Math Department (Thesis/2007-08)

Lattice points and the geometry of numbers

Under the direction of: Paul Friedman

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Consider the lattice in the plane formed by drawing in the lines $y=k$ and $x=j$ for all $j, k\in {\bf Z}$. Lattice points are points where these lines intersect; that is, points with integer coordinates.

We can investigate lots of questions. Here are some possibilities:

1. Lattice points and straight lines: Through how many lattice points does a line $y=mx+b$ pass? If a line doesn't pass through any, how close to a lattice point does it get? Are there infinite strips, or paths, between parallel lines that are lattice point-free? What about rectangles in the plane and lattice points?

2. Lattice points and the area of polygons: Pick's Theorem relates the area of a simple polygon $P$ whose vertices are lattice points to the lattice points inside and on the boundary of $P$. Can we discover and prove this theorem?

3. Lattice points in Circles: How many lattice points are in the interior and on the boundary of a circle centered at the origin and of radius $r$?

4. Minkowski's Fundamental Theorem and consequences: The theorem, loosely states, that a symmetric, convex set centered at the origin that has area greater than or equal to 4 contains lattice points (besides the origin). Using this theorem, we can approximate within a given tolerance, real numbers by rational ones.

5. And more...

This is a one-term thesis topic.

Prerequisite: Math 235 or 221 or Math 224 or permission of instructor.

Math Department web pages Created: 25 Apr 2007 Last modified: Apr 25, 2007 6:04:08 PM Comments to: math@union.edu

Thesis Topics - P. Friedman Thesis Topics for 2007-2008 Thesis Topics - K. Black