Curriculum Vitae of Hongru Zhao

Hongru Zhao

PhD Candidate

Email: avatarstates@gmail.com
Phone NO. : 218-461-6002  Site: www.meinv.me
in linkedin.com/in/hongru-zhao-b00156185

Education

MS in Applied and Computational Mathematics
University of Minnesota Duluth

Sep 2018 – May 2020
GPA: 4.0/4.0

BS in Statistics
Jilin University

Sep 2014-Jun 2018
Overall GPA: 3.76/4.0  Major GPA: 4.0/4.0 Ranking: 1/109

Experience

Graduate teaching Assistant
University of Minnesota Duluth
Aug 2018 – May 2020
● Conducted lectures for various UMD courses including Calculus I and Calculus III, mathematics lab.
● Assisted students with basic math, college algebra and precalculus problems.

Colloquium speaker on random matrix theory
University of Minnesota Duluth
Dec 05, 2019
● Discussed spectral empirical distributions for products of rectangular matrices

Graduate seminar speaker about Gaussian free field
Jilin University
Jun 17, 2019 – Jun 28, 2019
● Conducted lectures and designed materials for discrete and continuous Gaussian free field

Graduate seminar participant about Anderson model
Jilin University
Jun 2018
● Focused on discrete parabolic Anderson model, Feynman-Kac formula and asymptotics for the total mass.

Computational and applied mathematics summer school
Peking University
Jul 2017
● Learned about:
A mathematical introduction to machine learning;
An introduction of computational mathematics: from models to codes;
A Fast Algorithm for Algebraic Equations and Optimization Problems.

Research Interest

Random matrix theory
Statistical machine learning
SLE(Schramm-Loewner Evolution) and Gaussian free field
Fractional Brownian motion
Stochastic partial differential equations and stochastic analysis

Spectral empirical distributions for products of rectangular Matrices
Masters project
Jun – Sep 2019  Advisor: Yongcheng Qi
● Found transformation for eigenvalues for products of rectangular matrices, classified limiting distributions for empirical distributions, proved limiting theorem, and offered new examples.

Working Papers

Exponential stability for KDV equation
Research project
Mar 2019  Advisor: Zhuangyi Liu
● Proved exponential stability for KDV equation using C0-semigroup, ODE solving, and complex analysis technique.

Strengths

Presenting Reasoning Programming Chinese Mathematics intuition
MatLab Mathematica LaTex R C Python SPSS

Achievements

The First Class Scholarship(top 5%)
Jilin University
2017-2018, 2014-2015

The National Scholarship(top 0.2%)
Ministry of Education of the PRC
2016-2017, 2015-2016

The Zhao Yue scholarship(top 1%)
Mathematics School, Jilin University
2016-2017

Meritorious Winner(top 8%)
Mathematical Contest in Modeling, COMAP(the Consortium for Mathematics and Its Application)
2016
● Developed a microscopic collision model of space crafts

Self study & Freestyle exercise

■ Further study in Real Analysis Aug-Sep 2016
Reference:
 Real and complex analysis (Walter Rudin).
I have gone through more than half of the book which contains all chapters and all details about Real Analysis. I believe that I have a perfect understanding on this subject.
There is a proof of Euler reflection by using Cauchy’s integral theorem.

■ Further study in Functional Analysis Feb-Dec 2017
Reference:
 Functional analysis (Walter Rudin);
 A Hilbert Space Problem Book (Paul R. Halmos).
I nearly finished the Rudin’s book only except the Part2 distributions and fourier transforms. I believe that I have a great understanding on this subject. After finishing this book, I am eager to learn more about unbounded operators and to apply functional analysis technique to other subject. What’s more, more examples on Hilbert space made me have a better understanding on Functional Analysis.

■ A study of Probability in Functional Analysis view Feb-Jun 2018
Reference:
 Functional Analysis for Probability and Stochastic Processes (Adam Bobrowski).
I only finished going through the first 5 chapters. The purposes for reading this book is to understand how hard it is to construct the Brownian motion and to find the deep relationship between functional analysis and probability.

■ Self-study on Stochastic differential equations Jun-Aug 2018
Reference:
 Stochastic Differential Equations(Springer);
 Markov Chains (J. R. Norris 1998).
I only skimmed some part of these books.

■ Proof of Reynolds transport theorem Aug 2018
Reference:
 Reynolds transport theorem.
I was interested in this theorem since we could use it to derive the Navier-Stokes equation.

■ Further study in PDE Aug-Dec 2018
Reference:
 Methods of Mathematical Physics [Vol 1] & [Vol 2] (1989 R. Courant, D. Hilbert).
I was focused on Sturm-Liouville eigenvalue problem, orthogonal polynomials, asymptotic properties of Laplace operator and understanding of characteristic of PDE of first order in a geometric view.
I found an example of using hermite function to solve irregular Strum-Liouville problem.
Reference:
 Sturm-Liouville Theory and its Applications(M. A. Al-Gwaiz);
 An Introduction to Partial Differential Equations(Michael Renardy, Robert C. Rogers).
I was focused on the proof of Sturm-Liouville theorem.
Reference:
 Partial Differential Equations (Jurgen Jost);
 Harmonic Function Theory(Sheldon Axler Paul Bourdon Wade Ramey).
I was focused on the basic properties of harmonic functions, the generalization of maximum principle and Kelvin Transform.

Classical Perron-Frobenius theory and its generalization Sep 2018
Reference:
 Nonlinear Perron-Frobenius theory.
 Krein-Rutman Theorem on the Spectrum of Compact Positive Operators on Ordered Banach Spaces;
I was focus on the proof, generalization and its application on markov chain.

■ Self-Study on Differential Geometry Sep-Dec 2018
Reference:
 Calculus on Manifold (Michael Spivak).
I finished the whole book, but to be honest, I did not have a perfect understanding on differential geometry.
I wrote some note about Calculation on curvilinear coordinates, Christoffel symbols, gradient, divergence and laplacian.
I also recalled an example of Surface integral in spherical coordinates during my TA in cal III.

■ Proof of Brouwer’s fixed point and invariance of domain theorem Aug 2018
Reference:
https://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/

■ A further study on C0-semigroup Mar 2019
Advisor: Zhuangyi Liu
Reference:
 Optimal polynomial decay of functions and operator semigroups;
 Fine scales of decay of operator semigroups;
 Conditions on the Generator of a Uniformly Bounded C0-semigroup;
 Stability of C0-semigroups and geometry of Banach spaces;
 On continuous dependence of roots of polynomials on coefficients (Alen Alexanderian);
 Boundary Controllability of Korteweg-de Vries-Benjamin Bona Mahony Equation on a Finite Domain.
I finished my first project in which I offered a proof of the exponential stability for KDV equation.

■ Self-study on Random Matrix Apr-Jun 2019
Main Reference with online course:
 Introduction to Random Matrices Theory and Practice(Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)
https://www.bilibili.com/video/av24531710?from=search&seid=6296043717877600418
Related materials:
 Will a Large Complex System be Stable?
 Level-Spacing Distributions and the Airy Kernel(Craig A. Tracy, Harold Widom);
 The eigenvalue spectrum of a large symmetric random matrix.
I reorganize a proof of Joint Law of Eigenvalues and Eigenvector for Symmetry Random Matrix.
Reference:
 Introduction to random matrix theory(Todd Kemp);
 Lie groups, Lie algebras, and representations -an elementary introduction (2015 Hall, Brian C);
 Concentration of Measure and the Compact Classical Matrix Groups.

■ Summer research on Random Matrix May-Sep 2019
Advisor: Yongcheng Qi, Tiefeng Jiang
Reference:
 Spectral Radii of Products of Random Rectangular Matrices (2019 Yongcheng Qi, Mengzi Xie);
 Empirical Distributions of Eigenvalues of Product Ensembles(2019 Tiefeng Jiang, Yongcheng Qi);
 Spectral Radii of Large Non-Hermitian Random Matrices(2017 Tiefeng Jiang, Yongcheng Qi);
 Limiting empirical distribution for eigenvalues of products of random rectangular matrices (Xingyuan Zeng).
Using the technique in previous paper, I finish a paper which resolve the limiting spectral empirical distribution for  rectangular Ginibre ensemble in a general way.

■ Lecture and my own handouts about Gaussian Free Field Jun-Jul 2019
Advisor: Tiefeng Jiang
Reference:
 Topics on the two-dimensional Gaussian Free Field;
Conformally Invariant Processes in the Plane;

 Introduction to the Gaussian Free Field and Liouville Quantum Gravity Chap 1 Definition and properties of the GFF;
 Discrete Gassian Free Field & scaling limits;
 Gaussian free fields for mathematicians(Scott Sheffield);
Thanks to Prof. Tiefeng Jiang, I have an experience to offer a lecture to talk about Gaussian free field. After that, I integrated some handouts and my own thinking, thus I wrote my own handout for my lecture.

■ A survey on positive definite preserving transformations on symmetric matrix space Oct 2019
Reference:
 Positive definite preserving linear transformations on symmetric matrix spaces;
 Positive maps, absolutely monotonic functions and the regularization of positive definite matrices;
 Topics in Matrix Analysis, Chap 6;
I did a survey on the classification theorems on linear and non-linear transformations which preserves positive definiteness.

■ Basic on Wiener Space, Fractional Brownian motion, Fractional Laplacian, and Fractional Gaussian fields Jan 2019-
Advisor: Tiefeng Jiang
Reference:
 Analysis on Wiener Space and Applications;
 An Introduction to Analysis on Wiener Space. Chap 1 and Chap 2;
 On measurable norms and abstract Wiener spaces;
 A Modern Theory of Random Variation. With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration;
 The Malliavin Calculus and Related Topics. Chap 1 Analysis on the Wiener space and Chap 5 Fractional Brownian motion;
 Ten Equivalent Definitions of the Fractional Laplace Operator(Mateusz Kwasnicki);;
 Discretizations of the Spectral Fractional Laplacian on General Domains with Dirichlet, Neumann, and Robin Boundary Conditions(Nicole Cusimano, Felix Del Teso, Luca Gerardo-Giorda, and Gianni Pagnini);
 Fractional Integrals and Derivatives: Theory and Applications-Gordon and Breach Science Publishers (Stefan G. Samko, Anatoly A. Kilbas, Oleg I. Marichev);
 Fractional Calculus for Scientists and Engineers(Manuel Duarte Ortigueira);
 Hitchhiker’s guideto the fractional Sobolev spaces(Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci);
 Fractional Gaussian fields: a survey(Asad Lodhia, Scott Sheffield, Xin Sun, Samuel S.Watson);
 Introduction to Fourier Analysis on Euclidean space(Elias M. Stein, Guido Weiss)Chap IV Symmetry Properties of the Fourier Transform;
 Strong Approximation of Fractional Brownian Motion by Moving Averages of Simple Random Walks(Tama’s Szabados).