Research Summary of Zu Chongzhi Center Faculty Members

Research Summary

I am interested in probability and combinatorics, including randomized algorithms, random graphs and random tree, in particular:

Potential Signature Work Projects

Simulation of an Ethereum Network

Ethereum is the second most popular crypto-currency on the Internet. It is difficult to deduce the properties of the Ethereum network by observing internet traffics, since the neighbouring relationship in it is hidden. This project seeks to simulate a small Ethereum network locally over a period of time to capture the properties it, such as diameter of the graph, connectivity, etc. The ultimate goal is to design a random graph model which is better suited for model Ethereum networks.

The Horton-Strahler Number of Binary Trees

The Horton-Strahler number of a tree is a measure of its branching complexity. It has long been conjectured that in binary search trees, this number is of the order log(n)/log(3). The upper bound has been proved. But a proof for the lower bound is still missing.

Other Possibilities

Professor Chen’s research focuses on mathematical modeling and numerical methods used in finance, science, and engineering. Prof. Dangxing Chen’s main area of research is quantitative finance. In particular, his research focuses on the development of next-generation regulated machine learning models for financial data. It is essential that machine learning models used in finance are transparent, explainable, fair, confident, etc. A few examples of possible applications include credit scoring, anti-money laundering, fraud detection, etc. Additionally, he is interested in stochastic modeling and computational methods for analyzing return and volatility data. The second area of his research is the development of accurate, fast, and robust numerical algorithms for large-scale long-term simulations of partial differential equations and integral equations in science and engineering. Electromagnetism, quantum mechanics, and many-body problems are among the applications.

Prof. Chen is interested in mentoring Signature Work projects on topics related to his research interests in machine learning methods in fintech, stochastic modeling and computation of return and volatility data, numerical partial differential equations, and other related topics. For machine learning in finance, related projects will require good programming skills (Python, MATLAB, etc) as well as knowledge of numerical analysis and machine learning methods. Students should be able to write ML models and optimization algorithms by themselves, rather than using packages. The stochastic process in finance requires a thorough understanding of probability and stochastic calculus, how statistical methods can be applied in practice (such as hypothesis testing), and the use of numerical methods for modeling (Python, MATLAB, etc.).

Prof. Efstathiou’s research focuses on the study of dynamical systems, that is, systems that change in time. This is a very broad research field which uses ideas and techniques from many other mathematical domains. Prof. Efstathiou’s main research direction is the topology and geometry of integrable Hamiltonian systems where he uses ideas from differential geometry, group theory, algebraic topology, singularity theory, and more, to understand the interplay between dynamics and geometry in such systems. His second research direction is the study of collective dynamics in systems of coupled oscillators. Such systems are being used to model many types of interactions that occur in nature — pacer cells in the heart, cells in the brain’s suprachiasmatic nucleus, electric power grids, and more. Research in this direction combines dynamical systems theory with extensive numerical computations.

Throughout his career, Prof. Efstathiou has supervised undergraduate projects in several directions, including dynamical systems (solar sails, the Kepler problem, machine learning of dynamical systems), geometry (Lie groupoids, symplectic embeddings, string theory), topological data analysis, numerical methods, financial mathematics, the history of mathematics, and others. He is interested in mentoring Signature Work projects in topics related to his research interests in integrable Hamiltonian systems and in collective dynamics, and in other topics broadly related to dynamical systems, geometry and topology, mathematical physics, and numerical methods.

The research of Professor Gao takes an interdisciplinary approach, combining field, laboratory and computational studies, to uncovering the mechanisms behind environmental pollution and finding remediation solutions. His publications (several highly cited worldwide) focus on elucidating physicochemical mechanisms of urban air pollution as well as molecular nature, chemical transformation and physical transport of secondary aerosols.

In terms of utilizing computational approaches, his group has been examining spatial and temporal trends of air pollutants (including sulfur and nitrogen dioxides, key molecules in tropospheric air pollution). Several methods, including remote-sensing data (TROPOMI satellite, etc.), ground big-data analysis and Machine Learning simulation, have been used concurrently to examine the specific impacts of COVID-related events on air quality. Two DKU undergraduate students (applied math major, and data science major, respectively), now accepted into Univ. of Michigan and Carnegie Mellon Univ. for graduate studies, had worked on these as their Signature Work under Prof. Gao’s mentorship, and have presented findings at the prominent European Geoscience Union General Assembly.

Prof. Gao has also taken on a keen interest to formulate science-based environmental policies. His latest research involves developing a new scientific/policy framework to reduce plastics pollution, uncovering previously missing links between chemistry and policy analysis, ultimately furthering global agreements on ozone protection, climate mitigation and plastics reduction. This research has culminated in a publication at the prestigious Proceedings of the National Academy of Sciences of the US (PNAS) in 2021. Another DKU student, accepted into the Engineering graduate program at Stanford University, worked on this as part of her Signature Work under Prof. Gao’s mentorship.

Prof. Gao would welcome math or data science students, in addition to environmental science/policy (or other majors) students, to work on these projects or other ideas utilizing mathematical approaches to further understanding the environment.

A theoretical physicist by training, I am fascinated by the emergence of steady macroscopic features from fluctuating ensembles of microscopic degrees of freedom.

I am currently working mostly on out-of-equilibrium statistical physics. Statistical physics has provided deep research questions in mathematics and physics. In particular, Brownian motion at thermal equilibrium revealed the atomic nature of matter and triggered the development of the mathematical theory of stochastic processes.

However, living systems are not at thermal equilibrium: they exchange energy with their environment. In out-of-equilibrium statistical physics there are no new fundamental equations, but exact solutions can sometimes be found in simple models that can be transposed to multiple real-world systems.

The underlying universality classes can reveal rich integrable structures, giving rise to exact solutions. In particular, current developments on stochastic resetting reveal new stationary states of physical systems by putting them in contact with their initial states at random times.

In statistical physics, a model can often be approached from several complementary angles (including numerical simulations and exact solution). Interested students could explore Monte Carlo simulations of physical systems in their signature work, while working on a review of the underlying physical mechanisms.

**Pengzhan Guo**

Prof. Guo’s research interest is including methodology and applications in machine learning and data mining. With his educational background crossing mathematics and computational science, he focuses on addressing essential optimization problems in management and economics by developing/applying cutting edge data mining and machine learning algorithms as well as statistical models. His research appears in the leading journals and conferences in data mining. Specifically, his research falls into two categories: (1) application by using data mining and machine learning method and (2) methodology in data mining and machine learning.

As to the application part, his research interests is to develop data-driven models for addressing challenging real-world problems. The domains that his research covers include urban computing, human resource management, etc. In addition to the models and algorithms addressing specific business problems, he also develops several algorithms/models for solving fundamental machine learning tasks, which can be implemented in various applications.

Quantum science group at DKU led by prof. Myung-Joong Hwang explores a wide range of topics in the exciting field of quantum physics. Our group focuses on understanding quantum phenomena in engineered quantum systems with which quantum computers can be built. We also seeks to find ways to make our fundamental understanding of quantum physics useful for quantum technologies such as quantum computers and quantum sensors. Physical systems that we focus include ion-traps and superconducting quantum circuits. Our research is highly interdisciplinary lying at the interface of quantum information science, quantum optics, and condensed matter physics.

Prof. Huo’s research lies in the field of several complex variables which can be viewed as multivariable complex analysis. This branch of math studies holomorphic functions of several variables, the ones in complex variables and have power series expansions, and has interactions with other fields of math including differential equations, harmonic analysis, operator theory, and complex geometry. Prof. Huo’s primary research interest is about the Bergman projection and its related operators. He is particularly interested in understanding connections between properties of these operators and geometry and has used tools from dyadic harmonic analysis the study the Lp and weighted Lp regularity of the projection. A research project on related topics can involve various ideas and methods from simple integral/series computations to advanced techniques in harmonic analysis, special functions, and operator theory. He is also interested in topics from analysis in general and welcome discussions.

Dr. Lin Jiu’s main research areas include number theory and combinatorics, as well as probabilistic methods. He focuses on experimental mathematics methods and symbolic computations, also known as computer algebra. Major objects include sequences of numbers and polynomials, e.g. Bernoulli and Euler polynomials; special functions, such as Zonal polynomials, Riemann zeta-functions; combinatorial and probabilistic identities involving integer partitions, random walks, etc. Dr. Jiu’s also works on algorithms of symbolic integration and summations, For instance, in Summer 2022’s SRS program, he led a project on The Method of Brackets, of symbolic integration. Dr. Jiu’s current project also involves quantum algorithms, especially the applications in number theory and combinatorics.

Dr. Jiu has supervised several undergraduate projects, in various areas including random walk model, Hankel determinants, quantum variational algorithms, symmetric functions with applications. Besides topics related to his research areas, he is also willing to mentor Signature Work projects in other topics, such as modular forms, elliptic curves, etc.

Dr. Liu’s research interests lie primarily in the areas of functional analysis, operator algebras, and mathematical physics. Her research has been focused on algebras (families) of operators, including bounded and unbounded operators, affiliated with finite von Neumann algebras, in particular, the structure, commutativity, derivations, and reflexivity of these algebras. Her original research work has been published in prestigious mathematics and science journals, including two papers in the Proceedings of the National Academy of Sciences.

Dr. Liu enjoys working with students through Independent Study and Signature Work (SW) programs. Her SW projects are generally of inter-disciplinary and multi-disciplinary nature, feasible for undergraduate students (math and non-math majors) within 1-2 year time framework, aiming at fostering students’ general interest in research, improving their scientific writing skills, and preparing them for further academic and career advancement. Research topics and ideas from students are particularly welcome (e.g. DKU students work with her on analysis of spread of news through social media, which connects knowledge and perspectives from data science, psychology, behavioral science, social science, and communication study). She can be reached at zhe.liu372@duke.edu.

Prof. Ding Ma’s broad research interests are climate variability, atmospheric dynamics, and extreme weather events. The essential motivation for his research is to better understand and predict the behavior of the climate system, which has led to his focus on the largen-scale variability of the atmosphere, the related weather extremes, and their societal impacts. He focuses on the following overarching questions: 1) What physical mechanisms regulate the large-scale circulation variability and the related extreme weather events? 2) How can we improve the prediction skills for the variability, and in particular, the extremes? 3) How will these phenomena respond to climate change and what are the potential societal impacts?

Prof. Ma approaches these questions with a combination of (1) data analysis and (2) analytical and numerical solution of the governing equations of the atmosphere. The first research direction involves (a) statistical analysis of big datasets of observations and climate model output and (b) text mining of heterogenous sources of documents and social media. The other research direction is to solve the dynamical system of the atmosphere. He seeks to construct a hierarchy of numerical models of varying complexity, chosen so as to demonstrate and explain the observed behavior on various levels. This model hierarchy, an analogy to “model organisms” from bacteria to fruit fly to mouse in biology, leads to a firm understanding of the fundamental dynamics and conclusive answers to the questions proposed.

My research interests are in applied probability, statistics, graph theory, and combinatorics. Here are some problems I have been working on.

In statistics I am interested in problems related to the study of extremes, the branch of mathematics that deals with, for example, the distribution of the maxima or minima of random variables. A few months ago, the New York Times featured an article discussing the theoretical limit of the length of human life. This problem can be reduced to a problem of characterizing the limiting distribution of the maximum of a sequence of random variables. The mathematics one uses to study this problem is related to the mathematics one would use to study applications of extremes to DNA sequence analysis, or problems in number theory.

In probability/combinatorics, I have been studying properties of random binary matrices, that is, matrices where each entry can be 1 or 0 with probability p or 1-p, respectively. In neural networks, for example, there are instances in which one needs to investigate the existence of submatrices in which the number of 1’s in every column satisfies the majority rule. The question in this context is to find the limiting distribution of the number of such matrices as the size of the matrices goes to infinity.

Game theory has become an increasingly popular subject as it provides a unified way to investigate problems arising in different settings. Many games are now defined via a graph structure; a question I find interesting is how to understand how these structures affect the existence and configuration of Nash equilibria. Nash equilibria are configurations in which no player in a game can gain advantage by changing strategy.

In combinatorics, I am now working with a student on a problem called the sunflower problem. Suppose we have a collection of sets, where each set contains k elements. A sunflower is any group of r sets in the collection whose pairwise intersections all coincide. For a fixed k and fixed r, the problem consists of finding the smallest a collection must be in order to guarantee the existence of at least one sunflower. The simple case of k=4 and r =3 is a good starting point to appreciate the complexity of this problem.

My research focuses on general relativity, that is, Einstein’s theory of gravity, and its modifications and applications. This is a research area in differential geometry and mathematical physics overlapping with other branches of mathematics, such as geometric analysis and topology. I am particularly interested in geometrical properties of gravitational lensing, that is, the propagation of light under the influence of gravity which can be studied mathematically in various ways, including null geodesics in Lorentzian spacetimes and also Riemannian and Finslerian optical geometry. More recently, I have become involved in gravitational wave studies which provide an entirely new window on the universe. I am also interested in possible modifications of general relativity, especially a new approach called “constructive gravity”, which is based on a geometrodynamical technique to derive gravity theories so that they are causal, that is, having a well-posed Cauchy problem, by construction. Thus, optical effects of non-metric geometries, such as birefringence, can be studied as well, and this can yield powerful constraints on possible gravity theories beyond Einstein’s in turn.

Given the great recent progress in astrophysics, this research area has scope for Signature Work projects that are not only mathematically interesting but also timely and tractable, and I have supervised such projects on Riemannian optical geometry, black hole photon spheres, gravitational lensing, gravitational waves, and cosmology, at DKU. Finally, I have a strong interest in the history of mathematics and astronomy and have worked with experts in the history of science, and would support such interdisciplinary projects at DKU as well. Enthusiastic students considering a Signature Work research project in any of these areas are warmly encouraged to get in touch.

** **

Dr. Shixin Xu’s research focuses on integrating mathematics, life sciences and data sciences to solve challenging real-world problems such as investigating underlying mechanisms leading to cardiovascular and cerebrovascular diseases. The first theme of his research is modeling complex fluids systems in biological system like central nerve system and muscles. The second thrust of his research is developing efficient and structure-preserving numerical schemes for solving these model systems. The third theme of his research is related to machine learning. Data-driven models are proposed to predict the risks of chronical diseases, like Diabetic kidney disease. Physics-informed coupling data-driven models are used to predict health status of computer chips and DRAM.

Throughout his career, Dr. Xu has supervised undergraduate projects in several directions, including dynamical systems (Neuron-glia-extracellular coupling system), mathematical model (epidemic disease and noninvasive room occupation detection), machine learning (Stroke risk factor identification and Bitcoin price prediction) and others. He is interested in mentoring Signature Work projects in topics related to his research interests in mathematical modeling and data-driven model in biological and clinical problems, like debates and strokes.

Dr. Xiaoqian Xu’s research area is Partial Differential Equations, especially the differential equations about fluids. His main research theme is mixing for different kinds of fluid, active scalar equations, time-fractional differential equations and math biology. The main tool he used in research is applied analysis.

To work with Dr. Xiaoqian Xu as undergraduates, students will need to at least learn measure theory, functional analysis, Sobolev spaces first; or to learn numerical PDE and to be good at programming. It is also very welcome for students to contact Dr. Xu to have reading seminars to learn math of any areas in groups.

Prof. Zou’s research focuses on geometric deep learning, that is, deep neural networks that are not defined in Euclidean domains (e.g., graph neural network (GNN) and hyperbolic neural networks (HNN)). This is a very recent research field which has wide applications and inherits a lot of ideas from mathematics. The main mathematical techniques involved include applied harmonic analysis (particularly wavelet analysis and spectral graph theory), hyperbolic geometry, numerical linear algebra, topological data analysis, computational optimal transport. With a focus on developing novel models and algorithms, his research also highly relies on provable properties such as guarantees for invariance and robustness of neural operations.

Prof. Zou has supervised many DKU undergraduate projects. He is interested in mentoring signature work projects in topics related to geometric deep learning or robust / reliable / interpretable methods in machine learning. Focus of projects can be either theoretical or applied. Past undergraduate projects include: hyperbolic kernel convolution (for graph classification and machine translation); hyperbolic generative network (for molecular generation); interpretable anomaly detection (for industrial images); geometry-aware transformer (for cell segmentation); GNN-assisted domain adaptation (for phase retrieval); GNN-assisted authorship attribution; AI choreography and music generation; stock and portfolio price prediction; etc.