Projects

Book: Algebra, by Michael Artin

Prerequisites: Proofs and calculus. Multivariable calculus (e.g. MAT 203).

The main goal of this project is to cover the basics of groups, matrices, eigenvalues and eigenspaces, linear transformations, symmetry groups. We will emphasize applications such as modular arithmetic, the rotational symmetries of a cube/platonic solid, or Burnside’s lemma, which involves permutation groups. The student could also explore symmetry groups of plane tilings, such as deriving the crystallographic restriction, something that is not covered in a standard abstract algebra class. Another possibility is to bring in some applications to topology, such as describing restrictions on the fundamental group of a surface.

Book: Undergraduate Algebraic Geometry, by Miles Reid

Prerequisites: Proofs. A course in ring theory and abstract algebra (e.g. MAT 313, MAT 314) is highly recommended.

Algebraic Geometry at its core is the study of polynomial equations. These polynomial equations define shapes called algebraic varieties. This project will begin with the case of plane curves: polynomial equations defined on a suitable compactification of the complex plane, known as projective space. We will ’play’ with various toy problems, like how many cubics (curves defined by degree 3 polynomials) can pass through a given set of points, with the main aim to study Bezout’s theorem: a theorem that tells us exactly how many intersection points there are between two curves. Using this, we will be able to solve some fun numerical problems, such as Pascal’s mystic hexagon.

If time permits, we will study higher dimensional varieties, and hopefully end with a famous theorem: every smooth cubic surface contains 27 lines.

Book: Algebraic Topology, by Allen Hatcher

Prerequisites: Experience with metric spaces and groups.

What's the difference between a sphere and a doughnut? What about a circle and a punctured disk? The notion of homotopy, intuitively that of moving a path from one place to another, gives us a way of mathematically answering these questions. We will develop topological spaces and manifolds with the goal of studying the fundamental group of manifold. We will go through the van Kampen theorem and discuss covering spaces as well. If time permits a short introduction to homology is possible.

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: Category Theory in Context, by Emily Riehl

Prerequisites: A background in abstract algebra (e.g. MAT 313). Interested students will greatly benefit from a strong background in linear algebra (e.g. MAT 315) or topology (e.g. MAT 364).

As one pursues mathematics, a feeling of déjà vu can occur. The newest class covers a new “kind” of mathematical object and studies it intensively. Then, one moves on to study functions between objects of this type. Sets and functions, groups and group homomorphisms, vector spaces and linear maps, topological spaces and continuous maps: all are repetitions of this general trend. Moreover, similar kinds of constructions and proofs seem to reappear in these different contexts, always with a mostly similar appearance. For example, sets have a Cartesian product, vector spaces have the product space, and topological spaces also have a product space. Finally, many mathematical notions have some notion of “naturality” or “canonicality” to them. For example, a finite-dimensional vector space and its dual space are known to be isomorphic, but this isomorphism is not “natural”. This project will aim to introduce the body of mathematics which can explain all of these curious facts: category theory. We will introduce the concept of categories, discuss the process of diagram-chasing, and arrive at the idea of a limit. Other topics in category theory and relationships with other parts of mathematics may be discussed, according to interest.

Book: Clifford Algebras: an Introduction, D. J. H. Garling

Prerequisites: A course in linear algebra (e.g. MAT 310). A course in abstract algebra (MAT 313, MAT 314 encouraged)

The Clifford algebra associated to a given vector space with a quadratic form (like Euclidean space) is a very useful object to use to study the geometry of that quadratic form. Another way to say that this is a useful concept is to see that it comes up in multiple different contexts, like algebra, geometry, topology and physics. Our goal in this project is to get acquainted with this concept and its usefulness. We will aim at classifying the real and complex Clifford algebras, introduce the spin groups and study their relevance to the representation theory of orthogonal groups. This last fact comes up in physics, as discovered by Dirac, which we will also discuss.

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: Dynamic Programming and Optimal Control, Vol. II, 4th Edition: Approximate Dynamic Programming, by Dimitri P. Bertsekas

Prerequisites: A course in linear algebra (e.g. MAT 211). Exposure to real analysis (e.g. MAT 319) is a bonus.

How should rational agents plan for the future? This is the underlying question asked by the Federal Reserve, your neighbor’s Roomba, the electric utility company, and the AlphaZero chess player. Each of these problems features agents interacting with uncertain, evolving environments. Agents are tasked with finding policies that optimize their outcomes with respect to particular performance criteria. In this project, the mentee will explore the theory of Markov Decision Processes, the mathematical model at the heart of this class of optimization problems. They will prove the existence of optimal policies for finite and infinite horizon MDPs under the expected total discounted reward criterion, and study algorithms which directly compute these policies. Depending on the mentee’s interests and maturity, this project can either adopt a more theoretical flavor with analysis of additional performance criteria, or a more applied flavor with reinforcement learning.

Book: Mathematical Foundations of Neuroscience, by Ermentrout and Terman

Prerequisites: Calculus I-IV (e.g. MAT 203, 303).

In the 1950's, Hodgkin and Huxley experimentally derived a system of non-linear ordinary differential equations that, to this day, people use to understand the behaviors of neurons. The goals of this DRP will be to understand how these equations come about and the ways in which people understand the solutions to this equation. The former will take us through a tour of how differential equations are used in chemistry and physics. The latter will take us on a tour of some concepts in dynamical systems. (These are covered in Chapters 1 and 3 in the book). Students will then simulate a solution of the Hodgkin and Huxley equations by implementing a numerical ODE solver in MATLAB/Octave. Time permitting, based on student interest, additional chapters from the book can be explored.

Book: Machine Learning: A Probabilistic Perspective, by Kevin P. Murphy

Prerequisites: Linear algebra (e.g. MAT 211). Basic knowledge of probability will be very useful.

Machine Learning (ML) is foremost known as a tool for the industry; however, the theoretical aspects behind the tools of ML also encompass a very active area of research. Most of these foundations are better understood under the lens of probability.

In this project, we will learn enough probability to understand the whys and hows behind some of the basic algorithms of ML such as k-Nearest neighbors, Gaussian models, or Neural networks. Specific topics will depend on the interest of the student. If time permits it, and the student wishes to pursue doing so, we will implement (i.e. code) some working algorithm.

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.

Book: The Mathematics of Voting and Elections: A Hands-On Approach, by Jonathan Hodge and Richard Klima

Prerequisites: Calculus and Proofs

Currently, debates are raging about fairness in existing democratic systems. How powerful are the permanent members of the UN Security Council? Is gerrymandering and redistricting abuse an existential threat to democracy? Should election victors be declared based on plurality or are there other better methods to decide who won an election? Many of these extremely complicated questions can actually be addressed by mathematics! This project will discuss some of the ways to interpret these questions mathematically and will discuss some basic results. Depending on interest, discussions may involve linear algebra, probability, geometry, or optimization theory. Some classical results, such as the famous Arrow Impossibility Theorem,and some new topics, such as the efficiency gap (recently discussed at the US Supreme Court), may be included.