Collections of undergraduate research projects [closed]

I would like to compile a "big list" of undergraduate research projects in the following areas of mathematics:

calculus;
analysis;
abstract algebra;
linear algebra;
number theory;
geometry;
mathematical physics;
game theory;
graph theory, combinatorics, probability, statistics;
algorithms, computability, computational complexity, and other computer-science-related topics;
et cetera...

To be precise, I'm searching for some references (books, web-pages, online notes, etc.) that collect

the statement of the problems (which are the most important thing);
and preferably hints, guidance, or complete solutions (if they have been found) of the problems, and the actual complete projects.

I will start the list myself by mentioning the well-known book Student Research Projects in Calculus.

Solution 1:

I looked into the question and found some projects that may be of interest:

Calculus and Differential Equations

Simple Differential Equations and the Growth and Decay of Ice Sheets:

In this project we will re-visit and expand upon a project of John Imbrie of the University of Virginia and his daughter Katherine matching periodic earth temperatures reflected in ice cores to when the earth axis tilt wobbled and the planets relative annual position to sun. We will investigate how key aspects of the ice-age record (such as shifts in dominant periodicities) follow from simple ordinary differential equations capturing the essential physics of the growth and decay of ice sheets.

The Gamma Function:

The theory of the gamma function was developed in connection with the problem of generalizing the factorial function of the natural numbers. The gamma function is defined as a definite, improper integral, and the notion of factorials is extended to complex and real arguments. This function crops up in many unexpected places in mathematical analysis, such as finding the volume of an n-dimensional “ball”. In this project we develop and explore the basic properties of this function.

Conic Sections via Cones:

In this project we will work our way through Conics of Apollonius of Perga ca. 262 BC – ca. 190 BC. In this work Apollonius develops simple and not so simple properties of conic sections, many of which we now only know through calculus. We will also attempt to illustrate the propositions in Conics using the powerful mathematics software Mathematica.

Linear Algebra

Affine Transformations and Homogeneous Coordinates:

In this project we will look at geometric transformations using homogeneous coordinates and matrices. Affine transformations include translation, rotation, reflection, shear, expansion/contraction, and similarity transformations. This will show the student the relationship between high school geometry, linear algebra, and group theory. We will also illustrate properties in geometry and linear algebra using the powerful mathematics software Mathematica or MATLAB.

Approximation of Functions with Simpler Functions:

In this project we will use functions with "nice" properties to approximate other functions. An example that might be familiar to students is using polynomials to approximate certain functions via Taylor/Maclaurin series. Eventually we will look at families of so-called “orthogonal functions” and how they are used to approximate other functions. We will use the powerful mathematics software Mathematica to illustrate the approximations.

Least is the Best:

A common concern in industry is optimization: minimizing the cost, maximizing the profit, optimizing resource utilization, and so on. Students learn basic optimization techniques in calculus courses. But to what is it applied? What if the objective function is non-differentiable? What if variables are discrete? In this project, students can choose their preferred "no-so-nice" application and explore heuristic approaches to estimate the optimum and the optimizer.

Geometry

The kinematics of rolling. (Riemannian geometry/Non-holonomic mechanical systems):

On a smooth stone, draw a curve beginning at a point p, and hold the stone over a flat table with p as the point of contact. Now roll the stone over the plane of the table so that at all times the point of contact lies on the curve, being careful not to allow the stone to slip or twist. We may equally well think that we are rolling the plane of the table over the surface of the stone along the given curve. Mechanical systems with this type of motion are said to have "non-holonomic" constraints, and are common fare in mechanics textbooks.

Now imagine a tangent vector to the plane at p. This rolling of the plane over the surface provides a way to transport v along the curve, keeping it tangent at all times. The resulting vector field over the curve is said to be a "parallel" vector field. Show that there is a unique way to carry out this parallel translation. (Find a differential equation that describes the parallel vector field and use some appropriate existence and uniqueness theorem.) Let c be a short path joining p and q, whose velocity vector field is parallel. Show that c is the shortest path contained in the surface that joins p and q.

Whether or not you fully succeed, this mechanical idea will give you a concrete way of thinking about ideas in differential geometry that might seem a bit abstract at first, such as Levi-Civita connection, parallel translation, geodesics, etc. Also look for an engineering text on Robotic manipulators and explain why such non-holonomic mechanical systems are important in that area of engineering.

I don't know of many places where these things are explained in a simple way. Perhaps Geometric Control Theory by Velimir Jurdjevic is a place to start. In the engineering literature, A Mathematical Introduction to Robotic Manipulation is a particularly good reference.

Geometry in very high dimensions. (Convex geometry)

Geometry in very high dimensions is full of surprises. Consider the following easy exercise as a warm-up. Let $B(n,r)$ represent the ball of radius $r$, centered at the origin, in Euclidian n-space. Show that for arbitrarily small positive numbers $a$ and $b$, there is a big enough $N$ such that $(100 - a)\%$ of the volume of $B(n,r)$ is contained in the shell $B(n,r) - B(n,r - b)$ for all $n > N$.

Here is a much more surprising fact that you might like to think about. Let $S(n-1)$ denote the sphere of radius 1 in dimension $n$. (It is the boundary of $B(n,1 )$.) Let f be a continuous function from $S(n-1)$ into the real line that does not increase distances, that is, $| f(p) - f(q) |$ is not bigger than $| p - q |$ for any two points $p$ and $q$ on the sphere. ($f$ is said to be a "1-Lipschitz" function.) Then there exists a number $M$ such that, for all positive $a$, no matter how small, the set of points $p$ in $S(n-1)$ such that $| f(p) - M |>a$ has volume smaller than $\exp(-na^2 / 2 )$. In words, this means that, taking away a set with very small volume (if the dimension is very large), $f$ is very nearly a constant function, equal to $M$.

This is much more than a geometric curiosity. In fact, such concentration of volume phenomenon is at the heart of statistics, for example. To make the point, consider the following. Let $S(n-1, n^{0.5})$ be the sphere in n-space whose radius is the $\sqrt{n}$. Let f denote the orthogonal projection from the sphere to one of the $n$ coordinate directions, which we agree to call the x-direction. Show that the part of the sphere that projects to an interval $a < x < b$ has volume very nearly (when $n$ is big) equal to the integral from $a$ to $b$ of the standard normal distribution. (This is easy to show if you use the central limit theorem).

For a nice introduction to this whole subject, see the article by Keith M. Ball in the volume Flavors of Geometry, Cambridge University Press, Ed.: S. Levy, 1997.

Hodge theory and Electromagnetism. (Algebraic topology/Physics)

Electromagnetic theory since the time of Maxwell has been an important source of new mathematics. This is particularly true for topology, specially for what is called "algebraic topology". One fundamental topic in algebraic topology with strong ties to electromagnetism is the so called "Hodge-de Rham theory". Although in its general form this is a difficult and technical topic, it is possible to go a long way into the subject with only Math 233. The article "Vector Calculus and the Topology of Domains in 3-Space", by Cantarella, DeTurck and Gluck (The American Mathematical Monthly, V. 109, N. 5, 409-442) is the ideal reference for a project in this area. (It has as well some inspiring pictures.)

Another direction to explore is the theory of direct current electric circuits (remember Kirkhoff's laws?). In fact, an electric circuit may be regarded as electric and magnetic field over a region in 3-space that is very nearly one dimensional, typically with very complicated topology (a graph). Solving circuit problems implicitly involve the kind of algebraic topology related to Hodge theory. (Hermann Weyl may have been the first to look into electric circuits from this point of view.) The simplification here is that the mathematics involved reduces to finite dimensional linear algebra. A nice reference for this is appendix B of The Geometry of Physics (T. Frankel), as well as A Course in Mathematics for Studentsof Physics vol. 2, by Bamberg and Sternberg.

Symmetries of differential equations. (Lie groups, Lie algebras/Differential equations)

Most of the time spent in courses on ODEs is devoted to linear differential equations, although a few examples of non-linear equations are also mentioned, only to be quickly dismissed as odd cases that cannot be approached by any general method for finding solutions. (One good and important example is the Riccati equation.) It turns out that there is a powerful general method to analyze nonlinear equations that sometimes allows you to obtain explicit solutions. The method is based on looking first for all the (infinitesimal) symmetries of the differential equation. (A symmetry of a differential equation is a transformation that sends solutions to solutions. An infinitesimal symmetry is a vector field that generates a flow of symmetries.) The key point is that finding infinitesimal symmetries amounts to solving linear differential equations and may be a much easier problem than to solve the equation we started with.

Use this idea to solve the Riccati equation. Choose your favorite non-linear differential equation and study its algebra of infinitesimal symmetries (a Lie algebra). What kind of information do they provide about the solutions of the equation? Since my description here is hopelessly vague, you might like to browse Symmetry Methods for Differential Equations - A Beginner's Guide by Peter Hydon, Cambridge University Press. It will give you a good idea of what this is all about.

Riemann surfaces and optical metric. (Riemannian geometry/Optics)

Light propagates in a transparent medium with velocity $c/n$, where c is a constant and n is the so called "refractive index" -- a quantity that can vary from point to point depending on the electric and magnetic properties of the medium. For a given curve in space, the time an imaginary particle would take to traverse its length, having at each point the same speed light would have there, is called the "optical length" of the curve. Therefore, the optical length is the line integral of n/c along the curve with respect to the arc-length parameter. According to Fermat's principle, the actual path taken by a light ray in space locally minimizes the "optical length". It is possible to use the optical length (for some given function n) to defined a new geometry whose geodesic curves are the paths taken by light rays. This is a particular type of Riemannian geometry, called "conformally" Euclidian. All this also makes sense in dimension 2.

One of the most famous paintings of Escher show a disc filled with little angels and demons crowding towards the boundary circle. What refractive index would produce the metric distortions shown in that picture?

A fundamental result about the geometry of surfaces states that, no matter what shape they have, you can always find a coordinate system in a neighborhood of any point that makes the surface conformally Euclidian. Why is this so? (This will require that you learn something about so called "isothermal coordinates".)

Analysis

Failure of von Neumann's inequality.

Von Neumann proved that if A is a contractive matrix (has operator norm $\leq 1$) and $p(z)$ is a complex polynomial, then $p(A)$ has operator norm bounded by the supremum of $p$ on the unit circle. A two variable version of this result is true (Andô's inequality) but the three variable version is false. Counterexamples can be shown to exist either through probabilistic arguments (i.e. a random polynomial will fail the inequality) and there are also a few examples constructed through ad hoc methods. This project would involve trying to construct more interesting families of counterexamples to the three variable von Neumann inequality in order to understand "how badly" the inequality fails.

Multilinear Bohnenblust-Hille inequality

This is a different kind of inequality for polynomials. Multilinear polynomials satisfy an inequality bounding certain little $l^p$ norms of their coefficients by the supremum norm of the polynomial. This project would also involve looking for interesting examples to test the sharpness of known versions of this inequality.

Algebra

algebra project 1

If $a_1,a_2,...a_n$ are integers with $gcd = 1$, then the Eulidean algorithm implies that there exists a $n \times n$-matrix $A$ with integer entries, with first row $= (a_1,a_2,...,a_n)$, and such that $\det(A) = 1$. A similar question was raised by J.P. Serre for polynomial rings over a field, with the a's being polynomials in several variables. This fundamental question generated an enormous amount of mathematics (giving birth to some new fields) and was finally settled almost simultaneously by D. Quillen and A. A. Suslin, independently. Now, there are fairly elementary proofs of this which require only some knowledge of polynomials and a good background in linear algebra. This could be an excellent project for someone who wants to learn some important and interesting mathematics. (These results seem to be of great interest to people working in control theory.)

algebra project 2

A basic question in number theory and theoretical computer science is to find a "nice" algorithm to decide whether a given number is prime or not. This has important applications in secure transmissions over the internet and techniques like RSA cryptosystems. Of course, the ancient method of Eratosthenes (sieve method) is one such algorithm, albeit a very inefficient one. All the methods availabe so far has been known to take exponential time. There are probabilistic methods to determine whether a number is prime, which take only polynomial time. The drawback is that there is a small chance of error in these methods. So, computer scientists have been trying for the last decade to find a deterministic algorithm which works in polynomial time. Recently, this has been achieved by three scientists from IIT, Kanpur, India. A copy of their article can be downloaded from www.cse.iitk.ac.in A nice project would be to understand their arguments (which are very elementary and uses only a little bit of algebra and number theory) and maybe to do a project on the history of the problem and its ramifications.

VT past projects
UMD past projects
UC Berkeley past projects
NYU project ideas
NSF research experience for undergrads
Washington State past projects
Cornell REU
U of Mary Washington opportunities, conferences, and past projects
A few examples from Stanford and how to enroll if you are at Stanford
BYU mentor section at the bottom has projects ideas

Solution 2:

Not an answer, just one contribution.

The William's College SMALL summer program is an impressive model, at the high-end. "Around 500 students have participated in the project since its inception in 1988."

There are six areas this (2015) summer: Arithmetic Combinatorics (Leo Goldmakher), Combinatorial Geometry (Satyan Devadoss), Commutative Algebra (Susan Loepp), Geometry (Frank Morgan), Hyperbolic Knots (Colin Adams) and Number Theory & Harmonic Analysis (Steven Miller and Eyvi Palsson).

Here is a link for the project abstracts. Past projects have resulted in an impressive number of published papers.