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UrduGeometric Harmonic Analysis (GHA), the umbrella title for this five-volume series, was coined by the authors to label a specific area of mathematics at the crossroads of two well-established branches: geometry, classically understood as mathematics concerned with metric properties of the ambient space, and harmonic analysis, the branch of mathematics which studies a complex object by decomposing it into simpler building blocks and establishing patterns of behavior.
“GHA is not a textbook or a survey of the state of affairs in mathematics. It is an audaciously large program building a new generation of mathematical machinery capable of answering extremely difficult questions,” said Dorina Mitrea, who chairs the Department of Mathematics at Baylor. “Completing it has been an epic journey. As it stands, laying the foundations of a new genre of mathematics, this serves as a lighthouse which will guide future generations in the novel venues of research opened here.”
The monograph represents 15 years of research that started with one question: “Could the Divergence Theorem (aka the multidimensional version of the Fundamental Theorem of Calculus) be improved by allowing an optimal interplay between analysis and geometry?”
“Clarifying what kind of analysis a given geometric environment may support eventually became the broader theme of the series,” Dorina Mitrea said. “Much as the familiar laws of physics have to be reconsidered at the subatomic level, so too the classical results of mathematics need to be upgraded to remain effective as the settings to which they are applied become increasingly more general.”
A concrete example of practical significance where GHA marks essential progress is the scattering of acoustic and electromagnetic waves. As the obstacle responsible for scattering a wave is allowed to have an ever more pronounced degree of roughness, Dorina Mitrea said, the standard mathematical theory becomes incapable of describing the nature of this basic phenomenon.
When a hard mathematical question consistently eludes attempts to have it answered, it’s typically a sign that key tools and techniques are missing and need to be developed, she said. As a result of work in GHA, systems of partial differential equations arising in mathematical physics and engineering can now be solved in a class of regions that are allowed to develop notoriously hard-to-treat singularities like spiral points. These points are ubiquitous in nature, for example, the eye of a hurricane or the shape of a galaxy.
“Ultimately, we are solving what are called boundary value problems and partial differential equations with these tools that would have not been possible without developing that entire machinery and the entire theory,” Dorina Mitrea said.
ABOUT THE MITREAS
Drs. Dorina and Marius Mitrea joined the Baylor University mathematics faculty in 2019 after more than two decades teaching mathematics at the University of Missouri.
Dorina Mitrea, chair of the Department of Mathematics and professor of mathematics, was recognized recently as a 2024 Fellow of the American Mathematical Society, one of only four mathematicians from Baylor to be honored with the prestigious designation. Marius Mitrea also serves as professor of mathematics, with numerous research areas in partial differential equations, harmonic and Fourier analysis, geometric measure theory, complex and Clifford analysis, spectral theory, differential geometry, semigroup theory, and functional analysis.
Both Dorina and Marius were born in Romania, and they immigrated to the U.S. with their children in 1990. Listen to the Mitreas share their story of life and work together, the drive to teach, research and impact students, and the factors that brought them to Baylor on a recent episode of the Baylor Connections podcast.
Irina Mitrea, Marius’s sister on the mathematics faculty at Temple University, is a harmonic analyst working at the interface of this field with partial differential equations, scattering theory, geometric measure theory, several complex variables and validated numerics. She was recognized in the Class of 2015 of Fellows of the American Mathematical Society, and her research record earned a Von Neumann Fellowship at the Institute for Advanced Study at Princeton in 2014, and the Ruth Michler Prize from the Association of Women in Mathematics in 2008.
ABOUT THE COLLEGE OF ARTS & SCIENCES AT BAYLOR UNIVERSITY
The College of Arts & Sciences is Baylor University’s largest academic division, consisting of 25 academic departments in the sciences, humanities, fine arts and social sciences, as well as 11 academic centers and institutes. The more than 5,000 courses taught in the College span topics from art and theatre to religion, philosophy, sociology and the natural sciences. The College’s undergraduate Unified Core Curriculum, which routinely receives top grades in national assessments, emphasizes a liberal education characterized by critical thinking, communication, civic engagement and Christian commitment. Arts & Sciences faculty conduct research around the world, and research on the undergraduate and graduate level is prevalent throughout all disciplines. Visit the College of Arts & Sciences website.
ABOUT BAYLOR UNIVERSITY
Baylor University is a private Christian University and a nationally ranked Research 1 institution. The University provides a vibrant campus community for more than 20,000 students by blending interdisciplinary research with an international reputation for educational excellence and a faculty commitment to teaching and scholarship. Chartered in 1845 by the Republic of Texas through the efforts of Baptist pioneers, Baylor is the oldest continually operating University in Texas. Located in Waco, Baylor welcomes students from all 50 states and more than 100 countries to study a broad range of degrees among its 12 nationally recognized academic divisions.
