Mathematics Colloquium
Upcoming Colloquium
Geometric partial differential equations
Dr. Pei-Ken Hung, University of Minnesota
Friday, April 7, 2023
12:00pm via Zoom
Join 4/7 Zoom
Meeting ID: 910 8240 4817
Passcode: 073879
Abstract
In this presentation, I will provide an overview of geometric analysis, including major breakthroughs in this field over the past few decades and how partial differential equations (PDEs) were used to solve them. We will explore how the three types of PDEs (elliptic, parabolic, and hyperbolic) arise in various problems. Lastly, I will discuss my work in each of these three categories.
Bio Sketch
Pei-Ken Hung is a mathematician in the field of geometric analysis and general relativity. He is an assistant professor at the University of Minnesota. Dr. Hung earned a doctorate in mathematics from Columbia University in 2018, under the supervision of Mu-Tao Wang and Simon Brendle. Before joining the University of Minnesota, Dr. Hung has worked at MIT as the C.L.E. Moore Instructor.
The Mathematics Colloquium is a unique opportunity for students to learn about new developments in mathematics and what mathematics and statisticians do after they graduate. Hosted by the Department of Mathematics and Statistics at California State University, Long Beach, the weekly meetings invite guests from universities, research laboratories, and industry to present and discuss current topics in mathematics. All students are encouraged to attend.
Schedule
The schedule for Spring 2023 will be posted shortly after the start of the semester.
Dr. Pei-Ken Hung, University of Minnesota
Abstract
In this presentation, I will provide an overview of geometric analysis, including major breakthroughs in this field over the past few decades and how partial differential equations (PDEs) were used to solve them. We will explore how the three types of PDEs (elliptic, parabolic, and hyperbolic) arise in various problems. Lastly, I will discuss my work in each of these three categories.
Bio Sketch
Pei-Ken Hung is a mathematician in the field of geometric analysis and general relativity. He is an assistant professor at the University of Minnesota. Dr. Hung earned a doctorate in mathematics from Columbia University in 2018, under the supervision of Mu-Tao Wang and Simon Brendle. Before joining the University of Minnesota, Dr. Hung has worked at MIT as the C.L.E. Moore Instructor.
Dr. Cristian Oliva-Aviles, Genentech
Abstract
Biostatisticians play a key role throughout the lifecycle of drug development. They actively collaborate with several functions to design research studies, and to collect and analyze the data needed to draw conclusions about the safety and efficacy of a new drug molecule. Since clinical trials are generally designed to answer a specific question regarding a medical product, there cannot be clinical trials without drug molecules to test. In this talk, an overview of Genentech and its Data Sciences organization will be presented. In addition, the complexity of consistently manufacturing high-quality drugs will be shown to motivate the critical role that nonclinical biostatisticians play during the pharmaceutical drug development process.
Bio Sketch
Cristian has worked as a statistical scientist in the department of Pharmaceutical Development at Genentech since 2018, where he has been a member of the Nonclinical Biostatistics team. He was the coordinator of the 2022 Data & Statistical Sciences Summer Internship Program at Genentech. Cristian is a mathematician with a Ph.D. in Statistics from Colorado State University, with expertise in the areas of shape-restricted regression and survey sampling. The one thing Cristian enjoys the most of being a statistician is collaborating in multidisciplinary projects.
Dr. Xiangwen Zhang, UC Irvine
Abstract
Geometric flows have been proven to be powerful tools in the study of many important problems arising from both geometry and theoretical physics. Aiming to study the equations from the flux compactifications of Type IIA superstrings, we introduce the so-called Type IIA flow, which is a flow of closed and primitive 3-forms on a symplectic Calabi-Yau 6-manifold. Remarkably, the Type IIA flow can also be viewed as a flow as a coupling of the Ricci flow with a scalar field. In this talk, we will discuss the recent progress on this flow.
Bio Sketch
Xiangwen Zhang is an associate professor at UC Irvine, working in the field of geometric analysis and non-linear PDEs. Before joining UC Irvine, he was a Ritt Assistant Professor at Columbia University.
Gary Green, The Aerospace Corporation; SIAM Visiting Lecturer
Abstract
The Boeing Inertial Upper Stage (IUS) carried spacecraft from low earth orbits to their mission orbits from 1982 until 2004. Gamma Guidance provided the means of controlling the rocket engine firings in the presence of irregularities so that the spacecraft were injected into their desired orbits. I will discuss the basics of Gamma Guidance as well as selected collateral mathematics topics required for IUS success. Boeing analysts applied a number of techniques to solve a variety of problems in the face of challenging obstacles.
Nicholas Marco, UCLA
Abstract
Mixed membership models, or partial membership models, are a flexible unsupervised learning method that allows each observation to belong to multiple clusters. In this talk, we propose a Bayesian partial membership model for functional data. By using the multivariate Karhunen-Loève theorem, we are able to derive a scalable representation of Gaussian processes that maintains data-driven learning of the covariance structure. Compared to previous work on mixed membership models, our proposal allows for increased modeling flexibility, with the benefit of a directly interpretable mean and covariance structure. Our work is motivated by studies in functional brain imaging through electroencephalography (EEG) of children with autism spectrum disorder (ASD). In this context, our work formalizes the clinical notion of "spectrum" in terms of feature membership proportions.
Bio Sketch
Nicholas Marco is currently a 5th year Ph.D. student in the department of Biostatistics at UCLA, under the supervision of Donatello Telesca. His primary research focuses on Bayesian Methods for functional data analysis, applied to the field of neuroscience. Nicholas also collaborates with the Jonsson Comprehensive Cancer Center to create microRNA-based biomarkers to personalize cancer treatment. Prior to joining UCLA, Nicholas graduated from California State University, Long Beach in 2017 with a Bachelor of Science degree in Mathematics with an option Statistics.
Dr. Guofang Wei, UC Santa Barbara
Abstract
The fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schrödinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics. In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture that difference of first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D in the Euclidean space is greater than or equal to 3π2/D2. In several joint works with X. Dai, Z. He, S. Seto, L. Wang (in various subsets) the estimate is generalized, showing the same lower bound holds for convex domains in the unit sphere. In sharp contrast, in recent joint work with T. Bourni, J. Clutterbuck, X. Nguyen, A. Stancu and V. Wheeler (a group of women mathematicians), we prove that there is no lower bound at all for the fundamental gap of convex domains in hyperbolic space in terms of the diameter. Very recently, jointed with X. Nguyen, A. Stancu, we show that even for horoconvex (which is much stronger than convex) domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound. All necessary background information will be introduced in the talk.
Bio Sketch
Guofang Wei is a mathematician in the field of differential geometry and geometric analysis. She is a professor at the University of California, Santa Barbara. Professor Wei earned a doctorate in mathematics from the State University of New York at Stony Brook in 1989, under the supervision of Detlef Gromoll. In 2013, she became a fellow of the American Mathematical Society, for "contributions to global Riemannian geometry and its relation with Ricci curvature."
The research of Professor Wei has been concentrated on global Riemannian geometry--the interaction of curvature with the underlying geometry and topology which includes the study of the fundamental groups, comparison geometry, manifolds with integral curvature bounds, spaces with weak curvature bounds, the eigenvalue of Laplacian, and more.
Previous Colloquia
The Mathematics Colloquium Archive has the Colloquia from previous semesters.
Colloquium Committee
For Spring 2023:
- Dr. Xiyue Liao
- Dr. Antonio Martinez
- Dr. Lihan Wang
