Spring 2025 Program

Meet the Mentors

Joseph Breen

Postdoc in Math at the University of Iowa

Proposed project: Coloring in (Legendrian) knot theory

Description: Knot theory is a deep and powerful branch of topology. Most importantly, though, it is both fun and accessible, because it fundamentally involves drawing pictures and trying to visualize shapes. Solving problems and doing research in knot theory can often feel like playing a shapes-and-colors-based Sudoku-style game.

Regular knot theory is plenty interesting, but I am interested in a particular type of knot theory --- Legendrian knot theory --- where the diagrams and rules for manipulating them have some unusual properties (Click here for some pictures of Legendrian knots, due to Lenny Ng: https://sites.math.duke.edu/~ng/knotgallery.html). You will almost certainly never come across Legendrian knot theory in an undergraduate curriculum, but when presented in the right way, it is just as accessible and easy to play around with as regular knot theory. It is also part of deep and cutting edge math. For example, an active area of research is on the "normal rulings" of these knots, which are fancy ways to assign colors to a knot diagram, and something you can play around with yourself once you learn the rules. Click here for an example: https://celebratio.org/media/cunit/None_cunit_Figure3.png.

In this project, we'll learn some basics about knot theory, and depending on interest, can move into the colorful and game-like world of Legendrian knots and normal rulings.

Suggested Book: "The Knot Book" by Colin Adams, a freely available and friendly introduction to the subject. Depending on student interest, we can stay in The Knot Book, or branch out to other online resources to learn about Legendrian knots.

Prerequisites: Though not strictly necessary, some experience with multivariable calculus, differential equations, and proofs would be helpful.

Eric Chen

Assistant Professor in Math at the University of Illinois at Urbana-Champaign

Proposed project: Beyond Euclidean geometry

Description: The shortest path between two points in the flat plane is a line segment, but what about on a curved surface like a sphere, or something more complicated? And what does it mean to say that a path or surface is curved?

In this project, we will start by understanding important geometric properties of one-dimensional paths and two-dimensional surfaces, including length, area, and curvature. Afterwards our direction can be more open-ended – depending on student interest, we can proceed to topics such as: hyperbolic and spherical geometries, optimization problems (isoperimetric inequality, minimal surfaces/soap bubbles), or connections to physics (quantum computation, general relativity).

Suggested Book: "Differential Geometry of Curves and Surfaces" by Banchoff and Lovett

Prerequisites: Linear algebra, multivariable calculus. Some exposure to proofs is helpful but not necessary.

Marino Echavarria

PhD Student in Math at Brown University

Proposed project: Abstractions, Analogies, and Relationships in Mathematics

Description: Category Theory is a powerful theory that has propelled modern mathematical research, and is now a staple in the toolbox of many mathematicians. We will discuss some of the core principles of category theory in a variety of contexts, some even coming from real-world experiences!

Category theory is the study of objects, their relationships to one another, and analogies between things in different contexts. Classically, mathematicians used to study objects by probing them. For example, number theorists would study integers by playing around with them, eventually discovering things like prime factorization and theorems about special algebraic equations. Category theory instead says "Tell me who your friends are, and I'll tell you who you are.’’ This statement is actually captured in a theorem!

Categorical thinking also involves abstraction and analogy. What could the number system we use for our clocks possibly have in common with gluing two things together? It turns out they're both examples of an operation called a colimit. This process of abstraction involves zooming out so that you can see similarities in things that look like they have nothing to do with one another.

Suggested Book: “The Joy of Abstraction: An Exploration of Math, Category Theory, and Life,” by Category Theorist and Pop Science Writer Eugenia Cheng written for the general public. It isn't a standard textbook, so it reads easier while also keeping mathematical rigor intact. We can also use Emily Riehl's "Category Theory in Context" as a companion text.

Prerequisites: Familiarity with proofs, group theory, number theory (like modular arithmetic), and/or topology are appreciated but not necessary.

Tudor Popescu

PhD Student in Math at Brandeis University

Proposed project: A Friendly Introduction to Cryptography

Description: Would you like to learn about breaking codes and be a modern-day Turing? If so, this project is perfect for you. We will start by going over some number theory concepts, such as modular arithmetic, congruences, and quadratic reciprocity. Most of our time will be spent discussing public key cryptosystems, such as Diffie-Hellman and RSA, as well as some real-world applications. If time permits, we will talk about primality & factoring and about applications of elliptic curves.

Suggested book: "An Introduction to Number Theory with Cryptography" by Kraft and Washington

Prerequisites: A course on proofs is required. Prior knowledge of abstract algebra or number theory is a plus, but definitely not required.

Bailee Zacovic

PhD Student in Math at the University of Michigan

Proposed project: Symmetric functions: their combinatorics and applications

Description: Symmetry is a naturally-occurring phenomenon in our universe. It also pervades mathematics, and can be leveraged to extract patterns and enumerate objects of certain types. Functions can also exhibit symmetries; the "symmetric functions" are a special family which are invariant under reshuffling the input variables. The applications of symmetric functions in mathematics are abundant, especially in combinatorics, representation theory, and algebraic geometry.

This project will offer an introductory exploration of the space of symmetric functions, and one particularly nice basis of this linear space known as the Schur basis. On our quest, we will encounter playful combinatorial objects known as Young tableaux, graphs, permutations, and partitions, and investigate central results relating to these objects, including the Robinson–Schensted–Knuth correspondence and the Littlewood–Richardson rule. Time permitting, we will survey an active research area, such as chromatic symmetric functions and Stanley's open chromatic tree conjecture.

Suggested Book: "An Introduction to Symmetric Functions and Their Combinatorics" by Eric Egge

Prerequisites: Familiarity with linear algebra and/or multivariable functions would be helpful.