1) Approximation Theory and Approximation Practice

"Approximation" as a concept shows up everywhere in mathematics and computing.  Power series, Fourier series, and even epsilon-delta proofs in calculus are all examples; most algorithms for quadrature, lossy compression, or even machine learning are essentially attempts at approximating complicated functions by simpler ones.  Using Trefethen's "Approximation Theory and Approximation Practice", this project will explore "best" and "near-best" approximations with polynomials, trigonometric functions, and possibly rational functions, encountering applications along the way.

Reference: Lloyd Trefethen, "Approximation Theory and Approximation Practice", and possibly some textbook material for Fourier series.

Prerequisites:  Calculus, some Complex analysis, and ideally some programming background (any language is ok.) 

2) Numerical Solution of Differential Equations

Differential equations are one of the main tools used by scientists and engineers to model and simulate the physical world.  Their numerical solution is a field unto itself; this project will begin at the beginning (the Euler method for ODEs) and proceed from there.  Depending on the interest of the reader, the focus could involve implementing specific solvers, understanding convergence/error analysis, or a broad overview of many different methods for many different problems.

References: Burden, Faires, & Burden, "Numerical Analysis", and possibly other texts and online sources as necessary.

Prerequisites:  Calculus and brief exposure to ordinary differential equations.  With more background we can go deeper.  Experience programming or with Matlab opens many more options.

1) Differential Topology 

We will try to understand the basics of differential topology, the concept of a derivative and a tangent space, and work our way through famous results such as Sard's theorem, which paves the way for the existence of Morse theory, and the Borsuk-Ulam theorem. The aim is to get an intuitive picture of manifolds, and to learn the techniques and machinery necessary for problem-solving, proofs and research in topology. 

Reference: V. Guillemin, A. Pollack, "Differential Topology" 

Prerequisites: Point-set topology 

2) Morse Theory 

We can use certain differentiable functions on a manifold to sketch its shape and study its topology. Morse theory is a tool used in many important proofs in the study of manifolds, for instance the classification of all closed 2-manifolds. We will take a visual look at Morse functions and the foundations of Morse theory, and see it in action in a few famous proofs. 

References: L.I. Nicolaescu, "An Invitation to Morse Theory"; J. Milnor, "Morse Theory" 

Prerequisites: Point-set topology 

3) Knot Theory 

Knot theory is an old area of mathematics that is nevertheless very active in current research. Many problems can be understood visually and intuitively, and do not require heavy mathematical machinery. We will look at knots and knot invariants and try to understand some problems that still puzzle knot theorists. 

Reference: C. Adams, "The Knot Book" 

Prerequisites: None 

4) Basic Algebraic Geometry 

Modern algebraic geometry is deeply intertwined with algebraic number theory and heavily laden with category theory. We will attempt to learn algebraic geometry in a more visual and topological manner, with far fewer prerequisites. 

Reference: K.E. Smith et al, "An Invitation to Algebraic Geometry" 

Prerequisites: Point-set topology, linear algebra, some familiarity with ring theory will be useful

1) Evolutionary Processes 

Many natural systems are described by a set of ordinary differential equations. But once written, what do these equations tell us? Arnold's ODEs is a classic text that is both friendly but also introduces deeper topics like diffeomorphisms and smooth manifolds. 

Reference: V. I. Arnold, "Ordinary Differential Equations" 

Prerequisites: Some elementary coursework in real analysis and linear algebra. 

2) Probabilistic Properties of Deterministic Systems 

Chaos theory and dynamical systems are extremely important in applications, but when studying natural systems we often encounter plenty of noise, randomness and uncertainty. In this project, we will study how probability theory and dynamics interact, and we'll encounter several interesting connections to ergodic theory, linear algebra and fractals. 

Reference: Andrzej Lasota and Michael Mackey, "Chaos, Fractals and Noise" 

Prerequisites: Some experience in proofs and real analysis will be important. (If you have never seen measure theory before, we can build it up as we go.) 

3) Information Theory and Machine Learning 

Historically, information theory was developed in order to determine how to encode messages for communication. In modern times, information theory appears all over applied mathematics, and in this project we investigate why. We will introduce some classical information and coding theory, and we'll use that background to more deeply understand modern machine learning and statistical methods. 

Reference: David MacKay, "Information Theory, Inference and Learning Algorithms" 

Prerequisites: Some calculus and probability theory is greatly preferred.

The Sublime Primes 

Prime numbers are the beautiful, mysterious, and seemingly random building blocks of arithmetic; their distribution is intricately connected to Bernhard Riemann’s hypothesis — a fundamental, yet unsolved math problem. We will look at some of the history and ideas that led Riemann to his famous conjecture.

References:  Barry Mazur and William Stein, “Primes and The Riemann Hypothesis”; Jeffrey Stopple, “A Primer of Analytic Number Theory: From Pythagoras to Riemann”

Prerequisites: At least one course requiring proofs. A course in complex analysis is strongly recommended; some familiarity with Fourier Analysis is preferred, but not required. The main reference book is accessible to a wide range of learners; students may choose topics according to comfort level. 

Number Theory

Did you know that an odd prime number is the sum of two square numbers if and only if 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/log(N), where “log” denotes the natural logarithm? Have you ever wondered why that old “sum up the digits” trick detects divisibility by 3 and 9? Or been fascinated by the twin primes conjecture, the Goldbach conjecture, or the Collatz conjecture? These are all examples of questions and results from a branch of math known as number theory -- the study of properties and patterns of the natural numbers.

Modern number theory uses all sorts of tools to study these sorts of problems, from things like modular arithmetic, which a middle schooler could understand, to fancy tools from modern math like abstract algebra and complex analysis. In this broad and flexible project, we would focus on some part of number theory that you want to learn, and work through a textbook on the subject aimed at your level. For instance, we could discuss relatively elementary ideas such as modular arithmetic and divisibility, and how creative applications of these ideas are used in math competitions and in cryptography. We could learn about the p-adic numbers, which are a family of number systems different from the real numbers, where we can use calculus to do number theory, like finding integers x such that x^2-2 is divisible by a high power of 7. We could do a detailed study of how fancy tools from modern algebra or complex analysis let us prove special cases of Fermat’s last theorem or the Prime Number Theorem. We could even learn about some more “up-and-coming” tools, like Sieve Theory or Modular Forms, that have been used in recent breakthroughs. Or, if you have something else in number theory you want to learn that’s not listed here, I can mentor you through studying that.

References: G. H. Hardy and W. E. Wright, “An Introduction to the Theory of Numbers;” Neil Koblitz, “A Course in Number Theory and Cryptography;” John Conway, “The Sensual (Quadratic) Form;” Fernando Gouvêa , “p-adic Numbers;” G. J. O. Jameson, “The Prime Number Theorem;” Daniel Marcus: “Number Fields;” Stewart and Tall: “Algebraic Number Theory and Fermat’s Last Theorem;” Cojocaru and Murty, “An Introduction to Sieve Methods and their Applications;” Hester Graves, M. Ram Murty, and Michael Dewar, “Problems in the Theory of Modular Forms.”

Prerequisites: Whether you just love math and are ready to study mathematical proofs, have taken a semester or two each of real analysis, complex analysis, and abstract algebra, or are somewhere in between, I have ideas for a project at your level.

1) Topics in low-dimensional topology 

Possible topics include knot theory, surgery on three-manifolds, Floer theory, and hyperbolic geometry. 

References: Will depend on the topic and discussion with the student. Some standard references for introduction to some of these topics are: 

• The Shape of Space by Jeffrey Weeks 

• First Concepts of Topology by Chinn and Steenrod 

• The Knot Book by Colin Adams 

• Hyperbolic Geometry by James W. Anderson 

Not a mathematician and want to know what research in topology means ? Check out this page: https://web.stanford.edu/~cm5/topology.html 

Prerequisites: Basic undergrad math courses. 

2) Topics in contact geometry 

Possible topics include Legendrian and transverse knot theory and contact geometry in three-manifolds. 

References: Introductory Lectures on Contact Geometry by Etnyre (https://arxiv.org/abs/math/0111118v2)

Prerequisites: Point set Topology, some Algebraic topology and Differential geometry. 

3) Surface mapping class groups 

The symmetries of a manifold are described by its mapping class group, and the mapping class group (MCG) of a surface is a particularly important object. Possible topics include the finite presentations of MCG of finite type surfaces by generators and relations, and their construction, MCGs of infinite type surfaces. 

References: A primer on Mapping Class Groups by Benson and Farb

Prerequisites: Point set topology 

Curves, surfaces, and Gauss-Bonnet

Curvature is one of the most salient properties of curves and surfaces. There is both an extrinsic notion — curvature that measures how a surface twists in an ambient space — and an intrinsic notion — curvature that only depends on measurements taken "within" the surface. We'll learn the theory underlying the study of curves and surfaces, building up to the proof of the Gauss-Bonnet theorem. This theorem is an astonishing result relating the intrinsic curvature and geometry of a surface with its topology.

Reference: "Differential Geometry: Connections, Curvature, and Characteristic Classes" by Loring Tu 

Prerequisites: Familiarity with manifolds is preferred. Point-set topology, linear algebra is necessary. 

1) Complex Variables 

Calculus with complex variables may seem like it's more difficult than calculus with real variables, but it's actually the contrary. Calculus with complex variables is most often more streamlined and elegant than its real counterpart. This book approaches complex functions through the lens of harmonic functions, which effectively serves as a bridge between multivariable calculus and complex functions. The theory of harmonic functions will be developed first and will serve to motivate the parallel theory of complex functions.

Reference: Francis J. Flanigan, "Complex Variables: Harmonic and Analytic Functions"

Prerequisites: Single-variable calculus is essential. Multivariable calculus is strongly recommended; it will be reviewed at the beginning, but this may be too brief for a first encounter. A proofs-based course is lightly recommended; the proofs in this book, which are essential to the exposition, are not overly formal and thus are rather approachable. 

2) Fourier Analysis 

Fourier analysis is core to both the historical development of analysis and present day analysis. One only needs a glance at the Wikipedia page to see just how pervasive this subject is. This book develops the Fourier analysis from an elementary point of view, so one does not need a background in advanced areas such as complex analysis and Lebesgue integration. We’ll go through the genesis of Fourier analysis as it related to the study of partial differential equations, convergence and applications of Fourier series, and the Fourier transform. 

Reference: Elias Stein and Rami Shakarchi, "Fourier Analysis" 

Prerequisites: Proofs-based calculus, including epsilon-delta arguments, uniform convergence, a rigorous definition of the Riemann integral. These topics are usually found in an honors calculus course, advanced calculus course, or real analysis course.

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. 

Reference: Notes by Lasse Rempe 

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

1) Galois Theory 

The familiar quadratic formula (-b ± √(b^2 - 4ac))/2a expresses the roots of a general quadratic polynomial ax^2 + bx + c in terms of its coefficients (a,b,c). Notably, it does so using only basic arithmetic operations (addition, subtraction, multiplication, division) and one extra operation called "square root". It turns out there are analogous cubic and quartic formulas, although the expressions themselves become extremely complicated. However, there is no such quintic formula. More precisely, there is no mathematical formula, involving only basic arithmetic operations and application of radicals (square roots, cube roots, etc.), that expresses the roots of a general degree five polynomial in terms of its coefficients. In this project we will learn Galois theory, which studies the symmetries of roots of polynomials, in order to learn why no such quintic formula can exist. 

References: Harold Edwards, "Galois Theory"; Jörg Bewersdorff, "Galois Theory for Beginners, A Historical Perspective" 

Prerequisites: No prerequisites, but at least one proof based course is recommended. 

2) Arithmetic Beyond Z 

Algebraic Number Theory is about number systems that go beyond the usual integers. For example, consider the Gaussian integers Z[i] which contains all numbers of the form a+bi where a,b are usual integers, and "i" is a new number whose square is equal to -1. Something interesting happens when we pass from the usual integers to the Gaussian integers. The number 5 stops being prime! See: 5 = (2+i)(2-i). This immediately raises several questions: which prime numbers in Z stop being prime in the Gaussian integers? What about other number systems such as Z[sqrt(2)]? Studying questions like these will be a major theme of this project, and the answers have important applications to natural questions about the usual integers. 

References: Paul Pollack, "A Conversational Introduction to Algebraic Number Theory: Arithmetic Beyond Z"; Daniel A Marcus, "Number Fields" 

Prerequisites: Linear Algebra, Group Theory, Ring Theory, Galois Theory