(prerequisites: 2200-Mathematical Reasoning and Proof (or equivalent))

Combinatorics studies how different objects can be counted and arranged. This area has applications in many areas of pure and applied mathematics. This course will study some combinatorial properties of graphs, then use that knowledge to study certain problems in computer science. Possible topics include spanning trees, the traveling salesperson problem and sorting algorithms. Prerequisites: having taken Intro to Proofs. 

(Prerequesits: Formally, none; however, some familiarity with proofs would be helpful)

A graph (not the x-y plane kind) is a collection of points and lines connecting some of those points. A word is a finite string of symbols (letters). We will study word-representable graphs, graphs for which there exists a certain kind of word associated with them (unnecessary to understand now, but more precisely, they are graphs for which there exists a word on the vertex set of the graph such that an edge xy is present in the graph exactly when x and y alternate in the word). Along the way, we will learn about partially ordered sets and orientations on graphs, and how those are useful tools when determining if a graph can be represented by a word in this way, and if so, how long/short such a word must/can be.

(Prerequesits: Any student at any level is welcome, although for a student who has taken linear algebra or abstract algebra (or is willing to learn some of these) I would be able to jump into more advanced topics.)

The history of mathematics can be traced back to two primordial goals: the study of numbers and the study of shapes. What we call the Pythagorean Theorem had been discovered and rediscovered by civilizations thousands of years before Pythagoras was born. The quest to understand the structure behind shapes and forms culminated in Euclid's "The Elements", where starting from what he saw as reasonable axioms, Euclid developed a language of geometry using the logical structure of a proof. "The Elements" served as the most important mathematics textbook for thousands of years, and our modern approach to mathematics through proofs traces back to trying to understand shapes in their purest forms.

Eventually, people started to realize that what Euclid had assumed to be reasonable axioms did not need to be there, and by challenging his notions of structure new geometries were born, such has hyperbolic space and the projective plane. These geometries have been not only useful for science and engineering, but also in the arts, from linear perspective to the strange geometries of Escher.

As mathematical formalism advanced, so did our approach to geometry. People learned how to use algebra, the symbolic manipulation of structure, to talk about geometry. This approach led to a revolutionary shift in thinking thanks to the theory of ideals and modules introduced by Emmy Noether. Suddenly, the thousands-years-old study of the conic sections could be described through studying the ideal structure of polynomial rings, and this is only the beginning.

In this reading course we will talk about the formal approach to studying geometry in a historical light, starting with the straight-edge-and-compass constructions of Euclid, building towards non-Euclidean geometries, and possibly even beginning a discussion of algebraic geometry through polynomial rings and affine spaces. Often undergraduate mathematics classes focus on computation; hopefully this course will provide a chance to think about forms first and learn how the symbols arise later. The student's interest will in-part determine the focus of the course, be it advanced Euclidean proofs, non-Euclidean spaces, or using abstract algebra to connect geometry to modern mathematics.

Book: Geometry Through History, by Meighan I. Dillon