Complex Analysis
Complex analysis is a branch of mathematics that studies functions of a complex variable, where numbers have both real and imaginary parts. It focuses on functions that are complex differentiable, called holomorphic or analytic functions, which exhibit powerful and elegant properties such as infinite differentiability and representation by power series. Overall, complex analysis extends calculus into the complex domain and has deep theoretical beauty along with important applications in science and engineering.
Featured Publications
In 2021, Dan Reznik made a YouTube video demonstrating that power circles of Poncelet triangles have an invariant total area. He made a simulation based on this observation and put forward a few conjectures. One of these conjectures suggests that the sum of the areas of three circles, each centered at the midpoint of a side of the Poncelet triangle and passing through the opposite vertex, remains constant. In this paper, we provide a proof of Reznik's conjecture and present a formula for calculating the total sum. Additionally, we demonstrate the algebraic structures formed by various sets of products and the geometric properties of polygons and ellipses created by these Blaschke products.
Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping h and that of zh, we study various L2 norms for Tϕ(h), where Tϕ is the Toeplitz operator with symbol ϕ. In Theorem 2.1, given polynomials p and q we find a symbol ϕ such that Tϕ(p) = q. We extend some of our results to the polydisc.
Let Ω be a bounded convex domain in \(C_n\). We show that if \(φ∈C^{1}(\bar{Ω})\) is holomorphic along analytic varieties in \(bΩ\), then \(H^q_φ\), the Hankel operator with symbol φ, is compact. We have shown the converse earlier (Compactness of Hankel operators with continuous symbols on convex domains, \(\textit{Houston J. Math.}\) \((\textbf{46}(2020), 1005–1016)\) so that we obtain a characterization of compactness of these operators in terms of the behavior of the symbol relative to analytic structure in the boundary. A corollary is that Toeplitz operators with these nonvanishing symbols are Fredholm (of index zero).
The image area of the unit disk under (𝑧⋅ℎ)(𝑧) exceeds the image area under the holomorphic function ℎ(𝑧). In his book, Hermitian Analysis, J. D'Angelo precisely determines how this excess image area of the unit disk, Area (𝑧 ⋅ ℎ)(𝔻) − Area ℎ(𝔻), grows. In our work, we replace the multi- plier 𝑧 with a finite Blaschke product and observe that the excess area growth is a solution for the Dirichlet problem on the unit disk. We replace holomorphic functions with harmonic ones in the formulation and observe a new identity. Furthermore, we show that the excess area growth idea can also be implemented to some other domains conformal to the unit disk.
Let 1 ≤ q ≤ (n−1). We first show that a necessary condition for a Hankel operator on (0, q−1)-forms on a convex domain to be compact is that its symbol is holomorphic along q-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension (n − 1) (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level q if and only if they hold at all levels.
Let Ω be a bounded convex domain in \( \mathbb{C}_n, n ≥ 2, 1 ≤ q ≤ ( n − 1 ) \), and \( ϕ ∈ C (\bar Ω) \). If the Hankel operator \(H^{ϕ}_{q−1}\) on (0,q−1)–forms with symbol ϕ is compact, then ϕ is holomorphic along q–dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only `finitely many’ varieties, 1≤q≤ n, and \( ϕ ∈ C (\bar Ω) \) is analytic along the ones of dimension q (or higher), then \(H^{ϕ}_{q−1}\) is compact.
News
ETAMU Students Turn Data into Real-World Solutions at Alteryx Datathon
Students from across East Texas A&M University applied their analytical skills to real-world challenges during this fall's Alteryx Datathon
From ETAMU to Amazon: Computer Science Alum Shares Career Journey
East Texas A&M University alumna Tejaswini Atluri recently spoke with the campus ACM chapter about her new role as a software development engineer at Amazon.
Students Connect with Industry Leaders to Explore “The Future of Work in the Age of AI”
East Texas A&M recently hosted its first Industry Connect event in partnership with the Paris Economic Development Corporation, giving students look at how leading companies are using artificial intelligence to transform the modern workplace.
