# Math 4800 Undergraduate Research Topics

These courses provide a research experience in a familiar course setting. Topics vary every semester, but there is usually a Pure Mathematics and an Applied Mathematics oriented course every academic year. Enrollment in this class is usually by permission of the instructor only.

# Past Math 4800 (formerly Math 4950)

Fall 2017: Representation Theory of Finite Groups
When and Where: Mondays and Wednesdays 1:25PM-2:45PM in AEB 306
Class website
Description: Roughly speaking, group theory is the mathematical study of symmetry. Shapes in the plane may have rotational or reflectional symmetry; a collection of n objects can be permuted in n! different ways; a Rubik’s cube can be configured in roughly 43 quintillion different ways. Symmetries can be composed and in general the order of composition matters. The resulting mathematical objects - groups - have a rich structure. Understanding this structure can be quite challenging. In this course we’ll start with some down-to-earth examples of groups to build some intuition and the ability to do computations. Then we’ll dive into the basics of Representation Theory - a field that studies how to represent abstract groups as a collection of square matrices. We’ll see that the trace of these matrices is something very worthy of study - the character of the representation. Students should expect to work hard solving problems, doing computations, reading and presenting proofs in class. The course grade will be a combination of homework, class participation and an open-ended project.

Spring 2017 Math 4800: Graph Theory
Instructor: Thomas Goller
Class website
Description: This course, which is intended for advanced mathematics majors and computer science majors, is about graph theory. A graph is a simple mathematical structure that stores information about how a set of objects is connected. The definition of graph is so natural that although graphs arise frequently in mathematics and computer science, they rarely get the attention they deserve. That is about to change. We will construct graphs, prove theorems about graphs, and study algorithms that solve graph-theoretic problems relating to enumeration, subgraphs, graph coloring, finding routes, network flows, graph decompositions, and more. See Wikipedia’s page on graph theory for an introduction to these topics. Students will be encouraged to do graph computations in a programming language of their choice. Based on students’ interests, we will consider more advanced applications of graphs to linear algebra, group theory, topology, algebraic geometry, logic, complexity theory, theorem proving, data mining, and studying large networks.

Fall 2016: Random Walks with Algebraic Combinatorics, Instructor: Tom Alberts
Description: If you have ever found yourself in an unknown city then likely you performed some version of a random walk: not knowing which street to take next you randomly chose among the available options and then repeated. Although it’s a simple mechanism, the statistics of the walk produced in this way turn out to be ubiquitous across mathematics. If the geometry of the city has an underlying “group structure” then from a mathematical point of view the random walk process is particularly interesting. Commonly the statistical properties of the random walker are studied by simple counting arguments, and if one exploits the underlying group structure to do so then the tools of algebraic combinatorics become available. The course will start with the basics of simple random walk on integer lattices, with an emphasis on studying it through combinatorial ideas. Many simple and cute, but powerful, methods of counting will be used. We will then move into the study of random walks on graphs and groups and along the way encounter many interesting objects such as Young tableaux, the RSK algorithm, and the matrix­tree theorem. We will also briefly discuss the deep connections to probability, statistics, differential equations, geometry, and number theory.

Spring 2016: Network Inverse Problems, Instructor: Fernando Guevara Vasquez
Description: Networks can be used to model many physical phenomena such as electricity conduction and vibrations of an elastic body. We focus on the inverse problem, i.e. the question: Can one recover properties of the network from measurements made at a few nodes? A classic example is to recover the position and weight of beads in a vibrating string from measuring how the string responds to being plucked at one end. This class explores connections between physics, graph theory, partial differential equations, linear algebra and stochastic processes. Applications include medical imaging and geophysical prospecting.

Fall 2015: Algebra, Geometry and a little bit of String Theory, Instructor: Herb Clemens
Description: It turns out that String Theory in Physics is built on some very concrete algebra and geometry. In this course we will begin with the algebra and geometry that students bring to the course and inch our way toward the geometry of the other six dimensions that String Theory postulates in order to allow for an 'understandable' universe.

Spring 2015: Introduction to tropical geometry, Instructor: Tommaso de Fernex
Description: The purpose of this course is to give an introduction to algebraic curves and tropical curves. The origins of algebraic geometry lie in the study of zero sets of systems of polynomials. These objects are algebraic varieties, and they include familiar examples such as plane curves and surfaces in three-dimensional space. In tropical algebra, the sum of two numbers is their minimum and the product of two number is their sum. It makes perfect sense to define polynomials and rational functions over the tropical semiring. The functions they define are piecewise-linear. Also, algebraic varieties can be defined in the tropical setting. They are now subsets of $\mathbb{R}^n$ that are composed of convex polyhedra. Thus, tropical algebraic geometry is a piecewise-linear version of algebraic geometry.

Fall 2014: An inverse problem: finding boundary fields which produce breakdown, Instructor: Graeme Milton
Description: Most materials break down if the fields are high enough, this breakdown may be mechanical fracture or plastic yielding (for elasticity) or electrical shorting (in dielectric media) and in general one wants to prevent this. It is obviously important to know what boundary conditions necessarily lead to dangerously high internal fields. For a homogeneous body this is straightforward as one could solve for the internal fields, but what if the body is inhomogeneous, say containing two materials (or one material with holes) in an unknown geometry? Here we will explore this question and the ultimate goal of the course will be to produce a scientific paper on the problem, with the class contributing to the research and coauthoring the paper which will then be submitted to a scientific journal. Elementary analysis and numerical computation will be required, though it is expected that the class will have different strengths in different areas. Prerequistes are basic PDE theory and linear algebra.

Spring 2014: Selected Numerical Algorithms and Their Analysis, Instructor: Yekaterina Epshteyn
Description: Computational Mathematics is an esssential part of modern applied sciences. The course will provide an introduction to the research in the area of Numerical Analysis and Scientific Computing through lectures, students' presentations and projects. Example of Topics: Selected topics in numerical linear algebra, introduction to numerical methods for partial differential equations involving interfaces and irregular domains, meshfree approximation methods. The discussion on each topic will be self-contained.

Fall 2013: Inquiry Into Mathematics, Instructor: Steffen Marcus
Description:This course gives a problem-based introduction to the methods of mathematical research with a focus on topics within discrete mathematics, algebra, and geometry. Through a combination of lectures, problem-sessions and projects we will take a look at various examples of how mathematical theory can build from asking simple questions and generalizing. Topics will vary from week to week. A small sampling includes: special numbers, the map colour problem, incompleteness, projective geometry...

Spring 2013: Polynomials, Instructor: Aaron Bertram
Description: This course will be a mixture of algebraic geometry, number theory and some topics of a contemporary nature. It is about polynomials in one and several variables: their algebraic properties, the geometry of their real, complex and tropical solution sets, and the number theory of their rational and integer solution sets. Polynomials have fascinated mathematicians for thousands of years, and yet most people probably can't say a single intelligent thing about them. You will have lots to say after taking this course.

Fall 2012: The Mathematics of Disease

Fall 2011: Introdution to Tropical Geometry
Instructor: Tommaso de Fernex

Fall 2010: To A_D_E and Beyond; Dynkin Diagram and Classifications
Instructor: Aaron Bertram

Spring 2010: Symmetry & Goups
Instructor: Sarah Kitchen

Fall 2009: Metamaterials and cloaking theory
Instructor: Daniel Onofrei

Spring 2009: Graph Theory
Instructor: William Malone

Fall 2008: Random Walk: Modeling, Theory, and Applications
Instructor: Firas Rassoul-Agha

Spring 2008: Knot Theory
Instructor: Dan Margalit

Fall 2007: Metric Spaces, The Contraction Mapping Principle, Fractals & Other Applications
Instructor: Klaus Schmitt

Spring 2007: Fractals
Instructor: Elena Cherkaev

Fall 2006: Topics in Mathematical Finance
Instructor: Jingyi Zhu