1. Stochastic Differential Equations

   • A stochastic differential equation (SDE) is like a differential equation but you add 'random noise' into the equation. Because the noise is random, every time you solve a SDE, you get a different randomly-determined solution. The mathematics that we use to solve SDEs are super interesting, and SDEs are applied in physics, machine learning, and quantitative finance to do a lot of very cool things. The goal of this project is to get a sense of what SDEs are and how we can solve them. Based on the interests of the students, we may look at one or both of these textbooks, which I feel have really good exposition on the topic.

   • References: Applied Stochastic Differential Equations, by Särkkä and Solin;   Stochastic Calculus for Finance II: Continuous-time models, by Shreve

   • Prerequisites: Calculus and introductory linear algebra are required. Some familiarity with probability is greatly preferred. A course in differential equations would also be useful. 

2. Machine Learning with Kernels

   • In machine learning, kernels are functions that measure the similarity between two vectors. Support vector machines, or "kernel machines," have proven themselves to be a very robust, principled and theoretically grounded way to do machine learning, and their performance is often comparable to that of artificial neural networks. The theory of kernels is basically just geometry, and this leads us to methods that are as visually motivated as they are theoretically grounded. In this project I want to discuss what kernels are, and what are some of the key ideas in how we use kernels to solve machine learning problems. 

   • References: "Learning with kernels" by Smola and Scholkopf

   • Prerequisites: Calculus and introductory linear algebra are required. A basic course in set theory and proofs would be helpful for me to explain some of the details in this reading.

3. Generative Models and Unsupervised Machine Learning

   • Generative models are machine learning models that learn how to imitate or recreate data that is similar in pattern to the data used in training. For example, a generative model that is trained on a collection of dog photos will allow us to generate new photos of dogs that were not seen in the training set. Aside from being interesting in applications, generative models have a rich theory, borrowing concepts from topology, analysis and statistics. In this project, we will spend some time discussing the theory of machine learning and statistical inference, and building a bridge with the geometric ideas that are at play in the background.

   • References: "Geometry of Deep Learning: A Signal Processing Perspective" by Jong Chul Ye

   • Prerequisites: Calculus, linear algebra and probability are required. Any experience with statistical inference, machine learning or statistics is a plus.

Hausdorff Dimension

• We will read Kenneth Falconers book "Fractal Geometry: Mathematical Foundations and Applications", understand different notions of fractal dimension, their intuitive meaning and how they relate to each other. We will compute dimensions of various sets from the basic definitions and then prove theorems enabling us to compute many more examples of Hausdorff and other dimension quantities. 

• References: Kenneth Falconer "Fractal Geometry: Mathematical Foundations and Applications"

• Prerequisites: Real analysis, measure theory, basic topology  

Introduction to Optimal Transport 

• Optimal Transport was first formulated by French Mathematician Gaspard Monge in 1781. From the modest ambitions of moving piles of dirt efficiently, the role of optimal transport in applied mathematics has dramatically grown, with applications being found in image processing and fluid dynamics, to even understanding how the cells in our body differentiate! The book chosen is meant for applied mathematicians and enables focused theoretical discussions as well as cross-disciplinary projects.

• References: F. Santambrogio, "Optimal Transport for Applied Mathematicians"

• Prerequisites: Calculus and linear algebra. Familiarity with real analysis and differential equations is an optional bonus, but all levels in between are welcome.

1. Number Theory

   • Did you know that an odd prime is the sum of two square numbers exactly when it leaves a remainder of 1 when divided by 4? Or that the number of primes less than some big number N is roughly N/ln(N)? Have you ever wondered why that “sum the digits” trick detects divisibility by 3? Or been fascinated by the twin primes, Goldbach, or Collatz conjectures? These are all examples of questions and results from number theory -- the study of properties and patterns of the natural numbers.

Number theory uses all sorts of techniques to study these problems, from modular arithmetic (accessible to a middle schooler) to the fanciest modern tools. In this broad and flexible project, we would select a subfield to study based on your background and interests. We could discuss topics like modular arithmetic and divisibility and treat them as a training ground for reading and writing proofs -- that is, for thinking and communicating as a mathematician. Or use calculus to understand the distribution of primes. If you are comfortable with proofs and computer programming, we could study number-theoretic cryptography or modern factorization and primality testing algorithms. If you have experience with analysis, abstract algebra, or complex variables, we could study more advanced topics, such as the p-adic numbers and exotic distances on the rationals; number theory in rings other than the integers (e.g. ℤ[√2]); or the prime number theorem. Or, if you have ideas not listed here, I’d be happy to hear them!

• References: Hardy and Wright, “An Introduction to the Theory of Numbers;” Bressoud, “Factorization and Primality Testing;” Shahriari, “Approximately Calculus;” Gouvêa , “p-adic Numbers;” Stewart and Tall, “Algebraic Number Theory;” Jameson, “The Prime Number Theorem”

• Prerequisites: Whether you just love math, have taken two semesters each of algebra and real and complex analysis, or are somewhere in between, I have projects at your level.

2. Algebraic Geometry

   • Algebraic Geometry is the geometry of polynomials: we study spaces that arise as the solutions to polynomial equations and properties preserved by polynomial or rational maps between them. We can prove some amazing things about these spaces: in the plane, a curve of degree n and a curve of degree m will intersect in exactly m*n points, provided we count them correctly. Every cubic surface contains exactly 27 lines. “Holes” over the complex numbers influence the number of rational points.

Algebraic geometry is vast and typically only studied in grad school. This project will give you a first glimpse of the subject, tracing one of several possible paths based on your interest and background. We could follow the route I first did, playing with examples to develop an intuition for the fundamental idea of “projective space.” We could focus on working over the complex numbers and see what complex analysis tells us about curves. We could study the computational side of algebraic geometry, perhaps implementing algorithms in Python. Or, we could concentrate on commutative algebra, which underpins both algebraic geometry and algebraic number theory.

2. • References: Cox, Little, and O'Shea, "Ideals, Varieties, and Algorithms;" Kendig, “Elementary Algebraic Geometry;” Reid, “Undergraduate Algebraic Geometry;” Fulton, “Algebraic Curves”

   • Prerequisites: Complete comfort reading and writing proofs, and strong background in linear algebra. Courses in abstract algebra or complex analysis are helpful, not required. 

An Introduction to Mathematical Thinking Via Number Theory

• Number theory, the study of the patterns which emerge within the arithmetic of the counting numbers (1,2,3,...) has long been of fascination to mathematicians. Gauss famously said: "Mathematics is the queen of the sciences - and number theory is the queen of mathematics". Today, number theory forms the foundation of many of the cryptographic algorithms which are used to send secure messages over the internet. In this project we will study some beautiful and accessible results in number theory. Along the way, we will introduce techniques of mathematical proof as well as data visualization. If there is interest, we may implement some cryptography algorithms in python notebooks. 

This project will appeal to those who consider themselves visual or creative thinkers, and I welcome participants who do not identify as a "math person". 

• References: An Illustrated Theory of Numbers by Martin H. Weissman. 

• Prerequisites: Precalculus is required. Calculus is recommended.

Philosophy of Mathematics

• Pythagoras, Hypatia, René Descartes, Gottfried Leibniz, Bernard Bolzano, Willard Quine, Bertrand Russell, Alfred North Whitehead, Hilary Putnam, … the list goes on. These great minds made significant contributions to both the fields of philosophy and mathematics. From the preface to our text: “Philosophical conundrums pervade mathematics, from fundamental questions of mathematical ontology — What is a number? What is infinity? — to questions about the relations among truth, proof, and meaning. What is the role of figures in geometric argument? Do mathematical objects exist that we cannot construct? Can every mathematical question be solved in principle by computation? Is every truth of mathematics true for a reason? Can every mathematical truth be proved?” The author is himself both a mathematician and a philosopher, and also a former CUNY professor whose classroom I had the privilege of sitting in. I did my undergraduate degree in philosophy at Hunter College before going on to study mathematics. 

• Reference: Lectures on the Philosophy of Mathematics, by Joel David Hamkins 

• Prerequisites: Some experience with proofs.

Intro to Complex Dynamics

• Complex dynamics studies the orbits of points in the complex plane under iterates of holomorphic maps. We will introduce the Julia and Fatou sets of a map, as well as the famous Mandelbrot set, and study their properties. If time permits, we will read a short expository paper showing that the exponential map is chaotic. 

• References: Notes by Lasse Rempe 

• Prerequisites: Complex analysis and a proof-based real analysis course; some topology is useful

1. Measure Theory and Filtering

   • Noise ruins everything — it takes a signal that we’re interested in, and corrupts it to make analysis much more difficult. Filtering is the general task of determining a probability distribution over some “hidden” process X by observing a related signal Y. Not only is filtering one of the most important practical problems in the world of data, it’s also mathematically interesting, and a natural place to motivate and explore measure theory. If the student has not yet seen any measure theory/measure-theoretic probability, we’ll first spend considerable time with that formalism. We can jump straight to filtering if the student has seen measure theory and/or measure-theoretic probability. 


   • Reference: Measure Theory and Filtering, Aggoun & Elliott. 
 

   • Prerequisites: Real analysis. Experience with measure theory and/or measure-theoretic probability is a plus.
 

2. Gaussian Processes and Bayesian Optimization
 

   • Description: Gaussian processes are an infinite-dimensional generalization of the familiar, finite-dimensional Gaussian distribution. Interesting mathematical objects in their own right, the first part of this project would consist of an introduction to Gaussian processes and how they’re used in statistics machine learning. The second part of this project will look at an interesting application of Gaussian processes to optimization, often called “Bayesian optimization.” Depending on the student’s interests, this project can take a more applied flavor — coding Bayesian optimization and applying it to real-world problems — or a more theoretical flavor, taking a look at the corresponding “regret analysis.”


   • References: Bayesian Optimization, Garnett.


   • Prerequisites: Multivariate calculus, linear algebra, and introduction to probability or statistics.

3. Approximate Inference in Machine Learning


   • As machine learning models get larger, the cost of prediction does too. This is particularly challenging in probabilistic machine learning, where problems often involve solving an integral that’s not analytically tractable. Two particularly popular approaches to solve this are Monte Carlo methods and variational inference. Monte Carlo methods work by making random guesses and correcting them to approximate the integral in question. Variational inference works instead by approximating the integrands with functions we know how to integrate, and finding the solution “closest” to the intractable integral. In this project, we will take a broad view of this active area, and implement and compare a few methods that the student finds interesting.


   • References: Many are possible based on the student’s interests. We can start with Probabilistic Machine Learning: Advanced Topics by Murphy and branch out from there.
 

   • Prerequisites: Multivariate calculus, linear algebra, and introduction to probability or statistics. 

Topology of Numbers

• We will explore ideas in number theory through a topological viewpoint.  If you like visual thinking and arithmetic, this is the project for you!  Topics include Farey fractions, Farey diagram, Ford circles, continued fractions, quadratic forms, lattices, Conway's topograph, trees, modular group, hyperbolic geometry, mobius transformations, rational approximation, Pell's equation, quadratic irrationalities, Apollonian circle packings, Schmidt arrangements,  etc.  There are many topics in this circle of ideas which are all related to each other, and can be visualized.  

Note: If you like programming, we can incorporate that into this project as well!  Various coding projects can be done to visually understand certain ideas in number theory.

• References: The Sensual Quadratic Form by John Conway, Topology of Numbers by Allen Hatcher, various papers of Katherine Stange

• Prerequisites: Linear Algebra, knowledge of complex numbers. Helpful, but not necessary: Group Theory, Elementary Number Theory, Abstract Algebra, familiarity with computer programming