School of Science Department of Mathematics 40 Chromatic Polynomials of Graphs and Signed Graphs Supervisor: CHEN Beifang / MATH Student: JIA Yifei / MATH-FAM Course: UROP 1100, Spring UROP 1000, Summer This report investigates the chromatic polynomial and tension polynomial, with a particular focus on introducing the Ehrhart polynomial and employing geometric concepts to analyze their properties. We establish the reciprocity law for both the chromatic polynomial and the integral tension polynomial, revealing a connection between their values and the number of acyclic orientations in the graph. Additionally, we provide a combinatorial proof of the reciprocity law for the chromatic polynomial and extend this result to arbitrary partially ordered sets. Chromatic Polynomials of Graphs and Signed Graphs Supervisor: CHEN Beifang / MATH Student: SIN Tsz Yin / COGBM Course: UROP 1100, Spring Graphs has been a long used idea in mathematics and computer science. Hence, many researches have been conducted to observe different patterns of graphs. This UROP report is to document some research and self learning I have done in this UROP project under the topic of studying polynomials of graphs, in specific, flow and tension polynomials. The main goal of this project is to study the relationship between flow and tension polynomials and observe the patterns under different combination of flow and tension polynomials. Unfortunately, there are no meaningful results yet. Instead, two papers by Chen are read. This report summarizes the two papers by extracting the definition of flow and tension polynomials, explaining dual complementary polynomials, and concludes by mentioning ongoing work. Applications of Large Language Models in Special Sectors Supervisor: CHEN Kani / MATH Student: HAN Liuruo / DSCT Course: UROP 1100, Spring This project focuses on developing a specialized AI model using multi-agents structure, to enhance mathematical logic and reasoning capabilities of basic large language models. Through the integration of specialized agents and multi-agent systems, the AI assistant, can provide accurate solutions, detailed explanations, and deeper insights into mathematical concepts for relatively simple math problems.
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