2023 Counselor Seminars

Counting using Lagrange Inversion

The Normal Basis Theorem

Arithmetic Fuchsian Groups

Ramification Theory

Simplicial Homotopy Theory

Groups and Trees

Ramanujan Graphs

Distributions

Morse Theory (I-II)

Noether’s Theorem

Theta Correspondence

Representation Theory

Riemann Mapping Theorem

Resolutions

2022 Counselor Seminars

Art and Math
Pollock, Fractals, and Visual Aesthetics
Geometric Construction in Islamic Art

Homotopy Type Theory (HoTT)  (I-II)

Elliptic Curves
Intro to Group Cohomology
Weak Mordell Weil
Kummer Theory

De Rham Cohomology (I-IV)

Deligne’s Theorem
Derived functors
What is a Sheaf?
Algebraic Geometry
Etale Cohomology I-II
Modular Forms
Deligne I-III

Algebraic Topology (I-V)

Miscellaneous
Characteristic Classes
The Class Field Tower Problem
The Central Limit Theorem
Polynomials in Combinatorics
Intro to Category Theory
A Taste of Algebraic Geometry
De Rham’s Theorem
Quadratic Reciprocity
Kirby Calculus
Structure of Number-theoretic Graphs

2021 Counselor Seminars

Algebraic Number Theory
Cyclic Quartic Extensions of Fields
p-adics
Dedekind Domains I-IV
Galois Theory and Representation Theory
Extensions of Local Fields

Periods and Galois Representations in Algebraic Geometry and Arithmetic
p-adic Hodge Theory I-VI
Mumford-Tate groups/absolute Hodge classes I-II
Picard-Fuchs equations
Good reduction of abelian varieties

Commutative Algebra
Manifesting Modules
Noetherian Rings

Pósa Problem Solving (I-V)

Algebraic Topology
Fundamental Groups
Seifert-Van Kampen Theorem I-II
Covering Spaces I-II

Elliptic Curves: Geometry and Arithmetic
Riemann-Roch and Group Law of Elliptic Curves
Group Cohomology
Descent: Selmer and Sha

Graph of Groups (I-III)

Representation Theory of Lie Algebras (I-III)v

Probabilistic Combinatorics (I-II)

The Circle Method and Its Application to the Study of Rational Points (I-II)

Miscellaneous Topics
How Not to Give a Talk
Quotients in Algebraic Geometry
Type Theory
Introduction to Hyperbolic Geometry
Dynamical Systems
Generating Functions
Motivating the Definition of a Modular Form
Classifying Spaces and Equivariant Cohomology
Derived Functors
Topology
Introduction to Enumerative Geometry
Group Theory in Voting

2020 Counselor Seminars

Algebraic Number Theory (I-IV)

Algebraic Topology (I-IV)

Compactness in Metric Spaces (I-III)

Complex Analysis (I-IV)

Quantum Computation (I-VI)

Commutative Algebra and Algebraic Geometry
Affine Algebraic Sets
The Zariski Topology
Hilbert’s Nullstellensatz
Geometric Spaces and Spectrum of a Ring
Localization
Sheaves and Ringed Spaces
Morphisms of Ringed Spaces
An Overview of Schemes

Geometric Group Theory
Group Presentations and Cayley Graphs
Quasi-Isometries and the Milnor-Svarc Theorem
Milnor-Svarc Theorem and Ping-Pong Lemma
Groups acting on Trees
Ends of Groups

Machine Learning
Classification
Non-linearity
Generalization
Neural Networks

Perfectoid Fields
Definitions and Examples
Untilting and Finite Extensions
(phi, Gamma) - modules
Outlook on Perfectoid Spaces

Miscellaneous Topics
Biases and Products of Two Primes
Vector Calculus, Laplace’s Equation, and Hodge Theory
Introduction to the p-adics
Curves on Surfaces and Mapping Class Groups
Group Cohomology and Grothendieck Topologies
Model Theory
Perfect Dice
Mittag-Leffler Problems

2019 Counselor Seminars

Algebraic Topology
The Fundamental Group
The Seifert Van-Kampen Theorem
Covering Spaces and Galois Correspondence
Covering Spaces
Brown Representability Theorem

Class Field Theory
Quadratic Forms
Group Cohomology
Global Statements of Class Field Theory
Quadratic Forms, Lattices, and Ideals

Complex Analysis and Analytic Number Theory
Complex Analysis I-IV
Dirichlet’s Theorem on Primes in Arithmetic Progressions
The Prime Number Theorem I-II

Differential Forms
A Dip into Linear Algebra
Introduction to Differential Forms
Pullbacks and Integrating Forms

Functional Analysis
Zabrieko’s Lemma and Corollaries
Extending Linear Functionals
The Weak Topology and L^p spaces

Lambda Calculus
Introduction to Lambda Calculus
The Church-Rosser Theorem
Combinatory Algebra

Modular Forms
Introduction to Modular Forms
The Ramanujan Congruences
Algebraic Relations between Partition Functions and the j-function

Probability and Stochastic Processes
The Basics of Class Field Theory
Discrete Time Stochastic Processes
Brownian Motion and the Random Walk
Stochastic Calculus
Brownian Motion and Partial Differential Equations

Miscellaneous Topics
Basic Categories
Local/Global Principle for nth powers
Pathological Functions
Philosophies in Mathematics
Introduction to Hopf Algebras
p-adic Numbers
Introduction to Algebraic Number Theory
Introduction to Coq
Proving Theorems in Coq
Introduction to Mapping Class Groups
Congruences between Modular Forms
The Empirical and the Formal
p-adic Analysis
Structure Theorem for Finitely Generated Modules over a PID
The Laplacian and Number Fields
Farey Sequences (Distribution and Extension)