# 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.

**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.

** 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.

**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 of ℝnRn that are composed of convex polyhedra. Thus, tropical algebraic geometry is a piecewise-linear
version of algebraic geometry.

**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. Prerequisites are basic PDE theory and linear algebra.

**Computational Mathematics is an essential 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.**

**Description:****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...

**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.**

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