# Joint Analysis Seminar

Introducing The Joint Analysis Seminar! This seminar is co-organized by Ciprian Demeter, Norm Levenberg, and Michael R. Pilla. We’re kicking off the spring semester this Wednesday, January 27th. We will meet weekly at 4:15pm. The Zoom Room will open at 4pm. This is a joint seminar with the goal of having talks accessible to a general analysis audience. If you are interested, please contact me for joining information!

You will find the current schedule below:

Spring 2021:

01/27/2021: Alex Izzo, Bowling Green State University

Title: Convergence of Hulls of Curves

Abstract: It is known from work of Forstneric, Rosay, Low, and Wold that given a manifold M of dimension d<n, the set of polynomially convex, totally real embeddings of M in complex-n-space C^n of class C^s, is open and dense in the space of all totally real embeddings of M in C^n of class C^s in the C^s topology. For the case of polynomially convex simple closed curves, we establish stronger results regarding openness and denseness. We also show that under suitable hypotheses, the polynomial hull of the limit of a sequence of simple closed curves is the limit of the polynomial hulls. This is joint work with Lee Stout.

02/10/2021: Evangelos Nikitopoulos, UCSD

Title: It\^{o}’s Formula in Free Probability

Abstract: Free probability is a noncommutative analogue of classical probability theory — complete with a notion of independence, Brownian motion, etc. — that provides a framework for the study of large-$n$ limits of various $n \times n$ random matrix models. In 1998, P. Biane and R. Speicher defined a “free” analogue of the stochastic integral and proved a kind of It\^{o}’s Formula in this setting. In this talk, I shall motivate this formula using matrix-valued stochastic calculus, describe a generalization of the formula, and illustrate some useful consequences of the generalization. No background in free probability is necessary to follow the talk.

02/17/2021: Roozbeh Gharakhloo, Colorado State University

Title: Asymptotics of Bordered Toeplitz Determinants

Abstract: Among the numerous appearances of Toeplitz determinants in Mathematics, Physics, and Engineering problems, one of the most outstanding is the groundbreaking discovery of Kaufman and Onsager who established that the diagonal and horizontal two-point correlation functions in the square lattice Ising model have Toeplitz determinant representations. In another interesting development in 1987, Au-Yang and Perk expressed the next-to-diagonal correlations of the anisotropic Ising model in terms of a bordered Toeplitz determinant: a determinant with Toeplitz structure except for its last row or column. In my talk, after motivating the problem I will explain how a class of bordered Toeplitz determinants is encoded in the solution of the Baik-Deift-Johansson Riemann-Hilbert problem for biorthogonal polynomials on the unit circle. This approach is inspired by the 2007 work of N.Witte in which the connection of certain bordered Toeplitz determinants to the system of biorthogonal polynomials on the unit circle was found. Finally, I will show that by applying our general results to the Ising case, we can rigorously confirm the already heuristically known result that in the low-temperature regime, the next-to-diagonal correlations are the same as the diagonal and horizontal ones. This talk is based on joint work with Estelle Basor, Torsten Ehrhardt, Alexander Its, and Yuqi Li, where we independently employ Riemann-Hilbert, operator-theoretic and numerical methods to obtain the asymptotics of a class of bordered Toeplitz determinants as the size of the matrix tends to infinity.

02/24/2021: Ahmad Barhoumi, University of Michigan

Title: Kissing polynomials: The Story So Far

Abstract: Kissing polynomials, dubbed so for the peculiar behavior of their zeros, are a family of polynomials orthogonal with respect to an oscillatory, complex-valued weight. These polynomials were first considered in the development of a Gaussian quadrature rule to address highly oscillatory integrals. Since the weight of orthogonality is complex-valued, the quadrature nodes are not necessarily restricted to the real line, nor are we guaranteed n nodes! In this talk, I will introduce these Kissing polynomials and discuss several new results on their existence and asymptotic behavior.

03/03/2021: Ulises Fidalgo, Case Western Reserve University

Title: An Extension of Markov Theorems for Nikishin Systems

Abstract: We consider sequences of Hermite-Padé approximants for Nikishin systems. For such sequences we give the weakest uniform convergence condition found so far.

03/10/2021: Alan Legg, Purdue University Fort Wayne

Title: On the best uniform approximation to the checkmark function

Abstract: TBA

03/17/2021: Andrei Prokhorov, University of Michigan, Ann Arbor

Title: Behavior of rational solutions of Painleve III equation near zero

Abstract: TBA

03/17/2021: Zane Li, Indiana University, Bloomington

Title: TBA

Abstract: TBA

03/31/2021: Krystal Taylor, Ohio State University, Columbus

Title: TBA

Abstract: TBA

Fall 2020:

09/09/2020: Michael R. Pilla, IU Bloomington

Title: A Generalized Cross Ratio

Abstract: In this talk, we define a generalized cross ratio and determine some of its basic properties. In particular, by defining linear fractional maps in several complex variables, we have a class of maps that obey similar transitivity properties as in one variable, under some more restrictive conditions.

09/16/2020: Norm Levenberg, IU Bloomington

Title: Polynomials Associated to Non-Convex Bodies

Abstract: Polynomial spaces associated to a convex body $C$ in ${R^+}^d$ have been the object of recent studies. We first recall this  setting and then we consider polynomial spaces associated to possibly non-convex $C$. This is done in order to discuss quantitative  Runge-type polynomial approximation results using relatively sparse families of polynomials. Joint work-in-progress with Franck Wielonsky.

09/23/2020: Chris Judge, IU Bloomington (RESCHEDULED)

Title: On the Schwartz kernel theorem

09/30/2020: Yen Do, University of Virginia

Title: The number of real roots for random trigonometric polynomials: universality and non-universality of the variance

Abstract: We study the number of real roots of random trigonometric polynomials with iid coefficients of mean zero and bounded moments. We show that the variance of this number is asymptotically linear in terms of the expectation. This result extends a prior work of Bally, Caramellino, and Poly (where some smoothness conditions are required for the coefficient distributions). In particular, our methods work for discrete trigonometric polynomials. Joint work with Hoi Nguyen (Ohio State) and Oanh Nguyen (UIUC).

10/7/2020: Dominique Kemp, IU Bloomington

Title: A Weakening of the Curvature Condition in $R^3$ for $\ell^p$ decoupling

Abstract: The celebrated decoupling theorem of Bourgain and Demeter allows for a decomposition in the $L^p$ norm of functions Fourier supported near curved hypersurfaces $M \subset R^n$. In this project, we find that the condition of non-vanishing principal curvatures may be weakened. When $M \subset R^3$, we may allow one principal curvature at a time to vanish, and it is assumed additionally that $M$ is foliated by a canonical choice of curves having nonzero curvature at every point. We find that $\ell^p$ decoupling over nearly flat subsets of $M$ holds within this context.

10/14/2020: Ching Wei Ho, IU Bloomington

Title: The Brown measure of variables with semicircular imaginary part

Abstract: We are interested in computing the limiting eigenvalue distribution of the random matrix model of the form M_N+ i X_N, where A_N is deterministic, X_N is a Gaussian unitary ensembles. Computer simulation suggests this limiting eigenvalue distribution is expected to be the Brown measure of the sum of an arbitrary self-adjoint variable and an imaginary multiple of the semicircular variable. I will speak on the recent work with Brian Hall, where we computed this Brown measure explicitly. I will also specially look at the Brown measure corresponding to the special case where M_N has the form A_N+Y_N, where A_N is deterministic and Y_N is a Gaussian unitary ensemble which is independent of X_N and may have different variance from X_N. In this case, it was proved by Sniady that the Brown measure is the limiting eigenvalue distribution. These Brown measures are absolutely continuous, the support and density are closely related to the Hermitian random matrix M_N+X_N

10/20/2020: Ian Charlesworth, UC Berkeley (NOTE DIFFERENCE IN DATE)

Title: Free Stein Dimension

Abstract: Regularity questions in free probability ask what can be learned about a tracial von Neumann algebra from probabilistic-flavoured qualities of a set of generators. Broadly speaking there are two approaches — one based in microstates, one in free derivations — which with the failure of Connes Embedding are now known to be distinct. The non-microstates approach is not obstructed by non-embeddable variables, but can be more difficult to work with for other reasons. I will speak on recent work with Brent Nelson, where we introduce a quantity called the free Stein dimension, which measures how readily derivations may be defined on a collection of variables. I will spend some time placing it in the context of other non-microstates quantities, and sketch a proof of the exciting fact that free Stein dimension is a *-algebra invariant.

10/28/2020: Chris Judge, IU Bloomington

Title: On the Schwartz kernel theorem

11/04/2020: K. Hambrook, San Jose State University

Title: Explicit Salem Sets of Arbitrary Dimension in $R^n$

Abstract: A set in $R^n$ is called Salem if it supports a probability measure whose Fourier transform decays as fast as the Hausdorff dimension of the set will allow. We construct the first explicit (i.e., non-random) examples of Salem sets in $R^n$ of arbitrary prescribed Hausdorff dimension. This completely resolves a problem proposed by Kahane more than 60 years ago. The construction is based on a form of Diophantine approximation in number fields. This is joint work with Robert Fraser.

11/11/2020: I. Paata, North Carolina State University

Title: Rademacher and Enflo type coincide

Abstract: Pick any finite number of points in a Hilbert space. If they coincide with  vertices of a parallelepiped then the sum of the squares of the lengths of its  sides equals the sum of the squares of the lengths of the  diagonals (parallelogram law). If the points are in a general position then we can define sides and diagonals by labeling these points via vertices of the discrete cube ${0,1}^n$. In this case  the sum of the squares of diagonals is bounded by the sum of the squares of its sides no matter how you label the points and what n you choose. In a general Banach space we do not have parallelogram law. Back in 1978  Enflo asked: in an arbitrary Banach space  if the sum of the squares of diagonals  is bounded by the sum of the squares of its sides for all parallelepipeds (up to a universal constant), does the same estimate  hold  for any finite number of points (not necessarily vertices of the parallelepiped)? In the joint work with Ramon van Handel and Sasha Volberg we positively resolve Enflo’s problem. Banach spaces satisfying the inequality with parallelepipeds are called of type 2 (Rademacher type 2), and Banach spaces satisfying the inequality for all points are called of Enflo type 2. In particular, we show that Rademacher and enflo type coincide.

11/18/2020: Len Bos, University of Verona

Title: Optimal Prediction Measures

Abstract: Optimal measures for a compact set $K \subset R^d$ generalize the notion of optimal points for multivariate polynomial interpolation and coincide with Optimal Designs for polynomial regression from statistics. An optimal prediction measure deals with the problem of minimizing the variance of a least squares prediction by values interior to  the set $K,$ at a point exterior to it. We show that this problem is equivalent to the one of finding the polynomial of uniform norm one on $K$ which has maximal value at the exterior point. For $K$ the interval $[-1,1]$ and a real exterior point, the solution is known to be the classical Chebyshev polynomial. We use our equivalency theorem to find the polynomials of extremal growth for a purely complex external point. This is joint work with N. Levenberg, J. Ortega-Cerda and others.

12/02/2020: Martino Fassina, University of Vienna and Indiana University

Title: Sup norm Estimates for the Cauchy Riemann Equations

Abstract: It is a fundamental problem in several complex variables to find solutions to the inhomogeneous Cauchy-Riemann equations that satisfy some kind of estimates. Using tools from complex analysis in one variable, one can write an explicit integral operator that solves the inhomogeneous Cauchy-Riemann equations on domains that are products of one-dimensional domains. I will show that such a solution satisfies supnorm estimates. I will also discuss weak solutions and a possible approach to an old question of Kerzman and Stein which is still open. The talk is based on joint work with Yifei Pan.

12/09/2020: Hari Bercovici, Indiana University

Title: Super Convergence: An Exercise in Free Harmonic Analysis

Abstract: It has been observed that limit laws in free probability tend to be sharper in some ways than their classical counterparts. In this talk, I focus on the fact that weak convergence can often be upgraded to convergence of densities. This has consequences (which I will not touch upon) on the local study of eigenvalues for random matrices. I will however go into the fairly elementary complex analytic aspects of this work. Joint with C-W Ho, J-C Wang, and P Zhong.