Mathematical Sciences Courses

MA 590: Numerical Methods for Nonlinear Equations and Unconstrained Optimization - Professor Andrea Arnold

(offered Fall 2023)

Many real-world problems in science and engineering require robust methods for finding the solution to a system of nonlinear equations or the minimum value of a nonlinear, multivariable function. These two problems are connected and can be addressed using a similar suite of numerical approaches to find an approximate solution!

This course will introduce numerical methods for solving nonlinear systems of equations and unconstrained optimization problems. Topics will include: Newton’s method for nonlinear equations and unconstrained minimization; quasi-Newton methods; globally convergent modifications to Newton’s method and global strategies (such as line searches and trust region approaches); stopping, scaling, and testing criteria; secant methods; and nonlinear least squares.

Both analytical and computational aspects of the methods will be emphasized throughout. As the course progresses, students will implement the algorithms needed to build a robust quasi-Newton routine to address problems of this type. If you want to learn more about these algorithms and build your own code for potential application in your research, this is the class for you!

Recommended background: Knowledge of multivariable calculus and linear algebra, as well as familiarity with MATLAB or a higher-level programming language.

MA 590: Advanced Topics in PDE - Professor Chris Larsen

(offered Spring 2024)

This course will cover a selection of more advanced topics in PDE. These will include basic results in Sobolev Spaces, variational methods for studying PDE, properties of evolution equations, and possibly some measure-theoretic methods.

MA 590: Probabilistic Methods in Combinatorics and Graph Theory - Professor Adam Wagner

(offered Spring 2024)

Probabilistic combinatorics is a very active field of pure mathematics, with connections to other areas such as computer science and statistical physics. Probabilistic methods are essential for the study of random discrete structures and for the analysis of algorithms, but they can also provide a powerful and beautiful approach for answering deterministic questions. The aim of this course is to introduce some fundamental probabilistic tools and present a few applications. Topics covered will include first and second moment methods, random graphs, correlation inequalities, and the Lovasz Local Lemma. The emphasis will be on using probabilistic tools to prove a variety of theorems in combinatorics. (Prerequisite: knowledge of basic probability and graph theory will be very helpful.)

MA 4891: Topology - Professor Herman Servatius

(offered D-Term 2024)

Tired out lying around the Hausdorff? Is it it just a House with Two Rooms? Do you want to fill up your Klein bottle and take a tour of the Peano Curve? Can you face Alexander's Horned Sphere? How can you complete a project if you haven't toured the Projective Plane?

Topology is an essential part of a well rounded mathematical education. It is, on the one hand, the foundation of classical and functional analysis, and on the other, an abstract bridge providing a common language linking important aspects of discrete, combinatorial, and geometric mathematics.

In this course, we will develop both aspects of the subject, starting with set theory and the introduction of standard examples of topological spaces. We will study the various axiom systems for topological spaces, which accomplish the essential task of giving a discrete formation of continuity. We will investigate convergence is this abstract setting: both of sequences and sequences of functions. We will develop an understanding of the key notions of connectedness, compactness, countability. We will investigate the description of topological spaces via cell-complexes.

Other topics may be the fundamental group, the classification of surfaces, covering spaces and Cayley graphs. The textbook is: Topology, Pearson New International Edition, 2nd edition, James R. Munkres a classic in the field.