User dirk basson - MathOverflow

Which topics/problems could you show to a bright first year mathematics student?

2011-03-12T10:51:50Z

I am teaching a one semester course (January to June) to first year students pursuing various different degrees. Because there are students studying actuarial science, physics, other sciences, other management or business sciences etc., the course has to be a generic one. By this I mean that we teach Calculus almost exclusively. Sure, there are topics like the Binomial Theorem and general remarks of proving theorems, but students that are interested in mathematics don't find the material particularly interesting. What is more, I can relate to them since I found the first two years of university mathematics somewhat boring. This included calculus, linear algebra, convergent/divergent sequences, multiple integrals etc. Real analysis (except for sequences), complex analysis, abstract algebra, topology and even elementary number theory do not appear until the third year.

The sad thing is that many students take mathematics only up to second year and do not get to see any of the "cool"/"interesting" mathematics even though they might be interested in mathematics. So I wondered: Is there any way of introducing "interesting" mathematics to them? ("Them" - in particular first years, but this question is also relevant for second years, who might have a higher degree of maturity.)

Some things that I (and the other lecturers of this course) have thought of are:

Adding challenging questions to tutorials (e.g. IMC or Putnam, though these are harder than we would like)

Writing short introductory "articles" about fields or groups or perhaps the Euler characteristic (as an introduction to topology) etc. Of course, this is very idealistic, since one often doesn't have time or energy to do this.

Referring them to library books where some of these things are explained. Also, quite idealistic, but how many will actually go to the library.

The best solution is probably to combine the three. Have a question which has a strange answer or solution, which can be explained by some interesting mathematics. Shortly explain how this is done, and have a reference where the student can go if he is interested enough to pursue it further.

Are there any other ways of achieving this goal? Do you know of any questions to which this (combined) procedure can be applied?

Which areas of arithmetic algebraic geometry can be learned as "black boxes" and are there any references where they are treated as such?

2011-02-15T19:20:24Z

In Matthew Emerton's comment on Terry Tao's blog, he speaks about learning etale cohomology or the theory of Neron models as "black boxes". By this he means that you can learn what the theory is about and how to use it, without going into the detailed proofs of why they can be used.

Which theories (e.g. etale cohomology) can be learned as black boxes?

And where would one go (e.g. find lecture notes) to learn something like that?

Notes on something like this would ideally give you an idea of what is going on, give examples, and most importantly illustrate how they would be used to solve problems. I am mainly interested in arithmetic algebraic geometry and algebraic number theory, so I would especially like to know about "black boxes" in this direction, though "black boxes" in other areas might also be worth knowing about.

Answer by Dirk Basson for Video lectures of mathematics courses available online for free

2011-02-06T07:17:38Z

This might not fulfill the requirements of being a mathematics course, but I think that it is close enough. In 2006 the Clay Mathematics Institute hosted a Summer School in Arithmetic Geometry. The videos are great if you have a solid foundation in algebraic geometry already and wish to continue in the direction of arithmetic geometry .

Answer by Dirk Basson for how is the complexity to solve `$x^a=b(mod p)$` compared with the complexity of solving discrete logarithm

2011-01-29T14:31:10Z

If $a$ is relatively prime to $p-1$, you can solve for $c$ from $ac\equiv 1$ (mod $p-1$) and raise both sides to the power $c$. You obtain $x\equiv x^{ca}\equiv b^c$ (since $x^{p-1}\equiv 1$) (mod $p$).

If $a$ is not relatively prime to $p-1$, the congruence is not always solvable. However, there should be a least $d$ for which $ac\equiv d$ (mod $p-1$) has a solution and raise both sides to the power $c$. Then one has the congruence $x^d\equiv b^c$ (mod $p-1$). In most cases this will be simpler than the original congruence. The congruence has no solution if $b$ is not a $d$-th power modulo $p$. It should be clear that $d$ will be a divisor of $p-1$ (since it can be computed as the gcd of $a$ and $p-1$) and that $b$ will be a $d$-th power if $b^{p-1/d}\equiv 1$ (mod $p-1$).

That is as far as I can help.

Answer by Dirk Basson for Books you would like to read (if somebody would just write them...)

2011-01-24T11:36:09Z

Galois representations.

I know about Serre's Abelian $\ell$-adic representations and elliptic curves, but I am sure that a more general theory has been established since then. There are a few people who have notes on Galois representations on their web pages, but no book that I know of.

Answer by Dirk Basson for Is the ABC conjecture known to imply the Riemann hypothesis?

2010-04-29T22:09:06Z

Here is a link to many consequences of the ABC-conjecture.