Serre Reciprocity Conjecture

These are notes from C. S. Dalawat's lecture on the Serre Reciprocity Conjecture at H.R.I and from the draft of an article written by him.

The conjecture relates representations of Gal(Qbar/Q) and modular forms. The "level 1" case of this has been proved by Chandrashekhar Khare, "by carrying out a strategy he had worked out in collaboration with Jean-Pierre Wintenberger, and using results by Luis Dieulefait and others". This apparently breaks " a psychological barrier".

In the following we shall attempt to present, somewhat concisely, the statement of this conjecture. For a pedagogical article suitable for (advanced) high school students, click here (postscript format).

Let us get a few terms defined: Everyone knows what a field is;

• the field G of Gaussian numbers is the set {a+bi, a,b rational} and that Aut(G)=Z/2Z (observe that since polynomials are inv under Aut, i-> -i)

• the set of algebraic numbers A, i.e., those that are roots of polynomials also forms a field and what about Aut(A)=? - this is the question. For instance, all finite order elements in Aut(A) are order 2 !
  

A (sub)field is called Galoisian if the only numbers invariant under all of Aut are the rationals. (there is apparently a conjecture that every finite group G appears as the Aut of some galoisian subfield of A)

Finite Fields and Local Fields

An interesting Math paper

Unique Decomposition of tensor products of irreducible representations of simple algebraic groups

We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.

A trek idea

Last few dreams have been about trekking --
start from Bhatwari - Dayara - Dodi Tal - Darwa Pass - Yamunotri - Roopkund - Har-ki-dun - Borasu/Rupin- Sangla- Chitkul-Charang la - Charang-Thangi- Rekong Peo and out.

Sounds lovely - and its even doable - especially if one starts from Peo. If you stop at Har ki Dun its realistic even for me :-)

Quote

God may not play dice with the Universe, but there's something strange going on with the prime numbers! Paul Erdos

Some quotes

• It seems sometimes that through a preoccupation with science, we acquire a firmer hold over the vicissitudes of life and meet them with greater calm, but in reality we have done no more than to find a way to escape from our sorrows Hermann Minkowski in a letter to David Hilbert
  


• Andre Weil on how the charm of an analogy evaporates when you find a generalization that encompasses both terms:
  The day dawns when the illusion vanishes; intuition turns to certitude; the twin theories reveal their common source before disappearing; as the Gita teaches us, knowledge and indifference are attained at the same moment. Metaphysics has become mathematics, ready to form the material for a treatise whose icy beauty no longer has the power to move us.
  

Some interesting looking papers

• Background independent geometry and Hopf cyclic cohomology by Connes and Moscovici: The intriguing comment (from the abstract) is "the geometric framework that allowed us to treat the `space of leaves' of a general foliation provides a `background independent' set-up for geometry that could be of relevance to the handling of the the background independence problem in quantum gravity"

• From Primordial quantum fluctuations to the anisotropies of the CMBR : A detailed review - maybe useful someday.
  

• A Tractable Example of Perturbation Theory with a Field Cutoff: the Anharmonic Oscillator Perhaps something interesting about the simplest qft


• Between classical and Quantum : This is a review of how classical mechanics arises from Quantum mechanics ( h-> 0, or large number so that decoherence and consistent histories scheme holds)
  

There! I did it.

Didn't expect to succumb - but this could function like a refrigerator door - for those yellow post-its...

I would like to link to articles (with/out comments) which I find interesting.