# Senior Seminar Information (Class of 2023)

For the 2022-2023 academic year, the senior seminar topics are The Fundamental Theorem of Algebra, Infinite Series or the Generalized Stokes Theorem.

To ensure the senior seminar experience is an enriching experience it is necessary to keep class sizes relatively small and even. To help the department place students into seminars, each major who plans to take a senior seminar submits a proposal by NOON on the last day of classes of the winter semester of the junior year. Some details:

- The proposal is a LaTeX document, a template, to be filled out carefully by the student. The proposal should be approximately one page. It should describe which senior seminar you prefer to take, and why. For help with LaTeX, visit the What is LaTeX page.
- As a hypothetical example, here is a sample completed proposal in PDF format as Bernhard Riemann would have submitted it.
- By noon on the due date, the completed proposal is to be uploaded as a single PDF to the Senior Capstone Project Google form. The PDF file should have a useful, descriptive name. Riemann would’ve named his “
BernhardRiemannSeminarProposal .pdf”, for example. - The PDF file should have a useful, descriptive name. Riemann would’ve named his “BernhardRiemannSeminarProposal.pdf”, for example.
- It is a good idea for juniors to discuss the choice between thesis and seminar with faculty members before writing a proposal.
- The Department meets to consider all thesis and seminar proposals. The Department Chair will notify students of the results of the meeting by the middle of the short-term.
- The course descriptions for the 2022-2023 academic year senior seminars are below.

**FALL 2022**

**MATH 495L The Fundamental Theorem of Algebra.**

Over the centuries, there have been numerous proofs of the Fundamental Theorem of Algebra (FTA), which asserts that every polynomial of degree n must have at most n distinct roots over the complex numbers. The great German mathematician Carl F. Gauss (1777-1855) published no fewer than four different proofs of the result. While the name of the theorem foregrounds algebra, none of the known proofs is purely algebraic. Over the centuries, techniques from complex analysis, topology, and field extensions have been employed to give new proofs of the FTA. In this seminar, students explore some of these proofs where the methods are drawn from various subfields in mathematics. Prerequisite(s): MA

**WINTER 2023**

**MATH 495M Infinite Series.**

An infinite series is the sum of the terms of an infinite sequence. In calculus we encounter infinite series of real numbers, for example, the geometric series. This course focuses on infinite series of functions, beginning with an introduction to function series and convergence. Students explore power series, Laurent series, and trigonometric series, culminating with an in-depth examination of Fourier series. Fourier series have numerous applications to areas such as partial differential equations, signal and image processing, acoustics and econometrics, to name only a few. Based on student interests, they investigate one or more of the aforementioned applications of Fourier series using current research papers and texts in mathematics and computer software. Prerequisite(s): MATH 301. Instructor permission is required. [W3] K. Ott.

**MATH 495Z ****Generalized Stokes Theorem.
**

The famous theorems of Gauss, Green, and Stokes in multivariable calculus have many important applications in the study of electromagnetic fields, heat diffusion, fluid dynamics, and complex analysis. Moreover, they are all generalizations of the (one dimensional) Fundamental Theorem of Calculus. These results assert that a certain (k dimensional) integral over a region is the (k-1 dimensional) integral over the boundary of the region, culminating in the so-called Generalized Stokes Theorem. This seminar aims to explore this general form of the classical Stokes Theorem and related topics, including the concept of differential forms. Prerequisites: either M301 or M309. [P Wong]