2023-2024 Catalog

# MATH 395 Special Topics in Advanced Mathematics

Special topics in advanced mathematics selected largely by student interest and faculty agreement. May be repeated for credit.

The course will consist of advanced topics in differential equations not usually seen in either ordinary differential equations or partial differential equations, such as: delay differential equations and stochastic differential equations. Boundary value problems, numerical methods, and infinite series solutions. Prerequisite: MATH 340 or MATH 342 or permission of instructor.

Galois Theory

One learns in a first course in algebra that given an equation of the form ax^2 + bx + c = 0, one can always find solutions using the quadratic formula.  It is considerably harder to find comparable formulas to solve equations of this form if one allows degree 3 or degree 4 polynomials, but mathematicians found formulas for these in the 16th century.  It remained a mystery if one could solve an equation of the form f(x) = 0 if f(x) is a degree 5 polynomial until the work of a young mathematician named Evariste Galois in the early 19th century.  Unfortunately, Galois died in a duel at 21 years old without anyone accepting or understanding his work.  Galois proved that there is no formula to solve (in radicals) an equation of the form f(x) =0 for f(x) a general polynomial of degree 5 or higher.  The ultimate goal of this course will be to prove this result, but along the way it is necessary to develop the theory of rings, fields, and the connection between groups and field extensions known as Galois Theory.  It is possible to take this course concurrently with Math 320.

History of Mathematics

This seminar course is a survey of selected topics from the history of mathematics. Through exposure and access to primary historical sources and other materials, students will gain deeper insights into mathematical concepts they have seen before, be introduced to new mathematical ideas, and learn about the development and developers of mathematics in the past and present. The primary objective of the course is for students to synthesize various mathematical ideas from fundamental and advanced mathematics courses as a capstone experience for the major. Students will complete a project about an advanced mathematical topic with presentations in written and oral form at the end of the course. This course satisfies the Core Program Requirement: Pre-1800 (CPPE). Prerequisite: Any 300-level Mathematics course (may be taken concurrently) or permission of instructor.

Knot Theory

How knots are described mathematically, how one can distinguish different knots, create new knots, classify knots. Topics include: Reidemeister moves, links, knot colorings, alternating knots, braids, knots and graphs, knot invariants, mirror images, unknotting, number crossing, number applications to biology and chemistry. Prerequisite: MATH 210, MATH 212, or MATH 214 or permission of instructor.

Real Analysis II (Metric spaces and Lebesgue’s measure theory)

This course serves as a sequel to MATH 310 Real Analysis. The first part of this course  will cover the main topics surrounding the topology, theory, and applications of metric spaces and their morphisms. In particular we will study certain function spaces that arise as normed vector spaces. Completeness, separability, compactness, and connectedness are some of the themes that will appear throughout the course. Fundamental results in Analysis such as the Heine-Borel Theorem, the Baire Category Theorem, the Bernstein Approximation Theorem, the Stone-Weierstrass Theorem, and the Arzela-Ascoli Theorem will be proved and applied. The second part of this class will be devoted to developing the Lebesgue measure on the real line and the Lebesgue integral along with powerful convergence theorems. Time permitting we will study some Fourier series. Prerequisite: MATH 310 or permission of instructor.

2 or 4 units

### Core Requirements Met

• Mathematics/Science