(suggested by the mentors, you may also suggest topics of your own)
Algebra
Algebraic Topology
Approximation Theory
Brownian Motion
Calculus of Variations
Category Theory
Classical Differential Geometry
Combinatorics
Commutative Algebra
Complex Analysis
Computer-Assisted Mathematics
Elliptic Partial Differential Equations
Financial Mathematics
Fourier Analysis
Functional Analysis
Galois Theory
Geometric Measure Theory
Geometry
Graph Theory
Group Theory
Homological Algebra
Knot Theory
Lie Algebras
Logic
Mathematical Physics
Matrix Analysis
Measure Theory
Noncommutative Algebra
Ordinary Differential Equations
Partial Differential Equations
Probabilistic Combinatorics
Probability
Real Analysis
Representation Theory
Ring Theory
Set Theory
Sets of Finite Perimeter
Sobolev Spaces
Stochastic Partial Differential Equations
Theoretical Computer Science
Topological Combinatorics
Topology
Forcing
Set Theory
Mentor: Ben-Zion Weltsch
Text: Kunen, Set Theory: An Introduction to Independence Proofs
Description
An introduction to Cohen's method of forcing and independence of the Continuum Hypothesis.
Prerequisites: Familiarity with basic set theory and first-order logic.
Harmonic Functions
Analysis
Mentor: Samanthak Thiagarajan
Text: Evans, Partial Differential Equations
Description
We study some of the fundamental properties of harmonic functions, including the mean value property, the maximum principle, regularity, and unique continuation.
Prerequisites: Real analysis.
Homological Algebra
Geometry/Topology
Mentor: Jianing Zhang
Text: Charles Weibel, An Introduction to Homological Algebra
Description
We will study the basics of homological algebra, for example, chain complexes, functors, homology and/or cohomology together.
Prerequisites: Abstract algebra.
Infinitary Combinatorics
Set Theory
Mentor: Ben-Zion Weltsch
Text: Schimmerling, A Course on Set Theory
Description
An introduction to infinitary combinatorics (ordinals and cardinals, cardinal arithmetic, trees, graph colorings, club and stationary sets, filters and ideals, linear orders, etc.)
Prerequisites: Math 300.
Mathematics of the Ising Model
Probability
Mentor: Qidong He
Text: Chapter 3 of Sacha Friedli and Yvan Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction.
Description
The Ising model is a classic statistical-mechanical model of ferromagnetism (think of a real magnet). It is also one of the very few statistical-mechanical models with "realistic" interactions whose entire phase diagram can be mathematically derived. We will go through the definition of the Ising model and a complete derivation of its phase diagram.
Prerequisites: Mathematical maturity; prior exposure to discrete math would be helpful but is not required.