Research

Whitney problems

My dissertation research is on Whitney-type extension and selection problems and their connection to machine learning and non-parametric statistics. At the basic level, Whitney's problems ask for efficient and universal ways to fit a function to random data while minimizing energy. Here is an expository article by Charles Fefferman for an overview of Whitney's problems. I am particularly interested in how nonlinear constraints, such as positivity or convexity, preserve or alter the structure of these problems. 

Time-frequency analysis

I am also working on symmetric decompositions of nuclear operators. This stems from a problem posed by H. Feichtinger (and subsequently by C. Heil and D. Larson) which asks whether a positive-definite integral operator with M1 (Feichtinger algebra) kernel admits a rank-one decomposition series that is also strongly square-summable in M1. 

 Papers

1. Extension of flat nonnegative smooth functions by operators, in preparation

2. Roots, trace, and extendability of flat nonnegative smooth functions, International Mathematics Research Notices (to appear), 2024. arXiv:2306.16183 

3. (with C. Fefferman and G.K. Luli) C^2 interpolation with range restriction, Revista Matemática Iberoamericana, 39(2):649--710, 2023. arXiv:2107.08272 

4. (with C. Liang, Y. Liang, and G.K. Luli) Univariate Range-Restricted C^2 Interpolation Algorithms, Journal of Computational and Applied Mathematics, 425:115040, 2023.

5. (with G.K. Luli and K. O'Neill) Smooth selection for infinite sets, Advances in Mathematics, 407:108566, 2022. arXiv:2109.04905 

6. (with G.K. Luli and K. O'Neill) On the shape fields finiteness principle, International Mathematics Research Notices, online, 2021. arXiv:2010.09827 

7. (with G.K. Luli) Algorithms for nonnegative C^2(R^2) interpolation, Advances in Mathematics, 385:107756, 2021. arXiv:2102.05777 

8. (with G.K. Luli) C^2(R^2) nonnegative extension by bounded-depth operators, Advances in Mathematics, 375:107391, 2020. arXiv:2008.04962 

9. (with G.K. Luli) Nonnegative C^2(R^2) interpolation, Advances in Mathematics, 375:107364, 2020. arXiv:1901.09876 

10. Nonnegative Whitney Extension Problem for C^1(R^n), 2019. arXiv:1912.06327 

11. (with K. Xu et al.) A Transfer Function Approach to Shock Duration Compensation for Laboratory Evaluation of Ultra-High-G Vacuum-Packaged MEMS Accelerometers, IEEE 32nd International Conference on Micro Electro Mechanical Systems (MEMS), 676--679, 2019.

12. (with K. Xu et al.) Micromachined integrated self-adaptive nonlinear stops for mechanical shock protection of MEMS, Journal of Micromechanics and Microengineering, 28:064006, 2018.

13. (with K. Xu et al.) Micromachined integrated shock protection via a self-adaptive nonlinear system, 19th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), 524--527, 2017.

Talks

Loo-Keng Hua Lecture, Chinese Academy of Sciences, July 6-7, 2024, Beijing, China. 

Mathematics of Adversarial, Interpretable, and Explainable AI, Mathematics Research Communities (MRC) conference, June 23-29, 2024. Buffalo, NY

AMS Spring Southern Sectional Meeting, Special Session on Bases and Frames in Hilbert Spaces III, March 24, 2024, Florida State University

Research Interaction Team (RIT) on Applied Harmonic Analysis, February 19-26, 2024, University of Maryland

The 15th Whitney Workshop, August 14-25, 2023. UC Davis (Online).

Conference for early-career mathematical researchers, June 18-20, 2023, Shenzhen University.

CUNY GC Harmonic Analysis and PDE Seminar, March 10, 2023. CUNY Graduate Center. 

ICERM semester seminar, October 14, 2022. ICERM at Brown University.

Fall Fourier Talks 2022, Oct 6-7, 2022. Norbert Wiener Center at the University of Maryland (poster).

Harmonic analysis methods in geometric tomography, Sep. 26-30, 2022. ICERM at Brown University.

14th Whitney Workshop, May 20 - November 11, 2021. Thursdays 7:30 AM PST. UC Davis (Online).

Bay Graduate Math Conference, May 11, 2019, University of California, Berkeley.