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23 votes
4 answers
4k views

What information is contained in the Kazhdan-Lusztig polynomials?

The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations. For example the character of a simple module over a Lie algebra with Weyl group $W$ ...
16 votes
2 answers
539 views

Surprising appearances of Painlevé transcendents

What are some of your favorite examples of enumerative problems whose answer ended up being (related to) a solution to one of the Painlevé equations? I have seen examples from enumeration of classes ...
15 votes
5 answers
2k views

Striking existence theorems with mild conditions, and simple to state: more recent examples?

I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild ...
9 votes
4 answers
560 views

When is it easier to work projectively?

There are many instances in which theory over $\mathbb{C}$ is cleaner than theory over $\mathbb{R}$. For example, continuously differentiable functions over $\mathbb{R}$ are not necessarily twice ...