Analysis Projects

Book: The Banach-Tarski Paradox, by Grzegorz Tomkowicz and Stan Wagon

Prerequisites: A course in real analysis (e.g. MAT 319/320). Topology (e.g. MAT 364) and group theory (e.g. MAT 313) are not required, but are encouraged.

Can you imagine splitting a ball into five pieces, re-arranging them somehow, and be able to get two distinct balls identical to the original one? This can be done mathematically. This fact is called the Banach-Tarsky Paradox/Theorem.

To prove this, we need to use Measure Theory, understand some topological facts of the sphere, and make a brief detour into Group Theory. The goal of this project is to understand the proof of Banach-Tarsky’s Theorem.

Book: Convex Optimization, by Stephen Boyd and Lieven Vandenberghe

Prerequisites: A course in linear algebra (e.g. MAT 211).

Convexity is the secret sauce behind the success of modern optimization techniques, which have in turn generated myriad applications from circuit design to machine learning. Despite its widespread diffusion across the disciplines of engineering, finance, and computer science, convex optimization possesses at its center a startlingly elegant theory. Convexity is an algebraic property of Euclidean spaces which manages to interact cleanly with their topological structure. Duality of convex programs leads not only to deeper insights into the geometry of the problems but also to consequential optimization algorithms. In this project, the mentee will develop an acquaintance with convexity and learn to recognize problems which are convex in nature. They will study Lagrange duality and its many interpretations. They will also explore different algorithms for unconstrained and constrained optimization problems. Depending on the mentee’s interests and maturity, this project can either adopt a more theoretical flavor with an excursion into subdifferential analysis, or a more applied flavor with “hands on” computational problems.

Book: Elementary Differential Geometry (2nd Edition), by Andrew Pressley

Prerequisites: Multivariable calculus (e.g. MAT 203), and a proof based course (e.g. MAT 200).

The first goal is to carefully study curves in the plane and in space: definitions, revisiting the concept of the length of a curve, and reparametrization. The next goal will be to define curvature and understand it through the study of curvature in the plane and then in space. Finally, the ultimate goal will be to explore interesting theorems such as the isoperimetric inequality, the four-vertex theorem, or simply study very thoroughly an example (such as the helix) in a detailed manner. The latter will depend on the tastes and preferences of the student.

Book: Differential Geometry – Curves - Surfaces - Manifolds (2nd Edition) by Wolfgang Kuhnel

Prerequisites: Calculus III-IV (e.g. MAT 203, MAT 303), linear algebra (e.g. MAT 211), and real analysis (e.g. MAT 319/320). Complex analysis (MAT 342) is a bonus.

A study of curves in R^n: both the local and global theory, in order to see how the study of curves in 2D and 3D can generalize to many dimensions. This will allow us to study the local theory of surfaces at a more precise level. The goal will be to grasp well the fundamentals about the local theory of surfaces, and then study very carefully one specific type of surfaces of interest. These include but are not limited to: ruled surfaces, minimal surfaces, and hypersurfaces in R^{n+1}.

Book: Topology from a Differentiable Viewpoint, by John Milnor

Prerequisites: Point-set topology (e.g. MAT 364)

Project goal: The field of differential topology concerns the topological aspects of smooth manifolds and smooth mappings. It is a subject that began in the 50s which has evolved into various other subjects over the last several decades (gauge theory, 4-manifold theory, Seiberg-Witten theory, symplectic geometry, etc.). This project will situate the mentee with the fundamental basics of the field (tools, ideas, methods), while strongly emphasizing a geometric point of view. Time permitting, the project can cover more complicated topics, such as exotic spheres and cobordism theory.

Book: Nonlinear Dynamics and Chaos, by Steven Strogatz

Prerequisites: Linear algebra (e.g. MAT 211). Exposure to differential equations in some sense.

Biological systems are, more often than not, complex, interconnected, and non-linear. In addition, limited experimental data make the complete understanding of a given system difficult-to-impossible. With this in mind, many scientists turn to modeling of these systems as a way to understand them more fully. A prominent class of these models fall under dynamical systems, taking the form of a differential equation.

The purpose of this project will be to explore the field of continuous-time dynamical systems, using biological models as a guide throughout the process. We will look at phase-plane analysis, bifurcations, and introduce the concept of chaos. Supplementary topics that may be discussed, depending on the interests of the student (common modeling assumptions, integration techniques, interpretations of system characteristics in an intuitive way, etc).

Book: A Modern Introduction to Dynamical Systems, by Richard Brown

Prerequisites: A course in real analysis (e.g. MAT 319/320)

Continuous maps of the circle to itself are relatively simple to define and visualize. However, many systems of mathematical or physical interest are modeled extremely well by iterated application of such a map on a point on the circle. For example, asking how an initial point moves under the rotation map by some angle θ will lead us to an investigation of the continued fraction expansion, which is of great interest in number theory. Expanding on this will lead us to define the rotation number of a map and then to the stability of the solar system! On the other hand, the doubling map, which sends a point on the circle to a point with twice its angle from a reference point, is a fantastic example of a chaotic map. Investigation into this map leads to symbolic dynamics, connecting circle maps to linear algebra and information theory. Various connections with other parts of dynamical systems may be discussed, according to interest.

Book: The Poincare Half Plane: A Gateway to Modern Geometry, by Saul Stahl

Prerequisites: Calculus and proofs. Courses in linear algebra, topology, and complex analysis (e.g. MAT 310, 322, 342, 360, 362, 364) will allow the mentee to delve further into the material, but are not necessary.

In Euclidean geometry, one way to state the parallel postulate is to assume that given a line l and a point x not on that line, there exists exactly one line l’ containing x which does not intersect l.

For many years in the 18th and 19th centuries, mathematicians tried to deduce the parallel postulate from the other axioms of geometry, but a valid proof alluded them. Indeed, there are non-euclidean geometries (a fancy way of describing how to define the main objects in geometry like lines) which satisfy all the axioms of Euclidean geometry except the parallel postulate.

One such model is the hyperbolic ball or hyperbolic half plane. In hyperbolic space, the fastest way to get between two points may not be in a Euclidean straight line, but rather in certain circular arcs. The sum of the angles of a hyperbolic triangle is not necessarily equal to 180 degrees. In this project, we will study the basics of hyperbolic geometry in the planar setting, comparing theorems in Euclidean geometry and their hyperbolic counterparts. After getting through some basics, the direction will be up to the interests of the student.

Book: Fourier Analysis, by T. W. Korner

Prerequisites: A course in real analysis (e.g. MAT 319/320)

In the mid-1800s Joseph Fourier found that periodic functions can be realized as infinite sums of sines and cosines. This realization has made a profound impact on virtually every subfield of science and engineering. The mentee will begin by examining the sense in which these infinite sums converge, through the Dirichlet and Fejer kernels. After this, the student will delve into a number of applications of this theory, including applications to probability, Brownian motion, and dynamical systems.

Book: The Geometry of Fractal Sets, by Kenneth Falconer.

Prerequisites: A course in analysis (e.g. MAT 319/320). Enrollment in a class in measure theory (e.g. MAT 324) would be helpful, but is not necessary, as the minimal measure theory needed can be developed during the project.

We are used to the idea that points are zero dimensional, lines have one dimension, and planes have two dimensions. However, defining dimension of general sets in a careful way is not so easily done. Take for example, the Middle Thirds Cantor set, obtained by taking the unit interval and removing the middle third, and continuing this process indefinitely to the remaining intervals. What is the dimension of this set?

Such “fractal” sets occur naturally in analysis, probability, and dynamical systems, and mathematicians have developed several notions of dimension to study these sets. In this project, we will define various notions of dimension for sets in the plane, discuss some general properties of these dimensions, and use them to study interesting examples of fractal sets like the one described above. Possible directions for this project include constructing continuous yet nowhere differentiable functions, studying when fractal sets are a subset of a path in the plane, studying projections of fractal sets onto lines, or whatever other possible topic the student is interested in exploring. There are many possible directions.

Book: Office Hours with a Geometric Group Theorist, by Matt Clay and Dan Margalit

Prerequisites: Linear algebra (e.g. MAT 211). Some group theory would help but not required

Geometric group theory is a relatively new field concerning the geometric aspects of groups. It is a subject that began in the 80s that is an area of active research today. This project will start with a crash course on groups and eventually familiarize the mentee with the basic ideas of the field, while strongly emphasizing a geometric point of view. Time permitting, the project can cover more complicated topics, such as the mapping class group and hyperbolic geometry.

Book: Irrational Numbers, by Iven Niven

Prerequisites: Calculus and Proofs

You've likely heard that many common numbers are irrational, meaning that they can't be expressed by dividing two whole numbers. Some examples are √2, π, and e. But why are they irrational? In this project the mentee will prove that these numbers in fact are not rational and will learn about how to mathematically approach irrational numbers. In doing so, they will begin developing techniques from the more advanced mathematical subjects of analysis and algebra.

Book: Naive Lie Theory, by John Stillwell

Prerequisites: A course in linear algebra (e.g. MAT 211)

Lie groups and Lie algebras are mathematical objects that have both a geometric structure, and an algebraic one. The classical example of a Lie group is the set of real numbers - we can add and subtract real numbers from each other, but there is also a geometric interpretation of the real numbers as an infinite line. Broadly speaking, Lie theory is concerned with finding relations between these aspects.

We will study specific examples of Lie groups and Lie algebras, focusing on those which arise as symmetries of vector spaces, and examine how to express these objects as collections of matrices.

Book: Complex Analysis by Theodore Gamelin

Prerequisites: A course in real analysis (e.g. MAT 319/320) and a course in complex analysis (e.g. MAT 342).

Let f_c(z)=z^2+c be a polynomial, with c ∈ C. When studying the dynamics of polynomials, one is interested in the behavior of a point under successful iterations by f. The Mandelbrot set M is defined as the set of c so that the sequence of iterates,{0,c,c^2+c,(c^2+c)^2+c,...}, is bounded. Despite this simple definition, much is still unknown about the Mandelbrot set, and therefore much is still unknown about the iteration of even these simple quadratic polynomials! In this project, we will learn how to study the dynamics of polynomials, with an emphasis on specific examples, trying to avoid any difficult general theory where we can. We will see that we can partition the plane into a stable set called the Fatou set, and a fractal chaotic set called the Julia set. We will see how the dynamics of specific polynomials f_c relates to the Mandelbrot set.

After learning the basics, we can proceed in many directions. One direction is learning how to draw computer images of the Mandelbrot set, and of Julia sets of polynomials. We would also learn the mathematics of why our programs work.

These sets have an interesting fractal structure. Students could also study the theory behind Newton’s method, a technique learned in calculus courses to find the zeros of polynomials. Students could also learn about iterating rational or general entire functions, or learn the proofs and the complex analysis behind some of the several important tools we use in this area.

Book: Random Matrices, by M. L. Mehta

Prerequisites: Linear algebra (e.g. MAT 211), probability and statistics (e.g. AMS 310). This project can be tailored to applications to physics, in which case statistical mechanics (PHY 306) is necessary.

Random Matrix Theory (frequently abbreviated as RMT) is an active research area of mathematics with numerous applications to theoretical physics, number theory, combinatorics, statistics, financial mathematics, mathematical biology, engineering telecommunications, and many other fields.

Random matrices are matrices with their entries drawn from various probability distributions, which are called random matrix ensembles. Our goal is to study the eigenvalues of such matrices, which oftentimes have a rich mathematical structure when the matrices are large. For example, the spacing between eigenvalues are described by universal laws which are also found to describe the nontrivial zeroes of the Riemann Zeta Function.

During this project, we will cover a number of applications of RMT to mathematics or physics, depending on the interests of the student.

Book: Random Walk and the Heat Equation, by Greg Lawler

Prerequisites: Linear algebra (e.g. MAT 310), real analysis (e.g. MAT 319/320), and some knowledge of probability (at the level of AMS 311). A student interested in studying Brownian motion as part of this project must have taken or be enrolled in a course in measure theory (e.g. MAT 324).

There are two models for looking at the diffusion of heat inside of some medium. The heat equation is a deterministic partial differential equation which describes this diffusion. However, on the physical level, the diffusion of heat may be described in terms of “random” collisions of particles. The path of one such particle can be thought of as a random walk or Brownian motion. Such random processes have applications in several areas of science, engineering, and finance.

A discrete random walk can be thought of as a particle moving in the integer lattice, choosing a direction randomly at each stage. In this project, we will study discrete random walks in the plane and general euclidean space, answering questions such as how many times does a random walker return back to their starting point? We will see how this probabilistic point of view can enrich the study of a discrete version of the heat equation along with other areas of analysis. Depending on the student’s interests, we may branch off into further topics, such as more detailed analysis of discrete random walks, continuous random walks (Brownian motion), and other advanced topics in probability, such as martingales.

Book: Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDE, by Alex Kasman

Prerequisites: A background in partial differential equations (e.g. MAT 341)

When we imagine a particle in motion, we envision a single object with zero size which does not change as it moves, but rather holds itself together in a consistent way. However, when a wave hits the shoreline, it is not a single particle that arrives. Waves have nonzero size. Moreover, they constantly encounter disturbances around them, from uneven ground below to animals swimming, yet still hold their fixed shape without error. If we modeled these waves by a linear partial differential equation, such a stable solution – a “soliton” - would not be possible. However, for a certain class of nonlinear partial differential equations, these solitons are known to exist. Most famous among these is the Korteweg-de Vries equation. The explanation for this curious stability of solutions comes down to properties of this equation related to algebraic geometry. This project will investigate this shocking relationship between nonlinear PDE and algebraic geometry. To get to the core of this, we will introduce elliptic curves and the algebra of differential operators. This will bring us to a discussion of the Lax pair corresponding to this equation and, time permitting, the geometry of Grassmannian spaces. We will simultaneously learn how to use Mathematica to visualize and explore this subject.

Book: A Brief Introduction to Spectral Graph Theory, by Bogdan Nica or Spectral Graph Theory, by Fan R. K. Chung

Prerequisites: A course in linear algebra (e.g. MAT 211)

Graphs are mathematical objects used to represent societies and networks. The aim of spectral graph theory is to understand certain properties of graphs, by examining certain related matrices and their eigenvalues. For example, a classic theorem proved using spectral graph theory is the following: In a society where every two people have exactly one mutual friend, there is one person who is friends with everybody. This project can take on a more algebraic or a more analytic flavor depending on the interests of the student.

Book: Calculus on Manifolds, by Michael Spivak

Prerequisites: A course in multivariable calculus (e.g. MAT 203)

In a standard course in calculus in multiple variables one will encounter three theorems: Green's theorem, relating the flow of a vector field in a planar region to the circulation on the boundary; Stokes' theorem, relating the curl of a vector field on a surface to the circulation on the boundary; and the Divergence theorem, relating the divergence of a vector field in a 3-D region to the flux on the boundary. These are all special cases of what is known as Stokes' theorem on manifolds. This project will develop the concept of a manifold and that of a differential form. We will learn how to integrate these forms on manifolds and eventually prove the most general version of Stokes' theorem.

Book: The Symmetries of Things, by J. Conway, H. Burgiel, and C. Goodman-Strauss

Prerequisites: Only an enthusiasm for mathematics is required for this project. This project can very much be tailored to the mathematical background of the mentee.

Symmetries and symmetric patterns surround us in all aspects of our lives. We would like to be able to describe symmetrical patterns mathematically, and the first task will be to enumerate them. We will first discuss a new method, Conway’s enumeration, using four fundamental classifying features, seemingly non-mathematical in nature. We will then prove that this indeed classifies all possible planar patterns, using Euler’s theorem and some topology. The more advanced students will begin with spherical and frieze patterns, and will discover how group theory governs the possible symmetries that can occur. A more advanced student can go even further, venturing into hyperbolic space and flat universes.

Book: Topology, Geometry, and Gauge Fields: Foundations, by G. Naber

Prerequisites: A course in linear algebra (e.g. MAT 310). A course in real analysis (e.g. MAT 319/320).

This project can adapt itself to the student’s interests and background. We will start with some topology: introduction to topological spaces, homotopy and homology, and depending on the student this can be enough for the whole reading project. On the other hand, we can also move on to differentiable manifolds and Lie groups. The chosen book is especially approachable, because these are introduced in a more explicit, hands-on, way. The book then moves on to the application of all of these to the study of Gauge Fields, which lie at the heart of modern physics, especially particle physics, and which became a stage for deep collaborations between geometers and theoretical physicists.

Book: Variational Calculus and Optimal Control, by Troutman

Prerequisites: A course in real analysis (e.g. MAT 319/320).

Consider a function with multiple inputs and one output. For instance a function that takes in a point on the xy plane and outputs a single real number. In Calculus 3, presumably, we learned how to find minima of this equation: assuming our function is "nice" we check the partial derivatives. Imagine now we have an infinite dimensional space as an input (still only a single real number as output). The theory of the calculus of variations gives us tools to sensibly think about finding minima of these functions in some special cases. Troutman introduces the reader to this theory via its many applications (e.g. control, physics, minimal surfaces). In this DRP, we follow Troutman through Chapter 6 - which he estimates is an introduction to the basics. Topics include: examples of problems that may be addressed, appropriate notions of derivatives in this setting, a discussion of the restrictions we require of functions in this setting, and analysis of some necessary and sufficient conditions for finding minima. Time permitting, we may take more detailed looks at applications to mechanics and/or optimal control.