Geometry Course, Fall 2009

The first day of class, 8/31/09

We talked about the course syllabus, and about the history of geometry. A handout was given in class; if you missed it please refer to Lectures 26 and 27 of the book called Great Moments in Mathematics. "Homework" for the day is to read the handout and make a little summary of who the key mathematicians in the history of geometry were, and what their major contribution to geometry was. Bring your summary to the problem-solving session next week!

The second day of class, 9/1/09

We began the class by clarifying that Saccheri's hypothesis of the obtuse angle (see the handout from the last class for details) actually corresponds to the geometry of a surface of a sphere. We observed that "lines" in the spherical geometry "are" great circles, and that as such any two "lines" in spherical geometry intersect. Studying geometries in which every two lines intersect is really not all that far removed from "real life": artists and architects from Italian Renaissance were the first to study perspective. To illustrate the notion of perspective we made a drawing of a house in two-point perspective. For homework you should draw a house in three-point perspective (due Tuesday!). If you find you have troubles with drawing in perspective you may want to consult a book in the library called Perspective Made Easy. We also continued with our agenda of understanding the philosophy of math. The question we tried to answer is why Greek mathematicians insisted on deductive reasoning as opposed to a more "observational" approach to math. The answer perhaps lies in Plato's school of thought. We read a few paragraphs from Chapter 2 of Philosphy of Mathematics by J.R. Brown. Homework: try to get a sense for Plato's philosophy of math. You are not expected to write anything, only to be able to discuss it on Tuesday!

Introduction to axiomatizing geometry, 9/3/09

We had two visitors in class today, so we began by recapping what we did the first two days of class. We also digressed to talk a little bit about Kant's philosophy of Space and Time. If you missed the handout on Kant please consult the book called Space Through the Ages. The main goal of today's lecture was to get a better feeling for what organizing geometrical knowledge really entails. We used a toy example which arises from studying drawing in perspective. Please consult the lecture notes for further details. No homework was assigned today.

Introduction to axiomatizing geometry (Desargues' Theorem), 9/4/09

We continued to play will our toy example of axiomatic geometry. We spent most of the day applying Desargues' Theorem to situations arising from drawing in 3-point perspective. Please consult the lecture notes for further details. The homework assignment can be found at the end of the lecture notes; it is due on Tuesday 9/8.

Hilbert's Formalism, 9/10/09

In this lecture we study several examples of simple axiomatic systems according to the principles of Hilbert's Formalism. The discussion is partly based on the handout about Hilbert and Goedel from the book Philosphy of Mathematics by J.R. Brown. Here are some lecture notes on the subject. The homework problems are listed at the end of the lecture notes; the first draft of the assignment is due on Monday 9/14.

On Models, 9/11/09

The main goal of the lecture is to understand what it means for an axiomatic theory to be consistent, and how we go about showing this consistency. The supporting materials for the topic are: Section 1.2 of Judith Cederberg's "A Course in Modern Geometries", Charter 8 of Marvin Jay Greenberg's "Euclidean and non-Euclidean geometries: development and history", and the lecture notes. The homework assignment is given at the end of the lecture notes; its first draft is due on Monday 9/14.

Homework Day, 9/15/09

This was not quite my plan for the day, but we ended up spending a fair amount of time talking about the homework assignment due today. That's ok - being fluent in verifying something is an analytic model of geometric axioms is important!! At the end of the class we discussed Hilbert's legacy. Apart from his "Foundations in Geometry" and his subsequent work on formalism, a speech he gave on the current state of mathematics at a congress of mathematicians in 1900 ranks amongst the most influential things Hilbert had done. Here is a link to an article about the problems Hilbert posed at this congress. It was not uncommon around year 2000 for influential mathematicians to comment on the current state of mathematics. In class I handed out one article on the topic; it comes from a speech given by a former president of the American Mathematical Society. The speech refers to an earlier article which is written by one of the leading European mathematicians for the Report of the Senior Assessment Panel of the International Assessment of the U.S. Mathematical Sciences, National Science Foundation, March 1998. These are all highly recommended readings!! I pointed out that all these articles were written before 2003 which, as it turned out, is an important year in geometry! Your homework for the day is to find out what happened in 2003!!

On Current Trends in Geometry, 9/17/09

We talked about the most important development in geometry since the early 20th century: the Ricci Flow solution to the Thurston Geometrization Conjecture and ultimately - the affirmative answer to the century old Poincare conjecture. This event was not without any controversy. I handed out an article about the whole stink, and I gave a warning that the article is at least somewhat bias. In particular, certain mathematicians which are portrayed negatively in this article are to the best of my knowledge misrepresented. Following a little overview of what exactly happened, I tried to illustrate the whole idea of geometric flows in Riemannian geometry on an example. I am working on the notes on what I said in class (bad Iva!). We ended the class with a brief overview of Hilbert's (and essentially most recent) approach to studying Euclidean, synthetic geometry. Here is a handout on Hilbert's Axiomatization of geometry. The homework problems on the axiomatization are listed at the end of the handout.

Undefined geometric terms: betweenness and congruency, 9/18/09

There was a philosophical lesson to learn in this lecture: one needs to treat betweenness (for triples of points) and congruency (for two pairs of points) as undefined terms! Everything else however needs to be defined: including line segments, angles, triangles, congruency of angles and triangles etc.... A notion to take note of for the future is that of isometry. I am working on the notes on what exactly I said in class. Here is the link to the homework assignment. Also, make sure you pick up the take home "quiz" on the philosophy of studying mathematics and geometry.

First Theorems at the Level of Congruence - part I, 9/22/09

The synthetic part of last week's homework assignment did not go so well, and so we will spend an extra day this week on proving some basic theorems which hold true on the very primitive level at which we have incidence, betweenness, and congruency but no continuity or parallelism. Note that all the theorems we are going to prove this week also hold in hyperbolic geometry. Here are some notes on the subject. No new homework was assigned - you are to work on the problems you could not finish for the assignment due this week.

First Theorems at the Level of Congruence - part II, 9/24/09

This class was, in a sense, a continuation of the one from Tuesday. We mainly discussed the triangle inequalities. Do take note of the theorems we discussed in this class - we will use them often. Here are some notes on the subject and here is your homework assignment on triangle theorems.

First Theorems at the Level of Congruence - part III, 9/25/09

We continued with our plan for the week. One new notion we talked about is orthogonality. Here are some notes on the subject. No new homework was assigned. Consider this class as a wrap up of a topic - we move on to classifying isometries next time!

Discussing Isometries: Day 1, 9/29/09

The plan for this week is to introduce the basic isometries, get a good feeling for them, and prove a super-important classification theorem regarding isometries. On the first day we only define reflections, rotations, translations and glide reflections. Here is a link to some hand-written notes, and here is the link to the homework assignment.

Discussing Isometries: Day 2, 10/1/09

We digressed at the beginning of the class to talk about translations in this abstract setting. Up to this point everything we talked about in class matched with our intuition, and as long as we kept in mind what notions we get to work with there was little room for our intuition to misguide us. I brought to class a rough paper model of the hyperbolic plane geometry, which as an abstract geometry satisfies all the axioms and all the results we have thus far. However, translation on this paper model exhibited some curious features. These features will be properly addressed in the course once we get to the hyperbolic geometry, but I wanted to show to everybody where our intuition about translations might misguide us. The second part of the class we dedicated to an overview of the big theorem on classification of isometries (Fundamental Theorem of Synthetic Geometry?). Here is a link to some hand-written notes, and here is the link to the homework assignment. Parts of the assignment might be a little pre-mature and might require some familiarity with the content of the next class. Please note that the methods used in this homework assignment might be new to you. The primary purpose of the assignment is in practicing these potentially new techniques, and not in formally writing things we intuitively know.

Discussing Isometries: Day 3, 10/2/09

The goal of this class was to prove the classification theorem of isometries in all its details. The notes on the classification theorem can be found here. At the end of the class we also mentioned the idea of automorphism in general. This concept ties to the notion of an automorphism group, something we study in abstract algebra. Around 1870 Felix Klein proposed an approach to studying geometry based on the ideas of automorphism; Klein's speech on the topic became known as the Erlanger Program. Here are some notes on the Klein's Erlangen Program, but be warned that if you have not - or are not - taking abstract algebra the material might be too abstract. No new homework was assigned.

Measuring and Continuity 10/6/09

The main goal for the day was understanding what axioms of continuity say, what they intuitively mean, why we need them and where we use them. Here are some notes on the material. We will continue discussing these axioms and their consequences after the Fall Break!!

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