Fall 2022 Program

Meet the Mentors

Joanne Beckford

PhD Student in Math at Bryn Mawr College

Description: In geometry, the square, triangle and hexagonal tiles are the only three regular convex polygons that can cover a plane without any gaps or overlaps. Have you ever wondered, then, why honeybees choose honeycombs to have hexagonal cells, as opposed to square or triangular? What happens to the pattern when a honeybee makes a mistake during construction - can they just build their home using a squashed-looking convex pentagon? These questions and more can be considered to be a part of a broader category of packing problems, a class of optimization problems that involve packing objects (or at least attempting to pack them!) together into containers or spaces. In this project, we will explore tiling and packing problems related to semi-regular tilings, Penrose tilings, and tilings with irregular convex polygons, as well as applications to tiling optimization and the historic and cultural relevance that the Alhambra has to tiling. 

Suggested book: Tilings and Patterns by Branko Grunbaum and G.C. Shephard, 2nd edition.

Prerequisites: Experience with proofs, graph theory, and basic topology would be helpful, but ultimately this project is designed to be accessible to all who've had exposure to high school geometry and have an interest in seeing where art meets math!

Micah Henson

PhD Student in Applied Math at the University of Washington

Description: Mathematics can be used to solve real world problems. Researchers in many fields including economics, biology, computer science, and many social sciences rely on mathematics to gain deeper insights into their fields. For example, math models have helped us with the COVID pandemic. Social networks, marriage, climate change and much more can be studied using math models. In this project, you will be able to choose a topic you would like to explore with math models. We can go really in-depth into one model, or explore many different models. The choice is yours!

We will read from the book "Topics in Mathematical Modeling" by KK Tung.

Prerequisites: Some differential equations would be nice, but not necessary

Daniel Keliher

Postdoc in Math at the University of Georgia

Description: In this project, we will study how information gets transmitted reliably and correctly, and the various ways we can talk about the transmission of information. As an example, consider ISBN numbers. They're ten digit numbers that encode books. Online booksellers use them to handle orders. The last digit of an ISBN number is essentially determined by the preceding nine digits according to some rule. Using this rule, one can tell if an error has occurred while transmitting ISBN numbers. 

ISBN numbers are examples of “error detecting codes," i.e. codes that allow the receiver to detect if an error has occurred during transmission. Likewise, and perhaps amazingly, there are examples of “error correcting codes." These are codes that the receiver can use to correct errors. We’ll study some of the theory behind information transmission and codes, as well as many examples of codes. We’ll see how to measure the efficiency of codes, and the difficulties of producing efficient error detecting/correcting codes. This subject has many connections to computer science, number theory, and cryptography. There will be ample opportunities for diversions depending on the interest of the student. 

Suggested Book: Introduction to Coding and Information Theory by Steven Roman.

Prerequisites: Linear algebra. Some experience with finite fields or modular arithmetic would be helpful, but is not necessary.

Aaron Li

PhD Student in Math at the University of Minnesota

Description: Can you color a map of the world with four colors so that no countries that share a border are the same color? What is the most efficient way to distribute cell towers in a city? What's the best way to schedule classes and classrooms? Is this Sudoku puzzle solvable?

These seemingly disparate questions can be tied together with the concept of graph theory. They are all examples of graph coloring problems. Graph coloring is a really fun, visual area of math that is full of problems that are easy to state, but extremely difficult to solve. There are two potential directions for this project, depending on which the mentee would be more interested in:

1) We can approach this area from a more pure, theoretical perspective and examine bounds on the chromatic number (minimum number of colors needed to properly color a graph) and explore different families of graphs. We would see a lot of very beautiful and unique proofs that draw from algebra, combinatorics, and topology. This option would probably follow a book like "Chromatic Graph Theory" by Chartrand and Zhang.

2) We could also approach these problems from a more algorithmic standpoint. We would explore not just what an optimal number of colors would be, but also how to find it efficiently. This project would let mentees get a taste of some ideas that underpin tons of important applications in scheduling and computer science. This option would follow "A Guide to Graph Coloring: Algorithms and Applications" by Lewis.

Prerequisites: Some familiarity with proof-based math. Experience with discrete math or algorithms is a plus, but not necessary at all!

Alvaro Martinez

PhD Student in Math at Columbia University

Description: In how many ways can you knot a string? This innocent question is the beginning of a rich world of possibilities, and lies at the heart of the mathematical field of knot theory. Knot theory is an extremely visual area of mathematics where a lot of the work is done by manipulating diagrams, while simultaneously connecting with some of the most abstract fields of math. As a result, it can serve as a good introduction to advanced topics such as algebraic topology, group theory, differential geometry, and many more. If you are interested, we can use specialized knot theory software, in order to visualize and do computations with knots. This will be an open-ended project dictated by what you find intriguing and would like to explore.

Suggested Book: The Knot Book by Colin Adams.

Prerequisites: A course in abstract algebra (groups) or topology would be helpful, but not necessary.

Meet the Students

Elon Davis

Kristian Reed

Evelyn Rich

Tyrone Thomas

Naya Welcher