Joint PU/IAS Number Theory

In my recent work on a geometric proof of the endoscopic fundamental lemma for spherical Hecke algebras, there are many new features not present in its Lie algebra analogue originally proved by B.C.~Ng\^o. One of such new features is an asymptotic version of the fundamental lemma, formulated using the theory of Kashiwara crystals. It is at its core a purely combinatorial version of the original fundamental lemma and implies the latter when combined with results about multiplicative Hitchin fibrations. I will explain what the asymptotic fundamental lemma is and why it may be promising to study in a broader context like functoriality. The talk is meant to be made accessible.