# 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 |Manuscript 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

## Working Papers

**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.

**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).*