How to ask an important scholar to look at your tentative work, when you are just starting out.

How to ask an important scholar to look at your tentative work, when you are just starting out.

1. In effect you are showing your dirty linen in public, but presumably to an interested viewer.

2. The mathematician Robert Langlands (from British Columbia, Yale PhD) was starting out in his career at Princeton and he had spoken briefly to Andre Weil (professor at the Institute of Advanced Study) about some of his work. He followed up with a famous 16 page handwritten letter (good penmanship, by the way) describing what he was up to. Here is what Langlands wrote Weil--

Your opinion of these questions would be appreciated. I have not had a chance to think over these questions seriously and I would not ask them except as the continuation of a casual conversation. I hope you will treat them with the tolerance they require at this stage. 

If you are willing to read it [this letter] as pure speculation I would appreciate that; if not — I am sure you have a waste basket handy. ...

Eventually, Langlands' ideas crystallized in what is called The Langlands Program and The Langlands Conjectures. Some of its results were crucial to Andrew Wiles' work that led to the proof of Fermat's Last Theorem. 

One way of describing the Program is to say that it connects number facts with function facts, much as the factorials are connected to the sine function. (Recall 

sin x = x -x^3/3! + x^5/5! -x^7/7!

and n! = n(n-1)(n-2)... 1.) Studying the function's properties will tell you about the number facts, studying the number facts will tell you about the function's behavior.