(suggested by the mentors, you may also suggest topics of your own)
Algebra
Algebraic Geometry
Algebraic Topology
Analysis
Category Theory
Combinatorics
Commutative Algebra
Complex Analysis
Convex Analysis
Cryptography
Differential Equations
Differential Geometry
Differential Geometry of Surfaces
Differential Topology
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
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
A Soft and Motivated Introduction to Algebraic Number Theory
Number Theory
Mentor: David Herrera
Text: John Stillwell, Elements of Number Theory
Description
We will go through Stillwell's wonderful and basic introduction to (algebraic) Number Theory. This project is intended for the student who wants to explore number theory and learn about what are the types of questions that people have asked which led to the development of some concepts in Ring Theory (ideals, unique prime factorization, etc.)
This book is a great place to learn about the basics of Ring Theory for the student interested in taking 351 or who wants to see how the topics that one saw in 351 connect to natural number theory problems.
The exposition starts with basic problems in number theory (Pythagorean triples, continued fraction algorithm, modular arithmetic, Gaussian integers) and moves into more advanced topics (Euler totient function, rings, ideals, quotient rings, UFD, prime factorization) in discussing how these abstract ideas developed from natural questions about how to solve Fermat's Last Theorem in a few non-trivial cases.
This is a book that I learned from when I had difficulty with learning abstract algebra as an undergraduate. Although I later found Hungerford's book on Abstract Algebra to be a good book to learn the theory of the subject, Stillwell's book on Elements of Number Theory really makes some of the basic theory come to life in ways that few math textbooks do. (A very similar statement can be made about Stillwell's Elements of Algebra.)
Prerequisites: An introduction to proofs/mathematical reasoning.
Combinatorial Set Theory
Set Theory
Mentor: Ben-Zion Weltsch
Text: Andrew Marks, lecture notes
Description
We will explore combinatorial aspects of mathematical infinity: cardinality, trees, filters and ideals, cardinal arithmetic, infinite Ramsey theory, etc. Time permitting, connections between logic and combinatorics can be explored further: constructibility, Jónsson algebras, large cardinals, and forcing.
Prerequisites: Mathematical maturity.
Girth in Graphs
Graph Theory
Mentor: Pablo Blanco
Text: Reiher, "Graphs of Large Girth," Exoo and Jajcay, "Dynamic Cage Survey," etc.
Description
For this project, we will read about the girth parameter in graph theory, which is the length of the shortest cycle in a given graph, as well as related theorems and cage graphs. We may also do some reading on graphic matroids, if it comes up.
Prerequisites: Graph theory and basics of linear algebra (linear independence and such).
Mathematics of the Ising Model
Probability
Mentor: Qidong He
Text: Sacha Friedli and Yvan Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction.
Description
The Ising model is a classic statistical-physical model of ferromagnetism (think of a real magnet). It is also one of the very few statistical-physical models with "realistic" interactions whose entire phase diagram can be mathematically derived. We will go over this derivation. Time permitting, we can explore additional topics in the textbook, including liquid-vapor transitions, random surfaces, and the Mermin-Wagner theorem.
Prerequisites: Mathematical maturity; prior exposure to discrete math would be helpful but is not required.