Projects

Overview

• Data Analysis, Statistics, and Probability
• Interactive Visualizations with Mathematica
• Game Theory and Related Areas
• Randomness in Number Theory
• Calculus, Geometry, and Probability in n Dimensions

For upcoming projects and research opportunities see my Undergraduate Research Page.

Data Analysis, Statistics, and Probability

• General description: This project is the outgrowth of the Campus Honors (CHP) courses, Math 199 CHP: Probability and the Real World, and Math 199 CHP: Mathematics of Games, Sports, and Gambling, that I have been teaching for the past several years. Its goal is to seek out and explore interesting real-world problems of statistical flavor in sports, finance, society, and everyday life.
• Qualifications: The specific background and skills required depend on the particular project, but the projects typically require at least one of the following qualifications: (1) advanced statistical background (with multiple 400 level courses); (2) a high level of experience with data mining and machine learning through projects and/or courses; (3) expert-level programming skills in R or Python; (4) experience working in a Unix/Linux command-line environment and writing shell scripts.
• Past projects: Click on titles for links to posters.
  • Data Mining MSFE Program Admissions (Spring 2019). (Danjing Jiang, Peiyun Zhu)
  • Comparing University Rankings (Summer 2018). (Weiru Chen, Qianqian Li)
  • Data Mining Math Graduate Admissions (2017 - 2018). (Jianqiu Kong, Jiahao Zhu, Ruorong Yin, Hao Qiu, Pengyu Chen)
  • Analysis of AP College Football and Basketball Polls (Summer/Fall 2018). (Sam Guo, Qingqing Yang, Carson Zhang)
  • Distinguishing Cause and Effect (2017 - 2018). (Xinying Fang, Shuyu Wang; Eli Yu, Yuxuan Ren, David Zhang)
  • Statistical analysis of weather forecasts (2016 - 2017). (Mario Cannamela, Yajie Chu, Sijia Huo, Yantong Zheng, David Zhang, Yayu Zhou)
  • Zipf's Law (Summer 2016). (Daniyar Omarov, Khoa Tran, Laila Zhexembay)
  • Knowledge Spaces (co-advised by Dr. Michael Santana) (Summer 2016). (Qingci An, Madina Bolat, Jose Sanchez)
  • Statistical analysis of Daily Illini College Football Predictions (Fall 2015). (Qing Ma, Yayu Zhou)
  • Detecting non-randomness (2015 - 2017). (Tong Li, Jiarui Xu, Xusheng Zhang Victor Sui)
  • Benford's Law (Summer 2015). (Narken Aimambet, Chingis Matayev, Alisher Urazbekov)
  • The Fake Randomness Project (2014). (Jiachun Chen, Chad Franzen, Rishabh Marya, Robert Weber, Xin Wei)

Interactive Visualizations with Mathematica

• General description. The goal of these projects is to create interactive Mathematica-based visualizations of interesting mathematical topics and make these available to a broader audience through publication at the Wolfram Demonstrations website, http://www.demonstrations.wolfram.com. Topics will be chosen based on the backgrounds and interests of the participants, and will typically be at an advanced undergraduate or honors level. Past projects have covered topics such as the Coupon Collector Problem, the Ballot Problem, Lagrange Multipliers, Harmonic Functions, Random Walks based on digits of Pi and similar famous numbers, and Random Walk Solutions to PDEs. See the links below for projects reports and demonstrations created in past projects.
• Qualifications. Prior experience with Mathematica, or evidence of strong general coding skills, is highly desirable. Students are expected to be comfortable with using Mathematica by the start of the project. Students should also have a solid mathematics background, at the level of an A- grade or better in Calculus III.
• Past projects. Click on titles for project reports.
  • Interactive Visualizations in Calculus and Probability (Spring 2019). (Raymond Harpster, Tianli Li, Yikai Teng)
  • Interactive Visualizations in Calculus and Probability (Fall 2018). (Yuheng Chang, Baihe Duan, Yirui Luo, Yitao Meng, Cameron Nachreiner, Yiyin Shen)
• Wolfram Demonstrations created in past projects.

Game Theory and Related Areas

• General Description. The projects under this theme deal with mathematical questions arising in game theory, voting theory, and related areas at the interface of mathematics and economics.
• Qualifications. A strong mathematical background is essential, and additional background in economics, and especially game theory, is desirable. The projects are particularly aimed at students who are double majors in Math and Econ or Statistics and Econ. For many of the projects, strong coding skills in Python or R or Mathematica are also required.
• Past projects: Click on titles for project reports or posters.
  • The Mathematics of Poker-like Games (Fall 2018). (Kyle Begovich, Xiangyun Cao, Philip Dohm, Mengyang Zheng, Qiaoge Zhu)
  • The Mathematics of Poker (Fall 2017 - Spring 2018). (Tanner Corum, Jiayin Lu, Carl Tang, Alex Wang, Ajay Dugar, Kevin Grosman)
  • The Mathematics of Poker (Fall 2016 - Spring 2017). (Matt Romney (graduate student), Nick Brown, Junghyun Hwang, Ziyang Liu, Ki Wang, Yanxuan Wang, Ruoyu Zhu)
  • The Mathematics of Poker (Summer 2017) (Xinying Fang, John Haug, Sultan Muratov, Sara Rat, Blandon Su, Fan Wu, Ruoyu Zhu)
  • The Geometry of Voting (Fall 2015 - Spring 2016). (Matt Romney (graduate student), Daoyu Duan, Vivek Kaushik, Aubrey Laskowski, Zelin Li, Yukun Tan, Mengzhou Tang)

Randomness in Number Theory

• General description: Random-like behavior is ubiquitous in number theory. For example, the primes, the digits of pi or other famous constants, and the Moebius function and other number-theoretic functions, all appear to behave much like appropriately defined "true" random sequences. In this ongoing project we seek to explore such random features experimentally--for example, via large scale computations and geometric visualizations as random walks--and possibly also theoretically.
• Qualifications: The desired background depends on the specific project. For projects that fall mainly on the experimental side, strong coding skills in Python and/or C/C++ and/or Mathematica are essential. Experience with solving coding challenges from Project Euler would be a plus. For more theoretically oriented projects, students should have a strong general math background with A grades in key math courses, including Math 453 (Elementary Number Theory).
• Past projects: Click on titles for links to project reports or posters.
  • Experimental Number Theory (2016 - 2018). Mini projects at the interface of Number Theory and Computation, in the spirit of Project Euler. (Phillip Harris, Xinwei He, Thien Le, Yuchen Li, Zihe Wang, Yunyi Zhang, David Zhang)
  • Beatty Sequences, Exotic Number Systems, and Partitions of the Integers (joint with Prof. Ken Stolarsky) (Fall 2017 - Spring 2018). (Junxian Li (graduate student); Mark Cao, Weiru Chen, Xinxin Chen, Jared Krandel, Stephen Fan, Matthew Cho, Jared Krandel, Xiaomin Li, Yun Xie)
  • Randomness in leading digits of number-theoretic sequences (Fall 2015 - Spring 2016). (Junxian Li (graduate student), Zhaodong Cai, Yewen Fan, Matthew Faust, Kevin Kwan, Yuda Wang, Shunping Xie, Yuan Zhang)
  • Number-Theoretic Fourier Series (Spring 2015) (Junxian Li (graduate student); Yu Fu, Ryan Grady, Yuda Wang, Jia Yu)
  • Power series with number-theoretic coefficients (2014 - 2015). (M. Tip Phaovibul (graduate student), Yiwang Chen, Keran Huang)
  • Randomness in Number Theory: The Moebius Case (2013 - 2014). (M. Tip Phaovibul (graduate student), Yiwang Chen, Daniel Hirsbrunner, Dylan Yang, Tong Zhang)
  • Number-theoretic Random Walks (Spring 2013). (M. Tip Phaovibul (graduate student), Yiwang Chen, Wenmian Hua, Natawut Monaikul, Tong Zhang; Spring 2013)
  • The Quadratic Residue Random Walk (Fall 2012). (M. Tip Phaovibul (graduate student), Yiwang Chen, Yusheng Feng, Mateusz Wala)

Calculus, Geometry, and Probability in n Dimensions

• General description: In this project we seek out and explore interesting problems in n-dimensional space that are accessible at the calculus level, but rarely covered in standard calculus courses. Such problems are often motivated by applications to probability and statistics or other areas. They tend to have a broad appeal and make for ideal topics to present at outreach events. They also lend themselves naturally to geometric visualizations, and can make excellent candidates for creating interactive Mathematica-based visualizations.
• Qualifications: Students should have taken Math 241, preferably in the honors version, with a grade of A- or better, or be concurrently enrolled in an honors section of Math 241. Beyond strong calculus skills, there are no formal prerequisites, and the project can accommodate a broad range of interests, backgrounds, and majors. Experience with Mathematica visualizations is helpful, but not required.
• Past projects. Click on titles for project reports.
  • The Broken Stick Problem in Higher Dimensions (2014 - 2015). (Josh Adkins, Alex Page, Aditya Patel, Yuliya Semibratova, Yao Xiao, Yi Xuan, Rui Zhang)
  • Random Triangles (Fall 2013). (Zehan Chao, Liran Chen, Hao Gao, Mianfeng Liu, Gilad Margalit)
  • Intersecting Cylinders in n Dimensions (2012 - 2013). (Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Ananya Uppal)
  • The Broken Stick Problem with n Pieces (2012 - 2013). (Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Ananya Uppal)