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Friday, March 29, 2024
Program in Applied Mathematics Colloquium
Diffusion processes refer to a class of stochastic processes driven by Brownian motion. They have been widely used in various applications, ranging from engineering to science to finance. In this talk, I will discuss my experiences with diffusion and how this powerful tool has shaped our research programs. I will go over several research projects in the area of control, inference, and machine learning, where we have extensively utilized tools from diffusion processes. In particular, I will present our research on three topics: i) covariance control in which we aim to regulate the uncertainties of a dynamic system; ii) Monte Carlo Markov chain sampling for general inference tasks; iii) and diffusion models for generative modeling in machine learning.
Yongxin Chen is an Associate Professor in the School of Aerospace Engineering at Georgia Institute of Technology. He has served on the faculty at Iowa State University (2017-2018). Prior to that, he spent one year (2016-2017) at the Memorial Sloan Kettering Cancer Center (MSKCC) as a postdoctoral fellow. He received his BSc from Shanghai Jiao Tong University in 2011, and Ph.D. from University of Minnesota in 2016, both in Mechanical Engineering. He is an awardee of the George S. Axelby Best Paper Award of IEEE Transactions on Automatic Control in 2017 and the best paper prize of SIAM Journal on Control and Optimization in 2023. He received the NSF Faculty Early Career Development Program (CAREER) Award in 2020, the Simons-Berkeley Research Fellowship in 2021, the A. V. `Bal’ Balakrishnan Award in 2021, and the Donald P. Eckman Award for outstanding young engineer in the field of automatic control in 2022. His current research interests are in the areas of control theory, machine learning, and robotics. He enjoys developing new algorithms and theoretical frameworks for real world applications.
Place: Math Building, Room 501 https://map.arizona.edu/89
Monday, April 1, 2024
Statistics GIDP Colloquium
Title: Multiple Testing of Local Extrema for Detection of Change Points
Abstract: We propose a new approach to detect the number and location of change points in piecewise linear models under stationary Gaussian noise. Our method transforms the change point detection problem into identifying significant local extrema through kernel smoothing and differentiation of the data sequence. By computing p-values for all local extrema based on the derived peak height distributions of derivatives of smooth Gaussian processes, we utilize the Benjamini-Hochberg procedure to identify significant local extrema as the detected change points. The algorithm provides asymptotic strong control of the False Discovery Rate (FDR) and power consistency, as the length of the sequence and the size of jumps get large. Simulations show that FDR levels are maintained in non-asymptotic conditions and guide the choice of smoothing bandwidth. Compared to traditional change point detection methods based on recursive segmentation, our approach requires only one instance of multiple testing across all candidate local extrema, thereby achieving the smallest computational complexity proportionate to the data sequence length.
Tuesday, April 2, 2024
Analysis, Dynamics, and Applications Seminar
Traditional methods to detect chaotic trajectories include visualization and the computation of Lyapunov exponents. Both of these require long time computations of the trajectory and—for Lyapunov—its linearization. Lyapunov exponents converge slowly, if at all, and a number of Lyapunov indicators, including the “Fast” (FLI) and “Mean Exponential Growth” (MEGNO) have been developed to attempt to get around this. An alternative technique, “Frequency Analysis” was developed to indicate trajectories that lie on tori with dynamics conjugate to a rotation, thus giving an indictor for lack of chaos. A contrasting idea is the “0-1 Test” which maps the trajectory onto a random walk to show chaos.
I will discuss work with Evelyn Sander and Nathan Duignan that uses Weighted Birkhoff Averages as a test for chaos. Birkhoff’s ergodic theorem implies that when an orbit is ergodic on an invariant set, spatial averages of a phase-space function can be computed as time averages. However the convergence of a time average can be very slow. In 2016, Das et al introduced a $C^\infty$ weighting technique that they later showed gives super-polynomial convergence when the dynamics is conjugate to a rigid rotation with frequency that is sufficiently incommensurate (Diophantine). We show that the Weighted Birkhoff Average (WBA) gives a sharp distinction between chaotic and regular dynamics and allows accurate computation of rotation vectors for regular orbits. This allows one to find critical parameter values at which invariant tori are destroyed in Hamiltonian systems and symplectic maps. We apply these methods to circle, torus, area-preserving and three-dimensional angle-action maps. Comparisons with other techniques show that the WBA is more efficient. It also has the advantage of accurately computing physically relevant quantities like rotation vectors. We also show that the WBA can detect “strange non-chaotic attractors”, invariant sets that are geometrically strange but have zero Lyapunov exponents.
Place: Math Building, Room 402 https://map.arizona.edu/89
Algebra and Number Theory Seminar
In 1955, Richard Brauer, often regarded as the founder of modular representation theory, made one of the first of the so-called “local-global” conjectures in character theory and opened the door to an entire area of research. I’ll discuss the background of this conjecture, known as Brauer’s Height Zero Conjecture (BHZ), and its recent proof, which is joint work with G. Malle, G. Navarro, and P.H. Tiep.
Wednesday, April 3, 2024
Ph.D. Final Oral Dissertation Defense
Dissertation Title: Multi-Spacecraft Observatory Data Analysis Techniques: Uncertainty Quantification & Comparison
Advisor: Kristopher Klein, Lunar & Planetary Sciences
Thursday, April 4, 2024
Mathematics Colloquium
The rapidly growing demand in data science presents an exciting opportunity for math and statistics departments to redesign courses to benefit both traditional math and statistics majors as well as new students from other disciplines. Creating cohesive undergraduate curriculum entails weaving together content from different courses to steadily develop critical skills necessary to compete in today’s workforce. In this talk I will discuss how the Department of Mathematical and Statistical Sciences at the University of Colorado Denver is integrating data science throughout our undergraduate. I will demonstrate how students experiment with mathematical models in a virtual laboratory using interactive Jupyter notebooks to attract a wider range of students.
Event
Recruiting visit from Mercer/MarshMcLennan
If you have any questions, please let Cade (cademac1@arizona.edu), Bristol (bristoljohnson@arizona.edu), or Aqib (aqibadnan@arizona.edu) know.
Friday, April 5, 2024
Ph.D. Final Oral Dissertation Defense
Program in Applied Mathematics Colloquium
Hemostasis is the normal, healthy process in which a blood clot forms to stop bleeding in the event of an injury. Blood clot formation is a complex and nonlinear process that occurs under flow and on multiple spatial and temporal scales. Defects and perturbations in the hemostatic system can result in serious bleeding or pathological clot formation, but due to the complexity of the system, the responses to these changes and the underlying mechanisms are challenging to predict. Mechanistic mathematical models of blood clot formation and coagulation can elucidate biochemical and biophysical mechanisms, help interpret experimental data, and guide experimental design. In this talk I will briefly describe such models and show how our integrated mathematical and experimental approach has facilitated the discovery of previously unrecognized interactions within the clotting system.
Place: Math Building, Room 501 https://map.arizona.edu/89
Thursday, April 11, 2024
Ph.D. Final Oral Dissertation Defense
Monday, April 15, 2024
Ph.D. Final Oral Dissertation Defense
Tuesday, April 16, 2024
Analysis, Dynamics, and Applications Seminar
Patterns with almost periodic structures can be seen in so many natural contexts; as epidermal ridges on the tips of your fingers, cloud streets, convection rolls, on the growth tips of plants, sunflowers. Their behaviors can, in many situations, be captured by averaging over the local periodic pattern and writing what often turn out to be universal equations for macroscopic variables called order parameters. I will describe how these equations come about, what they express and how they describe both the evolution of the pattern and their defect structures. There are many open challenges, one of which I will discuss.
Place: Math Building, Room 402 https://map.arizona.edu/89
Wednesday, April 17, 2024
Ph.D. Final Oral Dissertation Defense
Thursday, April 18, 2024
Tuesday, April 23, 2024
Analysis, Dynamics, and Applications Seminar
A successful statistical description of most nonlinear systems is stymied by the lack of closure; the rates of change of lower order moments depend on moments of higher order leading to an infinite hierarchy. Most attempts to enforce an artificial closure fail. However, there is a natural asymptotic closure for fields of weakly nonlinear waves (think of a wind driven ocean surface). I will explain to you what (mild) premises are required to achieve this natural closure and the reasons it occurs. The corresponding kinetic equation for the energy density turns out to have very interesting stationary solutions corresponding to equipartition states and to finite flux solutions of Kolmogorov type, discovered originally by our good colleague Volodja Zakharov. But, despite many successes, the story is far from over as there are still open challenges, some of which I will tell you about.
Place: Math Building, Room 402 https://map.arizona.edu/89
Wednesday, April 24, 2024
Ph.D. Final Oral Dissertation Defense
Advisors: Co-Advisor: Xueying Tang, Dept of Mathematics
Co-Advisor: Cristian Roman Palacios, School of Information
Thursday, April 25, 2024
Friday, May 10, 2024
For details, see https://science.arizona.edu/academics/graduation-convocation