Counselor Mathematics

Counselor Mathematics

"I have learnt so much number theory and algebra from my three summers at PROMYS, it has been invaluably helpful to me in my studies... I found a great community and I now know that math is where my future lies." Molly Barker, Counselor 2019–2021

Counselors with Glenn StevensPROMYS is designed not only for the mathematical growth of students, but for the advancement of counselors as well. During the program, counselors devote considerable time to pursuing their own mathematical endeavors. They find new perspectives in the Number Theory lectures and problem sets, participate in advanced seminars, and interact individually with the faculty. In addition, they present minicourses (talks for students) and run counselor seminars (talks for counselors) on topics that interest them. We strive to provide these talented undergraduates with significant mathematical challenge from one summer to the next.

Minicourses

Counselor minicourses are talks designed by the counselors to present to students.

2021 Counselor Minicourses

How to Write a Proof
Introduction to Mechanism Design
Ramsey Theory
Knot Theory
Measure Theory
Differential Equations and Recursion
Tropical Arithmetic
LaTeX
Languages and Automata
Axiom of Choice
Proof Assistants
Cutie Pi: Leibniz, Euler, and Warmth from Hugs
Construction of R
Cryptography is Hard
Randomized Computations
Continued Fractions
Metric Space and Topology
Riemann Integration
T-shirt Talk: Voronoi Diagram
How to find roots of polynomials mod p
Polynomial Statistics
Erdös Problems
Law of Large Numbers

Seminars

Counselor seminars are talks designed by counselors to present to their peers.

2021 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)

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

2015–2020 Seminars

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)

2018 Counselor Seminars

Riemann Surfaces
Elliptic Integrals
Topology of Riemann Surfaces
The Klein Quartic
Sheaves and Analytic Continuation
Galois Theory and Riemann Surfaces
Form and Function
The Riemann-Roch Theorem
The Abel-Jacobi Theorem
Modular Curves

Algebraic Geometry
What is Algebraic Geometry?
Divisors and Intersections
Dimension and Moduli
Gröbner Bases

Algebraic Number Theory
Algebraic Number Theory I-IV
Artin Reciprocity I-II

Iwasawa Theory (I-III)

Elliptic Curves
Introduction to Elliptic Curves
Points of Order 13 on Elliptic Curves
Hasse Bound for Elliptic Curves

Topology
Morse Theory and Handles
Heegaard Splittings and Knots on the Torus
Dehn Surgery and Kirby Calculus
The Poincaré Homology Sphere
Dehn’s Lemma and an Application
Multi-Jet Transversality and Classification of Stable Immersions

Representation Theory of Finite Groups (I-II)

Intersection Theory in Algebraic Geometry (I-III)

Commutative Algebra
Introduction to Commutative Algebra
Introduction to Minimal Free Resolutions
Linear Resolutions of Edge Ideals

Measure Theory
Introduction to Measure Theory
Ergodic Theory and Continued Fractions I-II
Borel Measures

Random Matrix Theory (I-II)

Representation Theory of the Symmetric Group (I-III)

Matrix Lie Groups (I-II)

Rigidity Theory (I-II)

Tannaka Duality (I-II)

Miscellaneous Topics
An Introduction to Logic
Introduction to p-adics
Chemistry and Representation Theory
Holomorphic Functions and Complex Integration
Singular Value Decomposition
Complex Analysis
Probability and Expectation
An Overview of the Proof of the Prime Number Theorem
Hearing the Shape of a Manifold
SL(2,Z)
p-adic Modular Forms
A Quick Introduction to Linguistics
Groups and Zeta Functions
Review of Galois Theory

2017 Counselor Seminars

Algebraic Geometry
Affine Varieties
But dat Nullstellensatz tho
Projective Varieties
Morphisms: Putting the Fun in Function
Coordinate Rings and Rational Functions
Rational Maps and Blowups: Getting More Out of Morphisms
Localizations and Nakayama’s Lemma
Nonsingular Varieties
Curves
Divisors and Linear Systems
Embeddings into Projective Space
The Riemann-Roch Theorem
Riemann-Hurwitz and Applications
Reducing Elliptic Curves Modulo p I-II
An Overview of the Proof of Fermat’s Last Theorem

Complex Analysis
Holomorphic Functions and Contour Integrals
Cauchy’s Theorem
Cauchy’s Integral Formulas and Applications
Some Complex Theorems and Simple Analysis
The Residue Theorem and Applications

Characteristic Classes
Vector Bundles
Stiefel-Whitney Classes
The Euler Class and the Thom Isomorphism
Chern Classes
The Chern-Weil Homomorphism

Category Theory
Categories and Functors
Natural Transformations, Duality, and Equivalences
Duality, Equivalences
Representable Functors and the Yoneda Lemma
Universals and Limits
Adjunctions

Representation Theory and Combinatorics (I-II)

Riemann Surfaces (I-III)

Geometric Group Theory
Ping Pong and Quasi-Isometry
The Word Problem and Hyperbolic Groups

Miscellaneous Topics
The Ordinal Numbers
Monstrous Moonshine
Diophantine Approximations and Schmidt Subspace Theorem
Uniform Spaces
Propositional Logic and Stone Duality

2016 Counselor Seminars

Functional Analysis
Beginnings of Functional Analysis
Hilbert Spaces
Resolvent & Spectra of Bounded Linear Operators
Examples of Spectra, Holomorphic Functions, and the Spectral Radius
Compact Operators
Compact Operators: Self-Adjoint Operators
The Spectra of Bounded Self-Adjoint Operators
Proof of Spectral Theorem and Unbounded Operators
Linear Operators in Quantum Mechanics
Fredholm Alternative

Representation Theory
A Historical Introduction to Representation Theory: The Group Determinant
Introduction to Representation Theory: Maschke’s Theorem
Characters
Tensor Products and Induced Representations
Review of Concepts
Calculating Character Tables
The Tensor-Hom Adjunction
Burnside’s Theorem
Irreducible Representations of Sn
The Mackey Criterion and Frobenius Groups
Lie Groups, Lie Algebras, and Their Representations
A survey of the Schur-Weyl Duality and Schur Polynomials
Schur-Weyl Duality
Schur Polynomials
The Peter-Weyl Theorem

Miscellaneous Topics
Complex Multiplication, Part 0
Forcing and the Continuum Hypothesis I-IV
An Invitation to Extremal Set Theory
Intro to Mapping Class Groups
Chromatic Graph Theory
Matrix Decompositions
Complex Multiplication I-IV
Differential Topology
Green’s Functions
Morse Theory I-II
Normal Number Theorem
Random Matrices
A Brief History of Cyclotomic Fields
The Magic Queendom of Standard Young Tableaux
Spectra and Representability of Cohomology

2015 Counselor Seminars

Algebraic Topology
Point Set Topology I-III
Fundamental Groups
The Seifert van Kampen Theorem
Induced Maps and More
Covering Spaces
Simplicial Homology
Singular Homology
Exact Sequences
Cellular Homology
Eilenberg-Steenrod Axioms
Recitation
de Rham Cohomology and Introduction to Singular Cohomology
Singular Cohomology and Universal Coefficient Theorem
The Cup Product and Ring Structure on Cohomology

Algebraic Number Theory
Motivation for Algebraic Number Theory
Rings of Integers and Class Groups
Prime Splitting in Number Fields
Finiteness of Class Number

Modular Forms
Modular Forms I-II
Hecke Operators
Eichler-Shimura Theory

Miscellaneous Topics
Riemannian Geometry
Introduction to Classical Algebraic Geometry
Introduction to Modern Algebraic Geometry
Elementary Consequences of the Prime Number Theorem
Representation Theory
Extensions of Locally Compact Abelian Groups
The Birch and Swinnerton-Dyer Conjecture
The Antigeneric Blow-Up of the Category of Trimmed Artin Symbols
Nets, Ultrafilters, and Tychnoff’s Theorem
Infinite Graphs
Structure Theorem of Finitely Generated Modules over PIDs
Kähler Manifolds
Morse Theory
Inverse Limits
Schur Polynomials
Classical Algebraic Geometry, Part II