(suggested by the mentors, you may also suggest topics of your own)
Algebra
Algebraic Geometry
Algebraic Topology
Analysis
Analytic Number Theory
Calculus of Variations
Category Theory
Combinatorics
Commutative Algebra
Complex Analysis
Convex Analysis
Cryptography
Differential Equations
Differential Geometry
Differential Geometry of Surfaces
Differential Topology
Dynamical Systems
Elementary Number Theory
Elliptic Partial Differential Equations
Enumerative Geometry
Flows
Fourier Analysis
Functional Analysis
Functional Equations
Galois Theory
Geometric Analysis
Graph Theory
Group Theory
Homological Algebra
Knot Theory
Lie Algebras
Lie Groups
Linear Algebra
Logic
Matrix Analysis
Matroids
Measure Theory
Model Theory
Noncommutative Algebra
Number Theory
p-Adic Numbers
Parabolic Partial Differential Equations
Partial Differential Equations
Plane Algebraic Curves
Point-Set Topology
Probability
Real Analysis
Recursion Theory
Representation Theory
Riemannian Geometry
Set Theory
Statistical Physics
Symplectic Geometry
Theory of Inequalities
Sieve Theory
Number Theory
Mentor: Yung-Chiehe Hsieh
Text: John Friedlander & Henryk Iwaniec, Opera De Cribro
Description
Start with Eratosthenes-Legendre sieve and basic tools for sieves. A byproduct along the way is that the sum of reciprocities of prime numbers is infinite. Then move to Brun's sieve. As an surprising result of Brun's sieve, we'll deduce that the sum of reciprocities of twin primes is not infinite, but finite.
Prerequisites: Calculus.
Properties of Harmonic Functions
Analysis
Mentor: Samanthak Thiagarajan
Text: Lawrence C. Evans, Partial Differential Equations 2nd Edition
Description
An exploration of harmonic functions, starting with complex analysis and working up to general setting (Laplacian equals zero). Properties discussed could be regularity, mean-value property, and unique continuation.
Prerequisites: Real analysis.