Hao Shen

Hao Shen

  Assistant Professor
  Department of Mathematics
  University of Wisconsin-Madison
  Email: hshen3 at wisc or pkushenhao at gmail
  Office: Van Vleck Hall 619

Short Bio: PhD 2013 (Princeton, Advisor:Weinan E);
Postdoc 2014-2015 (Warwick, Mentor: Martin Hairer);
Ritt Assistant Professor 2015-2018 (Columbia, Mentor:Ivan Corwin).


  Here are some of the previous courses I have taught

Teaching in Spring 2021

  Math 431, Intro-Theory of Probability (online)
(Syllabus, course materials, homework are also posted on Canvas.)

Instructor Email/Preferred Contact hshen3@wisc.edu
Math 431 is an introduction to the theory of probability, the part of mathematics that studies random phenomena. Topics covered include axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, how and when to estimate probabilities using the normal or Poisson approximation, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.
Requisites MATH 234 or 376 or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program

The course grade will be based on homework assignments (20%), two midterms (20% each), and a cumulative final exam (40%).

REQUIRED TEXTBOOK: Anderson, Seppalainen, Valko: Introduction to Probability.

There are two midterm exams and a cumulative final exam. The midterm exams are online evening exams.
Exam #1: Wednesday, March 3, 5:30PM-7:00PM.
Exam #2: Wednesday, April 7, 5:30PM-7:00PM.
Final Exam: TBA.

HOMEWORK & OTHER ASSIGNMENTS Weekly homework, hand-in online on Canvas.

Weakly schedule

Week 1. Axioms of probability, sampling, review of counting, infinitely many outcomes, review of the geometric series (Sections 1.1-1.3).
Week 2. Rules of probability, random variables, conditional probability (Sections 1.4-1.5, 2.1).
Week 3. Bayes formula, independence, independent trials (Sections 2.2-2.4).
Week 4. Independent trials, birthday problem, conditional independence, probability distribution of a random variable (Sections 2.4-2.5, 3.1).
Week 5. Cumulative distribution function, expectation and variance (Sections 3.2-3.4).
Week 6. Gaussian distribution, normal approximation and law of large numbers for the binomial distribution, (Sections 3.5 and 4.1-4.2).
Week 7. Applications of normal approximation, Poisson approximation, exponential (Sections 4.3-4.5).
Week 8. Moment generating function, distribution of a function of a random variable (Sections 5.1-5.2).
Week 9. Joint distributions (Sections 6.1-6.2).
Week 10. Joint distributions and independence, sums of independent random variables, (Sections 6.3-7.1).
Week 11. Expectations of sums and products, exchangeability (Sections 7.2-8.1).
Week 12. variance of sums, Sums and moment generating functions, (Sections 8.2-8.3).
Week 13. covariance and correlation, Markov and Chebyshev inequalities, (Sections 8.4-9.1).
Week 14. Law of large numbers, central limit theorem, Review (Sections 9.2-9.3).


I am interested in stochastic partial differential equations, and its interaction with quantum field theory, statistical mechanics, interacting particle systems and geometric flows.

Publications and Preprints - in reversed chronological order

  1. Large N limit of the O(N) linear sigma model in 3D. (With Rongchan Zhu, Xiangchan Zhu)
  2. Langevin dynamic for the 2D Yang-Mills measure. (With Ajay Chandra, Ilya Chevyrev, Martin Hairer)
  3. Large N limit of the O(N) linear sigma model via stochastic quantization. (With Scott Smith, Rongchan Zhu, Xiangchan Zhu) Ann. Probab. accepted.
  4. Scaling limit of a directed polymer among a Poisson field of independent walks. (With Jian Song, Rongfeng Sun, Lihu Xu) J. Funct. Anal. accepted.
  5. Stochastic Ricci Flow on Compact Surfaces. (With Julien Dubédat) Int. Math. Res. Not. accepted.
  6. Some recent progress in singular stochastic PDEs. (With Ivan Corwin) Bulletin of the AMS. 57.3 (2020): 409-454.
  7. Local solution to the multi-layer KPZ equation. (With Ajay Chandra and Dirk Erhard) J. Stat. Phys. (2019) Vol 175, Issue 6, pp 1080-1106
  8. The dynamical sine-Gordon model in the full subcritical regime. (With Ajay Chandra and Martin Hairer)
  9. Stochastic Telegraph equation limit for the stochastic six vertex model. (With Li-Cheng Tsai) Proc. Amer. Math. Soc. 147 (2019), 2685-2705
  10. Stochastic PDE limit of the Six Vertex model. (With Ivan Corwin, Promit Ghosal and Li-Cheng Tsai) Comm. Math. Phys. (2020), pp.1-94
  11. Stochastic quantization of an Abelian gauge theory. Comm. Math. Phys. accepted.
  12. Open ASEP in the weakly asymmetric regime. (With Ivan Corwin) Comm. Pure Appl. Math. 71(10), pp.2065-2128.
  13. Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits. (With Hendrik Weber) J. Funct. Anal. Vol 275, Issue 6, (2018), 1321-1367
  14. Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem. (With Ajay Chandra) Electron. J. Probab. Vol 22 (2017), paper no. 68.
  15. ASEP(q,j) converges to the KPZ equation. (With Ivan Corwin and Li-Cheng Tsai) Ann. Inst. Henri Poincaré (B) Probab. Stat. (2018), 54, No. 2, 995-1012.
  16. Weak universality of dynamical Φ4_3: non-Gaussian noise. (With Weijun Xu) Stoch PDE: Anal Comp (2017).
  17. A central limit theorem for the KPZ equation. (With Martin Hairer) Ann. Probab. 45(2017), no. 6B, 4167-4221.
  18. The dynamical sine-Gordon model. (With Martin Hairer) Comm. Math. Phys. 341 (2016), no. 3, 933-989
  19. The strict-weak lattice polymer. (With Ivan Corwin and Timo Seppäläinen) J. Stat. Phys. 160(2015), no. 4, 1027-1053
  20. Exact renormalization group analysis of turbulent transport by the shear flow. (With Weinan E) J. Stat. Phys. 153 (2013), no. 4, 553-571
  21. Mean field limit of a dynamical model of polymer systems. (With Weinan E) Sci. China Math. 56 (2013), no. 12, 2591-2598
  22. A renormalization group method by harmonic extensions and the classical dipole gas. Ann. Henri Poincaré 17 (2016), no. 4, 861-911
  23. Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. (With Weinan E and Arnulf Jentzen) Nonlinear Anal. 142 (2016), 152- 193
  24. PhD Thesis: Renormalization Theory in Statistical Physics and Stochastic Analysis (Advisor: Weinan E)

Grants and Awards

My research is mainly supported by the following grants:
NSF CAREER DMS-2044415 (2021-2026);
NSF DMS-1954091 (2020-2023);
NSF DMS-1712684 / DMS-1909525 (2017 - 2020)