# Capstone Courses

## MTH 396 Prerequisites:

MTH 396 Prerequisites: Completion of Tier I Writing Requirement, MTH 309, MTH 310, and MTH 320 (or the honors equivalents, or approval of department) and approval of the department. Typically the department expects a cumulative GPA of at least 2.0 and an average of at least 2.0 across MTH 309, MTH 310 and MTH 320. Note: Email notification will be given once your override has been issued.

## MTH 496 Prerequisites:

Completion of Tier I Writing Requirement and approval of the department. Typically the department expects students to have completed MTH 309, MTH 310, and MTH 320 (or the honors equivalents) with cumulative GPA of at least 2.0 and an average of at least 2.0 across MTH 309, MTH 310 and MTH 320. Additional prerequisite courses may be required and can be found in the descriptions below. Note: Email notification will be given once your override has been issued.

**MTH 496-001, Fall 2024: Topological Data Analysis**

**Instructor: Guowei Wei**** **

Description: Computational topology has actively facilitated the integration of research efforts across computational geometry, algebraic topology, combinatorics, data analysis, and various related scientific domains. Notably, the last decade has witnessed significant growth in the field, particularly in the realm of data science. The utilization of topological methods in data analysis, previously dominated by statistical approaches, has had tremendous success in dealing with intrinsically complex data, such as that arising from biological science. This course aims to provide a foundational understanding of essential topological concepts relevant to topological data analysis, such as topological invariants (e.g., Betti numbers, Euler characteristic, Wu characteristic) and topological objects (e.g., simplicial complex, cell complex, path complex, directed flag complex, cellular sheaf, directed graph, hypergraph, hyperdigraph, etc.). It involves proficiently grasping a selection of topological formulations, including homology, singular homology, homotopy, topological persistence, persistent Laplacians, persistent Dirac, persistent Mayer homology, and interactions. The course also introduces topics in topological deep learning (TDL), an emerging paradigm in data science.

Prerequisites: CSE 231 or CMSE 201 and departmental approval

**MTH 496-002, ****Fall 2024****: History of Mathematics **

**Instructors: Bob Bell and Abe Edwards**

Description: The course will survey the development of mathematics from ancient to modern times. Students will learn about mathematical developments by engaging with excerpts from primary sources and by reading historical accounts of mathematicians and their discoveries (in a textbook format). In a typical week, students will work through a Primary Source Projects developed by the TRIUMPHS group (see https://nscoville.github.io/website/TRIUMPHS%20Available%20PSPs.pdf) and the class will discuss assigned readings from the textbook. Each student will give a proposal and final presentation

Department Approval Required

**MTH 496-001, Spring 2025: Bilinear forms**

**Instructors: Igor Rapinchuk **

Description: The objective of this course will be to study some aspects of the theory of bilinear forms and its applications. Bilinear forms generalize the usual dot product encountered in multivariable calculus. More precisely, the dot product is an important example of the general notion of a positive definite bilinear form; in analogy with the dot product's use in R^3, positive definite bilinear forms enable one to talk about distances and angles in arbitrary (possibly infinite-dimensional) vector spaces. In particular, in a vector space equipped with a positive definite bilinear form, one can construct a very convenient basis called an orthonormal basis, which has numerous applications in pure and applied settings. For example, from an algebraic point of view, determining the Fourier series of a periodic function essentially boils down to constructing an orthonormal basis consisting of trigonometric functions in a certain function space with respect to an appropriate bilinear form (of course, there are also various analytic questions of convergence that need to be considered). Although positive definite bilinear forms are closest to our geometric intuition, one can in fact establish a number of structural results for fairly general bilinear forms, resulting in a theory that has close connections to a number of disciplines, including algebra, number theory, analysis, geometry, numerical analysis, and physics, to name just a few. We will also consider linear transformations preserving various types of bilinear forms and develop their spectral theory. Time-permitting, in the last part of the course, we may highlight some of the applications of bilinear forms to the representation theory of finite groups.

Prerequisites: MTH 411 (or concurrent enrollment) and department approval

**MTH 496-002, Spring 2025: Topic in Optimal Control Theory**

**Instructor: Son Tu**

Description: This course introduces the optimal control problem, delving into its fundamental principles and its link to the Hamilton-Jacobi-Bellman equation. We will delve into the interaction between nonlinear optimization and control problems, exploring various facets of the theory, including dynamic programming, variational calculus, and Pontryaginâ€™s maximum principle. The curriculum incorporates numerous examples and applications of the theory within continuous systems. Towards the conclusion of the course, attention will shift to the study of numerical search algorithms and, time permitting, discrete systems.

Recommended background: CSE 231 or CMSE 2\01

Prerequisites: MTH 235 or MTH 340 and department approval