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) Archive
Cancer is the failure of regulation of cell replication in multicellular organisms. Understanding and treating cancer thus requires linking processes within individual cells with their environment, including non-cancerous cells, physical barriers, resources, and the immune system. It is through a sick, twisted version of evolution that the barriers inhibiting growth and spread are overcome. Treatment seeks to halt or reverse this process without causing undue damage to the host.
We will work together to build mathematical models of several key cancer processes, none of which are fully understood either biologically or mathematically, and then focus the second half of the semester on projects.
Polynomials are central to the study mathematics. For example, we learn in Calculus that Taylor polynomials are used to approximate exponential, logarithmic, and trigonometric functions. Behind these simple mathematical objects is a rich theory which unites many branches of mathematics. In this course we will explore analytic, algebraic, geometric, and number theoretic properties of polynomials in hopes of connecting the dots between seemingly different branches of mathematics.
Description:In modern day finance, with intrinsic nonlinearities in the models and vast amount of data sets available, machine learning (ML) is destined to transform the financial world as we know it, ranging from customer services and security measures. In this course we will discuss two particular data analysis subjects closely related to traditional quantitative financial analysis: portfolio selection and algorithmic trading. We will begin with a quick survey of a wide variety of data structures available and the challenges presented, and the basic notions of machine learning tools. The nature of finance makes it particularly difficult for standard machine learning tools to apply and yield successful results consistently. The rate of failure in financial ML is rather high and we would like to explain the reasons and provide clues to recognize the shortcomings. One area we would like to address is the assessment of values of strategies, and another is the detection of structural breaks. Regarding models, we will discuss the basic ideas in cross-validation and backtesting. For asset allocation, we will discuss approaches beyond the traditional quadratic optimizers that can compute a portfolio on ill-degenerated covariance matrix that is quite practical in reality. .
Description: The advancement of neuroscience techniques has allowed collection of vast and increasingly complicated sets of data. What can we learn from them? What are useful ways to analyze and present them? In this course we will explore methods of data analysis that are employed in modern neuroscience. We will learn about relevant techniques through a combination of reading, presentations, discussions and solving problem sets. We will then apply these techniques to actual data collected in neurophysiological and behavioral experiments. Students should expect to work hard solving problems, doing computations, reading and presenting in class. The course grade will be a combination of homework, class participation and a project.
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.
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.
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.
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.
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.
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 ofthat are composed of convex polyhedra. Thus, tropical algebraic geometry is a piecewise-linear version of algebraic geometry.
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