# 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

PROMYS 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.

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.

**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

**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 *n*th 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