(suggested by the mentors, you may also suggest topics of your own)
Algebraic Geometry
Algebraic Number Theory
Algebraic Topology
Analysis
Analytic Number Theory
Complex Analysis
Computer-Assisted Mathematical Problem Solving
Cryptography
Descriptive Set Theory
Elliptic Curves
Enumerative Combinatorics
Functional Analysis
Graph Theory
Group Theory
Lie Algebra
Low Dimensional Topology
Measure Theory
Model Theory
Partial Differential Equations
Ramsey Theory and Extremal Combinatorics
Random Graphs and the Probabilistic Method in Combinatorics
Riemannian Geometry
Set Theory
Undergraduate Analysis
Grid Homology
Geometry & Topology
Mentor: Timothy Bates
Text: Grid Homology for Knots and Links by Ozsvath, Stipsicz, and Szabo
Description
Grid homology is a modern invariant for knots and links. Grid diagrams are a combinatorial tool encoding a knot in 3-dimensional space. From the grid diagram, one obtains chain complexes whose homology give invariants of the knot represented by the grid diagram. These invariants are strong and can detect many geometric phenomenon of the knot such as the 3-genus, and give bounds on the slice genus and the unknotting number. By the end of the program, the student will understand the definition of grid homology, understand the proof that it is an invariant of knots and links, and its ability to give bounds on the slice genus, and be able to prove the Milnor conjecture for torus knots.
Prerequisites: Algebra (must have seen groups, rings, and modules), Background in topology is welcome but no prior knowledge of knot theory is required.
Power Series in More Generality
Analysis
Mentor: Ryan Mc Gowan
Text: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, Lars V. Ahlfors, (1979), McGraw-Hill Education (ISBN-13: 978-0070006571) and further texts as necessary
Description
The notion of a power series, while simple, reveals a lot of deep properties of analytic functions. In one complex variable, for instance, a function having a convergent power series about a point allows one to conclude it is "differentiable" in a complex sense. The goal of this reading project is to understand this correspondence, some simple applications of it in the one variable case, and then examine the much more complicated scenario if one moves up to several complex variables. There are both analytic and algebraic methods to examine this problem, depending on the interests of the student.
Prerequisites: Real analysis. Complex analysis is advantageous but not essential