(suggested by the mentors, you may also suggest topics of your own)
Algebra
Algebraic Geometry
Algebraic Groups
Algebraic Number Theory
Algebraic Topology
Analysis
Analytic Number Theory
Arithmetic Geometry
Category Theory
Combinatorial Game Theory
Combinatorial Set Theory
Combinatorics
Commutative Algebra
Complex Analysis
Computability Theory
Computer-Assisted Mathematical Problem Solving
Cryptography
Descriptive Combinatorics
Descriptive Set Theory
Differential Geometry
Differential Topology
Discrete Geometry
Dynamical Systems
Elementary Set Theory
Functional Analysis
Galois Theory
Geometry
Group Theory
Harmonic Analysis
Homological Algebra
Lie Algebras
Logic
Mathematical Quantum Mechanics
Measure Theory
Model Theory
Noncommutative Algebra
Number Theory
Partial Differential Equations
Probabilistic Combinatorics
Probability Theory
Real Analysis
Representation Theory
Surreal Numbers
Theoretical Computer Science
Analytic Number Theory a là Riemann
Number Theory
Mentor: Michalis Lolis
Text: Multiplicative Number Theory by Davenport
Description
Functional equation of zeta and Dirichlet L functions, zero-free region, PNT with error term
Prerequisites: Basic complex analysis
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
Maximum Principle and its Application
Analysis
Mentor: Junyoung Park
Text: Chapter 1, 2 of Elliptic Partial Differential Equations by Qing Han and Fanghua Lin, Chapter 1 of Extrinsic Geometric Flows by Ben Andrews, Bennett Chow, Christine Guenther, May Langford,
Description
Maximum principles is one of the most powerful methods in the study of second order elliptic / parabolic PDE. By going through (parts) of the mentioned text, the student will be able to understand what the maximum principle is, and how it can be used to study solutions to elliptic / parabolic PDE.
Prerequisites: Strong grasp of calculus. Real analysis is also useful.