New Faculty Spotlight:
Hung-Hsu Chou
I am Hung-Hsu Chou and I am thrilled to join the department of mathematics at University of Pittsburgh. My current research focus is machine learning, particularly in developing theoretical guarantee through implicit bias/regularization and neural tangent kernel. I have a bachelor’s degree in physics from USCB and PhD in math from UCLA, and hence a wide range of interests in physics, math, and data science. Here is a link to my website and research: https://sites.google.com/view/edwardchou/home
I have lived in 16 different places and traveled around the world. I enjoyed meeting and learning from people. Since I moved here, I already met some great people and had conversations that motivate me to think outside of my box. I would be happy if I can bring not only the knowledge, but also the open attitude to the community. Machine learning is evolving, but so are we, and I would like to believe that we are better than machines.”
Bruno Poggi
Originally from Portoviejo, Ecuador, Bruno Poggi Cevallos studied for his PhD degree in Mathematics at the University of Minnesota - Twin Cities under the direction of Svitlana Mayboroda. His doctoral thesis titled ''Boundary value problems for second-order elliptic equations and related topics'' garnered the Outstanding Thesis Award of the Year 2021 from the Department of Mathematics at the University of Minnesota.
From June 2021 to July 2024, he held a postdoctoral research position at the Universitat Autonoma de Barcelona, working in Xavier Tolsa's group. His research interests fit broadly within an intersection of harmonic analysis, partial differential equations, geometric measure theory, and mathematical physics. A lot of his recent work has centered around boundary value problems for elliptic partial differential equations and their subtle connections to the geometry of domains in Euclidean space. He is also heavily interested in the harmonic analysis of the Schrödinger operator and the exponential decay of its eigenfunctions. Here is a link to his professional website.
During his free time, he enjoys composing and playing music for the piano, as well as playing board games with his family and friends.
New Undergraduate Director
The Department of Mathematics has a new Undergraduate Director, Mellon Professor Thomas Hales, who took over the role starting in the Fall term of 2024. Dr. Jason DeBlois served as the Undergraduate Director from 2022 to 2025. We appreciate Dr. DeBlois's dedicated commitment and exceptional service, especially his professional handling of all the responsibilities.
We are grateful to Dr. Hales for stepping into this role and for his leadership of the Undergraduate Committee as we navigate both current and future challenges in our undergraduate program.
Research
Advancing the Predictability Horizon of Fluid Motion
The Research Projects of William Layton
Dr. Layton has published 200 + refereed papers (cited about 9000 times), 9 mathematics books and is on the editorial boards of many journals in computational and applied mathematics. His research has been continuously funded by the National Science Foundation and other agencies. He has advised numerous MS students and undergraduate student researchers and directed 43 PhD students of high accomplishment. Many of his PhD students have won awards for their high-quality thesis research, have gone on to tenure track positions at research universities and are now producing their own excellent PhD students. PhD students of his PhD students are advising their own successful PhD students! Other highly successful students have advanced at National Labs and in industrial research labs.
The overarching theme of his research is to advance the predictability horizon of fluid motion, especially turbulent flows. This problem is central to challenges in science, technology and engineering faced by human life. Fluids transport and mix heat, chemical species, and contaminants. Accurate simulation of turbulent flow is mathematically intractable yet essential for safety critical prediction and design in applications involving these and other effects. Turbulent flow prediction in science, engineering and industry requires the use of turbulence models. His current research has 3 objectives: increasing accuracy of these models, decreasing model complexity and developing a promising algorithmic idea for computer solution of models- all while working on compelling problems addressing human needs.
Modeling turbulence presents challenges at every level in every discipline it touches. 2-equation Unsteady Reynolds Averaged Navier-Stokes models are common in applications and also the ones with the most incomplete and mysterious mathematical foundation. They have many calibration parameters, work acceptably for flows similar to the calibration data set and require users to have an intuition about which model predictions to accept and which to ignore. This means improving the models is highly challenging on every facet and improvements are impactful in applications. The research’s model analysis addresses model accuracy, complexity and reliability. Even after modeling, greater computational resources are often required for their computational solution, addressed in the research’s algorithm development. A recent algorithmic theme is the synthesis of deterministic laws and data with the vision:
deterministic laws AND data > deterministic laws OR data
Torsors, Electromagnetism & Higgs Bundles
Research Projects of Roman Fedorov
The study of torsors (also known as principal bundles) began in the early 20th century by physicists as a formalism to describe electromagnetism. Later, this was extended to encompass strong and weak interactions, so that torsors became a basis for the so-called Standard Model - a physical theory describing all fundamental forces except for the gravitation. The standard model predicted the existence of various particles, the last of which, called the Higgs boson, was found in a Large Hadron Collider experiment in 2012.
In the 1950's Fields medalist Jean-Pierre Serre recognized the importance of torsors in algebraic geometry. In his 1958 seminal paper he gave the first modern definition of a torsor and formulated a certain deep conjecture. The first part of this project is aimed at proving this conjecture, which is among the oldest unsolved foundational questions in mathematics.
The second part of the project is related to the so-called Higgs bundles, which can be thought of as mathematical incarnations of the Higgs bosons. More precisely, the PI proposes to prove a certain duality for the spaces parameterizing Higgs bundles. This duality is a vast generalization of the fact that the Maxwell equations describing electromagnetic fields are symmetric with respect to interchanging electrical and magnetic fields. The duality is a part of the famous Langlands program unifying number theory, algebraic geometry, harmonic analysis, and mathematical physics. This award will support continuing research in these areas. Advising students and giving talks at conferences are going to be part of the proposed activity.