Category Archives: geometry

Vector question

Dear Professor.

I want to ask you a maths question. Hope you don’t mind.

What is the difference between

|C| = \frac {A \cdot B} {|B| },

and

C = \frac {A \cdot B} {|B|^2} B ?

Where C is the projection of A on B.

I understand that the first expression is the magnitude of C, which is the length of the projection, am i right?

However, I’m uncertain about the second expression and to what significance it holds. Please enlighten me regarding this.

Thank you very much! Your help is much appreciated.

Take care!

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Reply:

You are right about the length of the projection. Perhaps the best way to see the meaning of the second expression is to write it as:

C=(A \cdot (\frac{B}{|B|}))\frac{B}{|B|}

Here is the point: Given any vector W, I believe you know how to visualize its projection along any directed line L. For simplicity, I will assume that W forms an acute angle with the direction of the line. The projection is then the shadow of W you see when you shine a light on L along a perpendicular angle. It is an elementary exercise with a triangle to see that the length of this projection is exactly

W\cdot e_L,

where e_L is a *unit vector*, that is, a vector of length 1, in the direction of L. You should think of a unit vector as encoding ‘pure direction.’ So to get a vector of magnitude r in the direction of the unit vector E, you can just dilate E by a factor of r to get

X=rE.

Conversely, to change any non-zero vector V into the unit vector in the same direction, you divide by its magnitude to get

\frac{V}{|V|}.

Note now the key fact that the projection of W itself, not just its magnitude, is given by the formula

(W\cdot e_L)e_L

since the projection is clearly in the direction of e_L and this expression gives such a vector with just the right magnitude.

To return to your question, given any two vectors W and V, you can project W in the direction of V. Now, the unit vector in the direction of V is exactly

\frac{V}{|V|}.

So the projection of W in the direction of V is nothing but

(W\cdot (\frac{V}{|V|}))\frac{V}{|V|}.

By the way, your question is a very good one. It’s very important to understand the geometric meaning of basic expressions like this. Otherwise, everything degenerates into a bunch of formulas.

Student reports, end of August

Because of a rather hectic travel schedule, I was slow in putting up the reports that were submitted assiduously by Zhe and Alex. I apologize.

Zhe’s reports on the book `Riemann’s Zeta Function’ by Edwards:

Zhe’s report 4

Zhe’s report 5

Zhe’s report 6

Zhe’s report 7

Alex’s reports on elliptic curves:

Points of finite order (exercises)

Group of rational points

Mordell’s theorem

Student Report 01/07/08

Alex’s report 3

Student reports

Acyr’s report 2

Alex’s report 2

Summer projects for students

Three students are in the midst of summer reading projects under my supervision, Acyr Locatelli, Alex Tao, and Nikhil Mehrotra.

Acyr is reading Ideals, Varieties, Algorithms by Cox, Little, and O’Shea, a very nice introduction to computational algebraic geometry.

Alex is reading Rational Points on Elliptic Curves by Silverman and Tate. This book deals with integral and rational solutions to cubic equations like

y^2=x^3+x-1000.

This class of problems, seemingly limited in scope, turns out to have many remarkable attendant structures, making it one of the most important directions of contemporary number theory.

Nikhil is reading Goedel, Escher, Bach by Douglas Hofstadter, a great classic on consciousness, mathematics, and the possibility of artificial intelligence.

I will be posting here their weekly reports, in case someone else wants to follow along. For now, here are

Alex’s report 1

Acyr’s report 1

Kings lecture

Since arriving in the UK, I gave colloquium lectures at QMUL, Leeds, Durham, and Exeter. A `colloquium’ is when you lecture for about an hour to the whole mathematics faculty on the topic of your research. It’s considerably harder to prepare than specialized seminars (for example, the London-Paris number theory seminar) where you are speaking to people whose expertise is typically similar to your own. That is to say, the background of the audience is much more varied for a colloquium. Anyways, I acquired this year the bad habit of preparing my lecture at the last minute on the train. Probably because of this, the four colloquium talks turned out disastrously. I couldn’t stay focussed, and I think I lost everyone pretty soon after I started. This is in spite of the fact that I was constantly recycling material from the previous colloquium. Anyways, today I delivered a colloquium lecture at Kings, and my impression was that it was finally tolerable on the fifth try. So I’m putting a link here. Because these are just summaries of what was said, it probably looks no different from the previous lectures if you weren’t there, and maybe even if you were.

Kazuya Kato

The great arithmetician Kazuya Kato visited twice over the last few weeks, so I thought I’d use the occasion to recommend some writings. An undergraduate level textbook on number theory is

Number Theory I: Fermat’s Dream

published by the American Mathematical Society. It is short and covers fairly standard material, but contains many unusual insights. A research article that represents quite well Kato’s vision of number theory is

Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. Part I

available online if your institution subscribes to Springer. The background required for a genuine reading of the paper is rather extensive, but even without it, you can enjoy the introduction, the first few sections, and the closing remarks.

A main theme of the work is that the somewhat mysterious p-adic zeta functions and L-functions are the objects with direct relevance to the important problems of arithmetic geometry, while the usual complex functions are a sort of intermediary. It took me a long time to come to terms with this view, especially since I still don’t understand these functions (actually elements of some non-commutative algebra of measures) at all well, but it is eventually an essential component of my own thoughts about Diophantine geometry.

MK

Fundamental groups and Diophantine geometry

A few weeks ago, I gave a colloquium lecture at Leeds university and subsequently wrote up an exposition based on it. It’s still not entirely `popular,’ but may give a somewhat better sense than my previous remarks of at least a few ideas.

Let me know if there are some points on which you would like elaboration.

MK

Southwest Center for Arithmetic Geometry

It’s something of a constant concern of mine to put students of every level in touch with mathematical culture. So I thought I’d call your attention to the Southwest Center for Arithmetic Geometry where I did most of my public service work from 1998 to 2005. If you browse through, you’ll find ten years’ worth of surveys, projects, videos, etc. covering virtually all aspects of arithmetic geometry. They host an annual

Arizona Winter School on Arithmetic Geometry

for post-graduate students. Students doing an MSc or above at UCL could benefit from attending sometime. It’s a really great educational event and a chance to interact with first-rate mathematicians as well as students from all over the world.

MK

Non-commutative constructions

In June of next year, there will be a small workshop at UCL with the title

`non-commutative constructions in arithmetic and geometry.’

I will write later in more detail about the goals of the meeting, but here is a link to a preliminary web page.

MK