School of Science Department of Mathematics 39 Department of Mathematics Chromatic Polynomials of Graphs and Signed Graphs Supervisor: CHEN, Beifang / MATH Student: CHAN, Yuk Ming / MATH-IRE Course: UROP1100, Fall P.W. Kasteleyn showed that using the Pfaffian of a signed adjacency matrix of an directed graph oriented in certain fashion, the number of perfect matchings in the original undirected graph can be calculated. This is known as the "Pfaffian orientations" for planar graphs. On the other hand, G. Tesler introduced "crossing orientations" for calculating the number of perfect matchings of a nonplanar graphs. Here we give an overview of their work. Chromatic Polynomials of Graphs and Signed Graphs Supervisor: CHEN, Beifang / MATH Student: LI, Junru / MATH-CS Course: UROP1000, Summer Graph theory is closely related to the field of computer science. Spectral graph theory is a product of combining graph theory with linear algebra, which explores some properties of graphs by studying the eigenvalues and eigenvectors of their matrices. These theories play a crucial role in image processing, computer graphics, and other engineering fields. Therefore, understanding graph theory is useful for my future learning of algebraic topology and algorithms in the computer field. I will first introduce the basic concepts of graph theory and some graph algorithms summarized during the project process. In addition, since the Laplace matrix of a graph is the foundation and core of spectral graph theory, I will introduce the basic concept of the Laplace matrix of a graph. Statistical Analysis in Portfolio Construction Supervisor: CHEN, Kani / MATH Student: JIA, Tongtong / QFIN Course: UROP1100, Fall Over the past few years, cryptocurrencies have increasingly been discussed as alternatives to traditional fiat currencies. The interest of applying quantitative trading strategies in stock market has also extended to cryptocurrency market recently. Retrieving a set of 50 cryptocurrencies for a sample spanning 2019-2022, we look for statistical evidence to show the existence and significance of momentum factor among top performers. We scratched real-time data through price-tracking website for crypto assets, constructed features including market capitalization, volume, return in different time intervals, etc, and applied different machine learning models like Random Forest, Linear Regression. Final models’ result was able to prove the relative significance of momentum factors.

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