Past Projects

Mentor: Luke Barbarita

Algebraic Topology studies topological spaces through the scope of algebraic structures and tools, such as groups and homomorphisms. The goal of this reading course is to learn the foundational language and intuition for the subject. We will likely cover, time permitting, topics such as basic category theory, homotopy of maps, fundamental groups, homotopy groups, homology, and cohomology, with an emphasis on how algebraic invariants encode topological information. To do so we will be going through the first few chapters of Joseph Rotman’s An Introduction to Algebraic Topology, focusing on understanding definitions, key theorems, and some concrete examples. Other additional sources and/or topics may be added based off the interest of participants.

Mentor: Daniel Gotshall

Arithmetic dynamics generally studies patterns and properties of points formed from iteration of a polynomial or rational function starting at a given point in a field. Originally this area began by studying forward iterates over R and C, leading to the definitions of the complementary Julia and Fatou sets, which are simply the sets of points with chaotic and repetitive (preperiodic) behavior, respectively. This was then generalized to working over number fields, where one can form equivalence relations between pairs of rational functions and certain points. We will see theorems that can use a single member of the equivalence class to make conclusions on the dynamics of all maps in the class. Another topic will involve using “height functions” to quantify the size of the set of preperiodic points of rational maps under certain conditions. If time permits, we will study properties of the Galois extensions generated by adjoining certain preperiodic points. Current research is generally focused on the Galois theory arising from “preimages of points” (iterates done in the reverse direction).

Mentor: Roan James

The plan is to introduce the basics of polymer models. These can be seen as simple random walks (SRW) in an external medium/environment. We will start by recalling some of the basic properties of the SRW (transience/recurrence) and then proceed to define polymer models. If there is time, we can introduce the KPZ universality class.

Mentors: Zhihe Li and Ke Yu

In an effort to explore the foundations of harmonic analysis, we will begin with the basics of Fourier analysis and functional spaces to study key concepts in the subject. The first five chapters of Thomas Wolff’s Lectures in Harmonic Analysis provide a detailed introduction to essential topics such as Schwartz space, Fourier transforms, and its related properties such as Fourier inversion and Plancherel. By the end of this semester, we will apply our knowledge in harmonic analysis to study some results in uncertainty principles and restriction problems.

Mentor: Donovan Snyder

The area of “integrable probability” is the study of systems that can be studied and analyzed using algebra: sometimes complicated group and Lie theory, but often times more simple procedures. We’ll deal with those more simple procedures to look at examples such as ASEP, the Polymer, and Six-Vertex models and answer basic questions

Mentor: Siddharth Gurumurthy

The preliminary plan is to talk about the basic concepts in category theory: categories, functors, the Yoneda lemma, (co)limits and adjunctions with a bunch of examples. The plan is subject to change and will depend on the available time and the interests of the participant(s).

Mentor: Lippus Liu

One might find it easy to think about figures in a 2-dimensional plane. Imagining figures in a 3-dimensional space might also be not that hard. But what if we move to 4-dimensional spaces, or 5, 6, or even higher? Imagination in these cases turns out to be not very helpful. To solve this problem, topologists developed so-called “topological invariants”, which can normally be derived by just computations. In this reading group, we will learn about a particular example, that is the mapping degree, which can be used to distinguish mappings between high-dimensional figures.