Course descriptions: Number Theory and Geometry

Below are descriptions for Math 337: Number Theory and Math 343: Geometry if either were to run AY 26-27.

Number theory, at its core, delves into the surprisingly rich structure of a familiar and central object in mathematics: the set of positive integers. This investigation has engaged many human minds for several millennia, including the ancient Babylonians who inscribed Pythagorean triples on a clay tablet roughly 3800 years ago (such triples are integer solutions to the equation $x^2 + y^2 = z^2$). At times, number theory takes the integers on directly, asking questions about factorization, primality, or the distribution of numbers with some special property (e.g., prime, pseudo-prime, perfect, or amicable). At other times, number theory follows a slightly indirect approach, as in Diophantine geometry, where one asks whether systems of equations are solvable over the integers. A wonderful example of this concerns the equation $x^n + y^n = z^n$: for which positive integers $n$ does this equation have positive integer solutions? That this equation has no such solutions for $n$ at least 3 is better known as Fermat's Last Theorem, first conjectured by Pierre de Fermat in 1637. The conjecture remained open for centuries – it was finally settled by Andrew Wiles, building on the work of many others, in the 1990s. Beyond being a beautiful field of purely mathematical interest, there are applications of number theory to other fields, including cryptography.

Here are some of the major problems in number theory that we will solve in this course.

• Linear Diophantine equations. We will find all integer solutions to $ax + by = c$, when they exist.
• Pythagorean triples. We will find all integer solutions to $x^2 + y^2 = z^2$.
• The two square theorem. We will determine which natural numbers are expressible as a sum of two squares.
• The Pell equation. We will solve $x^2 - 2y^2 = 1$, and we will prove that $x^2 - ny^2 = 1$ always has a nontrivial solution (for any nonsquare positive integer $n$).
• Fermat's Last Theorem. We will prove that $x^3 + y^3 = z^3$ has no nontrivial solutions.
• The four square theorem. We will prove that every positive integer is expressible as the sum of four squares.
• Primes of the form $x^2 + 5y^2$. We will characterize the prime numbers $p$ such that $p = x^2 + 5y^2$ has an integer solution.

All of these problems are about the integers, and they are easy to state but usually difficult to solve. The unifying theme for the course is that we will be taking a Tour of Seven Rings, including the integers and a few of its relatives:

• $\Z$, the ring of integers.
• $\Z[i]$, the ring of Gaussian “integers” (the complex numbers with integer coordinates).
• $\Z_n$, the ring of integers modulo $n$.
• $\Z[\sqrt{n}]$, certain real quadratic “integer” rings (where $n > 0$ is not a square).
• $\Z\left[\frac{1+\mathbf{i} + \mathbf{j} + \mathbf{k}}{2}, \mathbf{i}, \mathbf{j}, \mathbf{k}\right]$, the ring of “integers” inside the real quaternions.
• $\Z[\zeta_3]$, a very well behaved ring of imaginary quadratic “integers” where $\zeta_3$ is a complex cube root of 1.
• $\Z[\sqrt{-5}]$, a slightly naughty ring of imaginary quadratic “integers.”

In the Gaussian integers and the last four rings, the concept of an integer is generalized. These rings all live inside fields; they play a role similar to the role played by $\Z$ inside the field of rational numbers $\Q$.

During the semester, we'll study a variety of Euclidean and non-Euclidean geometries using an elegant algebraic framework, first developed by Felix Klein in the late 19th century, that unifies them all. This framework is based on a study of transformations from the complex plane $\C$ to itself. Math 343 is a capstone course, so you will also work on a separate geometry-related project of your choosing. More details on the project will be forthcoming.

In a letter to his son János, the mathematician Farkas Bolyai wrote:

For God's sake, I beseech you, give it up. Fear it no less than the sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life.

What was Farkas talking about? Don't worry–he wasn't telling his son about his experience taking a geometry course at Gettysburg! His deep obsession was with proving that Euclid's fifth postulate–the famous parallel postulate–follows from the other four axioms of Euclidean geometry. Euclid, in his 13 book magnum opus The Elements (c. 300 BC), provides an axiomatic framework for the fairly rigorous development of what we now call Euclidean geometry. You are no doubt familiar with Euclidean geometry from high school, where you studied points, lines, angles, triangles, polygons, and congruence, among other notions. Underlying this planar geometry are certain basic assumptions. The axioms are (roughly in the words of Euclid):

1. One can draw a straight line from any point to any point.
2. One can produce a finite straight line continuously in a straight line.
3. One can describe a circle with any center and radius.
4. All right angles equal one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

The last axiom is far wordier and more complex than the others, and mathematicians attempted for centuries after Euclid to prove that it follows from the previous four axioms. Finally, in the early 19th century, Gauss (who never published his work, fearing that everyone would make fun of him), Lobachevsky, and János Bolyai independently discovered that the fifth axiom is in fact independent of the other four. They were able to construct a geometry called hyperbolic geometry that satisfies the first four axioms but not the fifth. It's easier to think about the parallel postulate using the following reformulation due to John Playfair:

Playfair's Axiom. Given a line and a point not on that line, there is exactly one line through the point that does not intersect the given line.

In hyperbolic geometry, `is exactly one line' is replaced with `are many lines'. In another non-Euclidean geometry, called elliptic geometry, `is exactly one line' is replaced with `are no lines'. In hyperbolic geometry, the sum of the angles in a triangle is always less than $180^\circ$, and in elliptic geometry this sum is always greater than $180^\circ$. The development of non-Euclidean geometry has had an enormous impact on mathematics and physics. It has a role to play in Einstein's theory of relativity and our understanding of the geometry of the very universe in which we live (a locally Euclidean space that may not be globally Euclidean).

In this course, we will work in a unified framework for studying Euclidean, hyperbolic, elliptic, and other geometries called Felix Klein's Erlangen Program. In this framework, congruence is the central concept. Congruent figures have precisely the same geometric properties; geometric facts about a given figure will also hold for all figures congruent to it. In the Euclidean geometry of the plane, two figures are congruent if there is some combination of translations, rotations, and reflections that takes one figure onto the other. Such moves are called transformations, and the set of all transformations that encode the concept of Euclidean congruence form what is called a group.

Varying the group of transformations leads to different notions of congruence and therefore to different geometries. For example, in Euclidean geometry without reflections (EGWR), only rotations and translations are allowed. Consequently, two triangles that are reflections of one another (and therefore congruent in ordinary Euclidean geometry) may not be congruent in EGWR. This distinction is meaningful if you want to view handedness as a geometric concept. Another geometry, called Möbius geometry, contains Euclidean, hyperbolic, and elliptic geometry as sub-geometries. The Erlangen Program provides an elegant organizational principle for studying almost every type of geometry out there.