Research

Classical Analysis
(Leader: )

Friday, April 19, 2024

Title: Rosette Minimal Surfaces and JS surfaces
Speaker: Jane McDougall, Colorado College
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

The study of Minimal Surfaces goes back several centuries, but such surfaces are an active area of research today, both in mathematics, and in the applied sciences; applications arise from area-minimizing properties of a minimal surface, and there is particular interest in periodic structures. Minimal surfaces are closely related to harmonic mappings, which are univalent (one-to-one) harmonic functions of one complex variable. Harmonic mappings do not preserve angles, and so behave differently from conformal mappings. Under certain conditions harmonic mappings do lead to minimal surfaces and by obtaining a new harmonic mapping we can sometimes obtain a new minimal surface. The Poisson integral formula is one way to obtain a harmonic mapping. A technique dating to 1984, known as the shear construction, is another. An “inspired application” circa 2017 led to the discovery of the Rosette Harmonic Mappings. We examine Rosette Minimal Surfaces, and some similarities and differences when compared with JS surfaces, which are also minimal surfaces but are obtained from the Poisson integral formula. Characteristic of a JS surface is its projection onto a polygonal-shaped harmonic mapping, where the graph approaches either \(+\infty\), or \(-\infty\) on any given bounding edge. A well-known example from the 19th century is the doubly-periodic Scherk surface. One particular Rosette Minimal Surface has commonalities with Scherk’s surface, but instead of being doubly periodic, it is triply periodic and so tiles three dimensional space.

Friday, April 12, 2024

Title: Minimal Zeros Revisited and the Shanks Conjecture
Speaker: Catherine Bénéteau
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

In the last decade, there has been a lot of work on optimal polynomial approximants, which are polynomials that indirectly approximate inverses of analytic function in various Hilbert or Banach spaces. In this talk, I will revisit some known work related to minimal zeros of optimal polynomial approximants in certain weighted Hilbert spaces of the disk. This problem is connected to orthogonal polynomials on the real line, Jacobi matrices, and some particular differential equations. It turns out that it also answers the weak Shanks Conjecture in the Hardy space of the bidisk, which relates to the location of zeros of certain least square inverse polynomials of several variables. I will examine all these connections and conclude with what can be said about the Shanks Conjecture.

Friday, April 5, 2024

Title: Extremal polynomials in Rogosinski-Szego estimate for the coefficients of typically real polynomials
Speaker: Alex Stokolos, Georgia Southern University
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

Famous Bieberbach conjecture claims the estimate \(|a_n|\le n\) for a univalent in the unit disc function \(f(z)=z+a_2z^2+\dotsb\). The conjecture was stated in 1916 by L. Bieberbach, and finally proven by L. de Branges in 1985. An important intermediate step was the proof for the real coefficients case, done independently by J. Dieudonne and W. Rogosinki in 1931. Univalent functions with real coefficients are typically real. In 1950, W. Rogosinski, G. Szegő got sharp estimates for the \(a_2\) and \(a_3\) coefficients of a typically real polynomial. However, the extremal polynomials were not found. In my talk, I will present the extremal polynomials. This is a joint work with D. Dmitrishin, D. Gray, and W. Trebels. The presentation will be accessible to everyone who took the standard Complex Analysis course.

Friday, March 29, 2024

Title: Almost optimal uniform polynomial approximation on convex sets in \(\mathbb{C}\)
Speaker: Liudmyla Kryvonos, Vanderbilt University
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

For the function \(f\), continuous on a convex set \(K\) in \(\mathbb{C}\) and analytic in its interior points, I will construct a sequence of polynomials that up to a constant gives optimal rate of approximation on the whole \(K\) and converges with geometric rate at the interior points. I will also discuss the limitations on the possible rates of convergence of almost optimal polynomials for sets with a different geometry.

Friday, March 22, 2024

Title: Random Acyclic Orientations and the Length of the Longest Path: Analytic Combinatorics as a Complex Analyst’s side hustle
Speaker: Erik Lundberg, Florida Atlantic University
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

Given a graph (in the combinatorial sense, i.e., a network) an acyclic orientation is an assignment of directions to each edge in a way that does not form any (directed) cycles. Peter J. Cameron posed the problem of studying the distribution of the length of the longest (directed) path when the acyclic orientation is random. We answer this question when the underlying graph is a complete bipartite graph (a graph with vertex set consisting of two parts such that every vertex in the first part is connected to every vertex in the second part). In addition to providing a generating function whose coefficients give the probability distribution of the longest path, we use Complex Analysis to show that the distribution is asymptotically Gaussian in the case of equal part sizes (tending to infinity). This is joint work with Jessica Khera. Most of the talk will be self-contained and accessible for students.

Friday, March 8, 2024

Title: Uniqueness problem in the algebra \(H^oo\) on nowhere dense sets of finite perimeter: a trip down the memory lane, Part III
Speaker: Dmitry Khavinson
Time: 4:00pm–5:00pm
Place: CMC 130

Friday, March 1, 2024

Title: Uniqueness problem in the algebra \(H^oo\) on nowhere dense sets of finite perimeter: a trip down the memory lane, Part II
Speaker: Dmitry Khavinson
Time: 4:00pm–5:00pm
Place: CMC 130

Friday, February 23, 2024

Title: Uniqueness problem in the algebra \(H^oo\) on nowhere dense sets of finite perimeter: a trip down the memory lane
Speaker: Dmitry Khavinson
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

The algebra \(H^oo\) of bounded, analytic in the domain, functions can be defined on an arbitrary compact set \(X\) in \(C\) as the weak \(*\) closure in \(L^oo(dxdy)\) of rational functions with poles outside \(X\). Surprisingly, although defined almost everywhere wrt the area measure, these functions have many properties similar to the usual \(H^oo\) algebra on bounded domains. In particular, on the so-called sets of finite perimeter, e.g., Swiss cheeses, those functions have the Cauchy integral representation. There are interesting open uniqueness questions that remain. No background on geometric measure theory or Banach algebras will be assumed, and the talk should be accessible to undergraduate students who have had one semester of complex and one semester of real analysis.

Friday, February 16, 2024

Title: Pfaff and Grassmann enter D-bar, Part II
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

Friday, February 9, 2024

Title: Spectral theory of soliton gases for integrable equations: some basic facts, recent developments and open questions
Speaker: Alexander Tovbis, University of Central Florida
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

We present a brief overview of the theory of soliton gases for integrable systems using algebro-geometric approach. We also discuss some related open problems and their connection with potential theory.

Friday, February 2, 2024

Title: Pfaff and Grassmann enter D-bar
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

Generalizing the notion of Pfaffian of a skew-symmetric, even-dimensional matrix, we relate the Laplace operator in two real variables to the matrix representation of the antiholomorphic derivative operator. This leads to the observation that the eigenvalue problem for the D-bar operator computes the Gaussian curvature of a two-dimensional Riemannian manifold, which implies the surprising result that the Liouville equation can be solved by Grassmannian calculus. Time-permitting, (far-reaching) applications of this result will be discussed.

Friday, January 26, 2024

Title: A survey in R.M.T., Part II
Speaker: Louis Arenas
Time: 4:00pm–5:00pm
Place: CMC 130

Friday, January 19, 2024

Title: A survey in R.M.T.
Speaker: Louis Arenas
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

As a statistical model, Random Matrix Theory (R.M.T.) has enjoyed success in modeling a variety of physical processes famously ranging from average spacings of energy levels in heavy nuclei atoms to the average spacing of public buses in Cuernavaca Mexico.

In this talk we focus on the simplest collection of random matrices called the “Gaussian Unitary Ensemble” where its inherit symmetry allows one to develop expressions involving Hermite polynomials. The Gaussian case of “Wigner Semi-Circle law“, which is a statement about the bulk distribution of eigenvalues as the size of the random matrix grows, follows almost immediately from these expressions. We additionally explain how the ensuing determinantal structure allows us to compute “gap probabilities” of the eigenvalues.

Capitalizing off this determinantal structure, in the sequel we will continue to survey facts about “Normal Matrix Ensembles” and summarize current joint work between Seung-Yeop Lee and myself.