Hao Shen
Hao Shen 


Short Bio:
PhD 2013 (Princeton, Advisor:Weinan E);
Postdoc 20142015 (Warwick, Mentor: Martin Hairer);
Ritt Assistant Professor 20152018 (Columbia, Mentor:Ivan Corwin).
CV.



Teaching
Here are some of the previous courses I have taught
Teaching in Spring 2021
Math 431, IntroTheory of Probability (online)
(Syllabus, course materials, homework are also posted on Canvas.)
INSTRUCTORS AND TEACHING ASSISTANTS
Instructor Email/Preferred Contact
hshen3@wisc.edu
OFFICIAL COURSE DESCRIPTION
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 PreMasters 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, 5:30PM7:00PM.
Exam #2: Wednesday, April 7, 5:30PM7:00PM.
Final Exam: TBA.
HOMEWORK & OTHER ASSIGNMENTS
Weekly homework, handin online on Canvas.
Weakly schedule
Week 1. Axioms of probability, sampling, review of counting, infinitely many outcomes, review of the geometric series (Sections 1.11.3).
Week 2. Rules of probability, random variables, conditional probability (Sections 1.41.5, 2.1).
Week 3. Bayes formula, independence, independent trials (Sections 2.22.4).
Week 4. Independent trials, birthday problem, conditional independence, probability distribution of a random variable (Sections 2.42.5, 3.1).
Week 5. Cumulative distribution function, expectation and variance (Sections 3.23.4).
Week 6. Gaussian distribution, normal approximation and law of large numbers for the binomial distribution, (Sections 3.5 and 4.14.2).
Week 7. Applications of normal approximation, Poisson approximation, exponential (Sections 4.34.5).
Week 8. Moment generating function, distribution of a function of a random variable (Sections 5.15.2).
Week 9. Joint distributions (Sections 6.16.2).
Week 10. Joint distributions and independence, sums of independent random variables, (Sections 6.37.1).
Week 11. Expectations of sums and products, exchangeability (Sections 7.28.1).
Week 12. variance of sums, Sums and moment generating functions, (Sections 8.28.3).
Week 13. covariance and correlation, Markov and Chebyshev inequalities, (Sections 8.49.1).
Week 14. Law of large numbers, central limit theorem, Review (Sections 9.29.3).
Research
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

Large N limit of the O(N) linear sigma model in 3D.
(With Rongchan Zhu, Xiangchan Zhu)

Langevin dynamic for the 2D YangMills measure.
(With Ajay Chandra, Ilya Chevyrev, Martin Hairer)

Large N limit of the O(N) linear sigma model via stochastic quantization.
(With Scott Smith, Rongchan Zhu, Xiangchan Zhu) Ann. Probab. accepted.

Scaling limit of a directed polymer among a Poisson field of independent walks.
(With Jian Song, Rongfeng Sun, Lihu Xu) J. Funct. Anal. accepted.

Stochastic Ricci Flow on Compact Surfaces.
(With Julien Dubédat) Int. Math. Res. Not. accepted.

Some recent progress in singular stochastic PDEs.
(With Ivan Corwin) Bulletin of the AMS. 57.3 (2020): 409454.

Local solution to the multilayer KPZ equation.
(With Ajay Chandra and Dirk Erhard)
J. Stat. Phys. (2019) Vol 175, Issue 6, pp 10801106

The dynamical sineGordon model in the full subcritical regime.
(With Ajay Chandra and Martin Hairer)

Stochastic Telegraph equation limit for the stochastic six vertex model.
(With LiCheng Tsai)
Proc. Amer. Math. Soc. 147 (2019), 26852705

Stochastic PDE limit of the Six Vertex model.
(With Ivan Corwin, Promit Ghosal and LiCheng Tsai)
Comm. Math. Phys. (2020), pp.194

Stochastic quantization of an Abelian gauge theory. Comm. Math. Phys. accepted.

Open ASEP in the weakly asymmetric regime.
(With Ivan Corwin)
Comm. Pure Appl. Math. 71(10), pp.20652128.

Glauber dynamics of 2D KacBlumeCapel model and their stochastic PDE limits.
(With Hendrik Weber)
J. Funct. Anal. Vol 275, Issue 6, (2018), 13211367

Moment bounds for SPDEs with nonGaussian fields and application to the WongZakai problem.
(With Ajay Chandra)
Electron. J. Probab. Vol 22 (2017), paper no. 68.

ASEP(q,j) converges to the KPZ equation.
(With Ivan Corwin and LiCheng Tsai)
Ann. Inst. Henri Poincaré (B) Probab. Stat. (2018), 54, No. 2, 9951012.

Weak universality of dynamical Φ4_3: nonGaussian noise.
(With Weijun Xu)
Stoch PDE: Anal Comp (2017).

A central limit theorem for the KPZ equation.
(With Martin Hairer)
Ann. Probab. 45(2017), no. 6B, 41674221.

The dynamical sineGordon model.
(With Martin Hairer)
Comm. Math. Phys. 341 (2016), no. 3, 933989

The strictweak lattice polymer.
(With Ivan Corwin and Timo Seppäläinen)
J. Stat. Phys. 160(2015), no. 4, 10271053

Exact renormalization group analysis of turbulent transport by the shear flow.
(With Weinan E)
J. Stat. Phys. 153 (2013), no. 4, 553571

Mean field limit of a dynamical model of polymer systems.
(With Weinan E)
Sci. China Math. 56 (2013), no. 12, 25912598

A renormalization group method by harmonic extensions and the classical dipole gas.
Ann. Henri Poincaré 17 (2016), no. 4, 861911

Renormalized powers of OrnsteinUhlenbeck processes and wellposedness of stochastic GinzburgLandau equations.
(With Weinan E and Arnulf Jentzen)
Nonlinear Anal. 142 (2016), 152 193

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 DMS2044415 (20212026);
NSF DMS1954091 (20202023);
NSF DMS1712684 / DMS1909525 (2017  2020)