I went to the three “big” talks in the morning.
Dawn Lott’s talk was about applied mathematics and its relation to the study of brain aneurysms; in particular the aneurysm model was discussed (partial differential equations with a time coordinate and stresses in the radial, circumference and latitudinal directions were modeled).
There was also modeling of the clipping procedure (where the base of the aneurysm was clipped with a metal clip); various clipping strategies were investigated (straight across? diagonal?). One interesting aspect was that the model of the aneurysm was discussed; what shape gave the best results?
Note: this is one procedure that was being modeled:
Next, Bhargava gave his second talk (on rational points on algebraic curves)
It was excellent. In the previous lecture, we saw that a quadratic curve either has an infinite number of rational points or zero rational points. Things are different with a cubic curve.
For example, has exactly one rational point (namely (0,0) ) but has an infinite number! It turns out that the number of rational points an algebraic curve has is related to the genus of the graph of the curve in (where one uses complex values for both variables). The surface is a punctured multi-holed torus of genus with the punctures being “at infinity”.
The genus is as follows: 0 if the degree is 1 or 2, 1 if the degree is 3, and greater than 1 if the degree is 4 or higher. So what about the number of rational points:
0 or finite if the genus is zero
finite if the genus is strictly greater than 1 (Falting’s Theorem; 1983)
indeterminate if the genus is 1. Hence much work is done in this area.
No general algorithm is known to make the determination if the curve is cubic (and therefore of genus 1)
Note: the set of rational points has a group structure.
Note: a rational cubic has a rational change of variable which changes the curve to elliptic form:
Weierstrauss form: where are integers.
Hence this is the form that is studied.
Sometimes the rational points can be found in the following way (example: :
note: this curve is symmetric about the axis.
is a rational point. So is . This line intersects the curve in a third point; this line and the cubic form a cubic in with two rational roots; hence the third must be rational. So we get a third rational point. Then we use to obtain another line and still another rational point; we keep adding rational points in this manner.
This requires proof, but eventually we get all of the rational points in this manner.
The minimum number of “starting points” that we need to find the rational points is called the “rank” of the curve. Our curve is of rank 1 since we really needed only (which, after reflecting, yields a line and a third rational point).
Mordell’s Theorem: every cubic is of finite rank, though it is unknown (as of this time) what the maximum rank is (maximum known example: rank 28), what an expected size would be, or even if “most” are rank 0 or rank 1.
Note: rank 0 means only a finite number of rational points.
I enjoyed many of the short talks. Of note:
there was a number theory talk by Jay Schiffman in which different conjectures of the following type were presented: if is some sequence of positive integers and we look at the series of partial sums, or partial products (plus or minus some set number), what can we say about the number of primes that we obtain?
Example: Consider the Euclid product of primes (used to show that there is no largest prime number)
etc. It is unknown if there is a largest prime in the sequence .
Another good talk was given by Charlie Smith. It was about the proofs of the irrationality of various famous numbers; it was shown that many of the proofs follow a similar pattern and use a series of 3 techniques/facts that the presenter called “rabbits”. I might talk about this in a later post.
Another interesting talk was given by Jack Mealy. It was about a type of “hyper-hyperbolic” geometry called a “Snell geometry”. Basically one sets up the plane and then puts in a smooth closed boundary curve (say, a line or a sphere). One then declares that the geodesics are those that result from a straight lines…that stay straight until they hit the boundary; they then obey the Snell’s law from physics with respect to the normal of the boundary surface; the two rays joined together from the geodesic in the new geometry. One can do this with, say, a concentric series of circles.
If one arranges the density coefficient in the correct manner, one’s density (in terms of area) can be made to increase as one goes outward; this can lead to interesting area properties of triangles.