About Me
Chinese Name: 姚何塬
Senior student (bachelor of Arts) at Columbia University
Affiliation: Department of Mathematics
Major: Mathematics-Statistics
Current Duties: Teaching Assistant for MATH GU 4032 Fourier Analysis; Teaching Assistant for STAT ** ****. (I am an anonymous grader and the course name will not be shown here until Jan. 2024. But you can find it in my CV. )
Previous Teaching and Teaching Assistantship: MATH UN 2500 Analysis and Optimization (Fall 2022, Spring 2023); STAT UN 1101 Introduction to Statistics (Fall 2022); STAT UN 2103 Applied Linear Regression Analysis (Spring 2023).
Contact: hy2704 AT columbia DOT edu
Research Interests
My current research interests are mainly influenced by Professor Victor H. de la Peña and Professor Ioannis Karatzas.
(Concentration) inequalities, with a focus on general dependence structure, related to decoupling methods, exponential bounds, and application to hypothesis testing.
Martingale theory and stochastic differential equation, with a focus on Langevin diffusion and trajectorial approach.
Self-normalization, with a focus on (generalized) U-statistics and applications on ratio statistics, including the Gini coefficient, Taylor's law, and squared coefficient of variation.
Hitting time, combined with time series, Itô diffusions, and survival analysis, with applications on climate and epidemiology.
Other topics, not involved in past research experience but in previous readings or classes, include:
Probability: Markov chain mixing time; Filtering; Random matrix; High-dimensional probability.
Theoretical statistics: Asymptotic statistics; Bayesian statistics; Non-parametric statistics.
Functional Analysis: Convex analysis; Optimal transport; Fourier analysis.
Some Notes I Designed or Took
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Decoupling and self-normalization:
With the help of Prof. Victor H. de la Peña and Haolin Zou, I am designing and formulating course notes and slides, whose topics are decoupling, self-normalization, (concentration) inequalities, and their corresponding applications in stochastic analysis, optimization, and machine learning. Those notes serve primarily as the course materials for a tutorial at the Artificial Intelligence Institute for Advances in Optimization (AI4OPT), Georgia Institute of Technology.
Planed Content:
1. Introduction to Decoupling
2. Probabilistic Inequalities
3. Complete Decoupling
4. Tangent Decoupling
5. Decoupling Inequalities for Generalized U-Statistics
6. Bernstein Inequality and Self-normalization
7&8&9. Self-normalization II
10. Applications to Optimization and Stochastic Bandit Problems
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MATH GU 4156 Advanced Probability Theory
I am taking and formulating lecture notes for the course MATH GU 4156 Advanced Probability Theory, designed by Prof. Ioannis Karatzas. These notes primarily serve for some students in the abovementioned tutorial who may not have prior familiarity with martingale theory.
Partial notes converted from the hand-written notes by Prof. Ioannis Karatzas.
1. Relationships between Measures 2. Conditional Expectation 3. Conditional Probability 4. Filtrations and Stopping Times 5. Martingales: Definitions and First Results 6. Martingale Convergence 7. Uniform Integrability 9. Square-Integrable Martingales 17. Brownian Motions
Selected Talks and Presentations
Invited Talk to Columbia Undergraduate Statistics Seminar (Nov. 2023)
Introduction to the Theory and History of Stochastic Analysis: Random Walk, Markov Chain, and Martingale.
Columbia Undergraduate Research Symposium (Oct. 2023)
An Alternative Proof, via Decoupling, of Hanson-Wright Inequality.
[Poster]Probability II Final Presentation (April. 2023)
Lozenge-Tiling Markov Chain: Lattice Paths, Contraction Property and Wilson's Method.
[Slides]Bounding Mixing Times of Lattice Path Markov Chains and Related Models: Utilizing Contraction Properties, Coupling, and Wilson’s Method.
[Essay]Fall 2022 Directed Reading Program Colloquium (Dec. 2022)
The Fredholm Alternative and Its Application.
[Slides]Polymath Jr. Program Final Conference (Aug. 2022)
Characterizing symmetric matrix distributions, Exponentiated Symmetric Matrix Distributions with Applications to Linear Inverse Problems.
[Slide]My News and Stories
Sharing the Impact of My Undergraduate Research Experience in GS website and CityU Joint Degree website,, Dec. 2023
Inducted into GS Honor Society, Nov. 2022
Joining Columbia Statistics Undergraduate Research Interns, May 2022
Awarded Columbia Math Integration Bee Finalists , Jan. 2022
Admitted to the Joint Bachelor’s Degree Programmes between Columbia & CityU, May 2021