# 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 this semester on Wednesday, September 9th and 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:

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