds if we work with \(L^p(\mathbb{R}^n, w)\), where \(w\) is a Muckenhoupt \(A_p\) weight. We discuss recent extensions of this result to the case where the commutator acts between two different weighted spaces, so \([b, T]: L^p(\mu) \rightarrow L^p(\lambda)\), where \(\mu\) and \(\lambda\) are \(A_p\) weights.

**10:30—11:10am**

We consider the motion of the coupled system constituted by a heavy rigid body with an interior cavity completely filled by a viscous liquid. We study the stability of the steady–state configurations and the long–time dynamics of the coupled system. In particular, we show that after an initial interval of time, whose length depends on the initial data as well as on the relevant physical parameters involved, every Leray–Hopf solution to the equations of motion, corresponding to a “large” set of initial data (with finite total energy), approaches to a (stable) steady–state at an exponentially fast rate.

**11:20am—12:00pm**

*Joint work with Kossovskiy, Huang, and Li*

We will discuss the embeddability problems of real hypersurfaces into hyperquadrics and spheres. In particular, we present a negative answer to a question concerning the embeddability of compact strongly pseudoconvex real algebraic hypersurfaces into spheres.

**12:00—2:00pm**

**2:00—2:40pm**

In this talk I will explain a method that allows, in certain settings, to derive series expansions (and hence asymptotics) of a sequence of orthonormal polynomials \( P_n(z) \), \( n=0,1,\ldots \). The important idea I want to convey is that everything is encoded in the reproducing kernel \( \sum_{n=0}^\infty P_n(z)\overline{P_n(\zeta)} \), and once this kernel is known, there is a rather direct way to extricate the asymptotic behavior of the polynomials. I will illustrate how the method works in the case of orthogonal polynomials over circular multiply connected domains.

**2:50—3:30pm**

*Joint work with Pavel Bleher*

Random matrices appeared for the first time in the works of Wishart, back in the 1920's, but came into spotlight after the studies of Dyson in the 1950's related to high energy physics. In the past thirty or so years, the theory of random matrices has attracted a lot of activity, partially in virtue of its connection to several areas of physics and mathematics, as for instance statistical mechanics, number theory, approximation theory, telecommunications, algebraic geometry and integrable systems, to mention only a few.

In this talk, we plan to discuss the relation between eigenvalues of large random matrices and zeros of orthogonal polynomials of high degree. As we will discuss at the beginning of our talk, when the matrix model is hermitian, it is now well–understood that the eigenvalue distribution and the distribution of zeros, in the appropriate limit, coincide and are described in terms of a weighted equilibrium problem.

However, when the random matrices under consideration are normal, the connection between eigenvalues and the zeros is given in terms of the so–called motherbody problem, and the dynamical evolution of them can be described in terms of the Laplacian growth.

**3:30—4:00pm**

**4:00—4:40pm**

The pseudo–ultraspherical polynomial of degree \(n\) is defined by \(\tilde{C}_n^{(\lambda)}(x) =(- i) ^n C_n^{(\lambda)}(ix)\) where \(C_n^{(\lambda)}(x)\) is the ultraspherical polynomial. We discuss the orthogonality of finite sequences of pseudo–ultraspherical polynomials \(\{\tilde{C}_{n}^{(\lambda)}\}_{n=0}^{N}\) for different values of \(N\) that depend on \(\lambda.\) We discuss applications of Wendroff's Theorem and use an identity linking the zeros of the pseudo–ultraspherical polynomial \(\tilde{C}_n^{(\lambda)}\) with the zeros of the ultraspherical polynomial \(C_n^{(\lambda')}\) where \(\lambda'= \frac12 - \lambda -n\) to prove that when \( 1-n < \lambda < 2-n,\) two (symmetric) zeros of \(\tilde{C}_{n}^{(\lambda)}\) lie on the imaginary axis.

**4:50—5:30pm**

*Joint work with M. Yattselev and the late H. Stahl*

We consider the problem of approximating a function \( f \) of one complex variable, analytic except over a polar subset of \(\mathbb{C}\) comprising only finitely many branchpoints all of which are algebraic, by a rational or meromorphic function with at most \( n \) poles, uniformly on a compact subset \( K \) of the domain of analyticity of \( f \). It is known that there exists a (essentially) unique compact set \( S \) outside of which \( f \) is single valued, to minimize the condenser capacity \( C(S,K) \). Subsequently, the optimal \( n \)–th root rate of approximation to \( f \) on \( K \) was shown in to be \( \exp\{-2/C(K,S)\} \). We explain in this talk that the normalized counting measure of the poles of a sequence of best approximants converges weak\(^*\) to the condenser equilibrium distribution on \( S \), and more generally that any sequence of optimal approximants in the \(n\)–th root sense has this property. The same results holds for best meromorphic (i.e. AAK) approximants. We dwell on the solution to the Gonchar conjecture on the degree of approximation by Parfenov, combined with some AAK theory and logarithmic potential theory.

**10:00—10:40am**

Let \( G \) be a domain in a complex manifold \( M \) of dimension \( n \). The core \( c(G) \) of \( G \) is defined to be the set of all points \( P \) in \( G \) such that the rank of the Levi form of \( F \) at \( P \) is less than \( n \) for every smooth bounded above plurisubharmonic function \( F \) in \( G.\) In this talk we make an overview of some recent results and some open questions related to this notion.

**10:50—11:30am**

*Joint work with Charles Chui and Hrushikesh Mhaskar*

Decomposition of signals into finitely many primary building blocks, called *atoms*, is a fundamental problem in signal analysis, especially when the instantaneous frequencies of the atoms are close together. In this talk, we describe a novel data–driven, local method to address this problem, called *SuperEMD*, which is in essence a clever adaptation and combination of the popular empirical mode decomposition (EMD) and the signal separation operator (SSO). The highlights of our discussion include a natural formulation of the data–driven atoms, a modified sifting process for EMD for real–time implementation with specific reference to handling boundary artifacts, motivation of the SSO, the description of our SuperEMD, and experimental results.

**11:40am—12:20pm**

*Joint work with Shahaf Nitzan and Michael Northington*

The classical Balian–Low theorem is a strong form of the uncertainty principle that constrains the time–frequency localization of Gabor systems that form orthonormal bases. We discuss a generalization of the Balian–Low theorem that provides a sharp scale of constraints on the time–frequency localization of Gabor systems under a weaker form of spanning structure associated with so–called exact \( C_q \) systems. Admissibility conditions on Fourier multipliers play an important role in the proofs and, as an additional application, yield sharp Balian–Low type theorems in the setting of shift–invariant spaces.