Math 40960: Topics in Geometry
Spring 2014

Instructor: Juan Migliore
Office: HAYE 236.
Phone: 631-7345

Office Hours:
Monday, 1:00 - 2:00
Tuesday, 1:00 - 2:00
Or by appointment.

Time and place of class: MWF 9:25-10:15, Hayes-Healy 125

Textbooks:

Course overview: The purpose of this course is to examine different kinds of geometries and how they are related. From the preface of Stillwell's book:

In this book, I wish to show that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them.

In our other primary source, we will follow the exposition of one of the great geometers of our time, Robin Hartshorne. According to Hartshorne,

Euclid does not adhere to the strict axiomatic method as closely as one might hope. Certain steps in certain proofs depend on assumptions that, however reasonable or intuitively clear they may seem, cannot be justified on the basis of the stated postulates and common notions ... These lapses in Euclid's logic lead us to the task of disengaging those implicit assumptions that are used in his arguments and providing a new set of axioms from which we can develop geometry according to modern standards of rigor.
This leads to different kinds of non-Euclidean geometries and connections to algebra, including finite geometries. We will not attempt to closely follow the historical development of the subject.

Finally, from personal bias, we will focus on projective geometry and the notion of duality. Some of the material will come from the second book by Hartshorne listed above. If time permits we will delve into algebraic geometry.

Examinations, homework and grades. There will be three problem sets and a final exam. The problem sets can be accessed below. THE PROBLEM SETS WILL BE UPDATED REGULARLY. I WILL LET YOU KNOW WHEN THE CURRENT VERSION IS IN FINAL FORM. DON'T ASSUME THAT THE LINK YOU SEE NOW IS IN FINAL FORM BEFORE I SAY THAT IT IS!!!

How you will be evaluated: Your course grade will be based on your total score out of 450, with points allocated as follows:

Homework and Reading: There will not be any homework assigned apart from the problem sets. In most of my lectures I will try to follow the material in the textbooks, but this will not always be the case, and we will not necessarily follow the order in the textbook. The problem sets will be based primarily on class lectures, but possibly also from the material in the chapters. Thus it is important to read the material as well as following the lectures. You are responsible for keeping up with the material and getting your problem sets in on time. You should always feel free to ask questions in and out of class about any of the material that you do not understand.

Honor Code Recall the Academic Code of Honor Pledge: ``As a member of the Notre Dame community, I will not participate in or tolerate academic dishonesty.'' Both the final exam and the problem sets are conducted under the Notre Dame Honor Code. You are not allowed to discuss the problem sets with your classmates, although you are strongly encouraged to come talk to me if you are having any problems.