Abstract:

Number theory is the study of completely discrete things that can be counted or numbered, while topology is the study of continuous things and of properties that are not changed by deformation. So it is perhaps surprising that questions of pure topology lead to surprisingly subtle questions of number theory.   In this talk I will sketch several instances of this phenomenon, starting with Euler's discovery of a discrete invariant (now called the Euler characteristic) of a topological space can be computed by triangulating it any way, and then move on to more sophisticated examples: the "Betti numbers" of a manifold (roughly speaking, the number of k-dimensional "holes"  for each integer k up to the dimension of the manifold), the appearance of Bernoulli numbers in connection with so-called exotic spheres in higher dimensions, the appearance of Dedekind sums (another simple number-theoretical object going back to the 19th century) in the study of 4-dimensional manifolds, and most recently a whole series of discoveries relating "quantum invariants" of knots to various deep number-theoretical concepts (units, K-theory, Bloch groups, Habiro ring,...). This last is ongoing research in collaboration with Stavros Garoufalidis in SUSTech. The talk will be at a very general level and will not require any prior knowledge.  In particular, rough explanations of all of the various technical words occurring above will be given. 

In the Math Center seminar talk that I will give on December 6, I will tell more about one of the specific questions appearing above, namely, the question of the possible Betti numbers that can occur for smooth n-dimensional manifolds.  The answer here turns out to involve elementary but quite hard number theory, and is quite amusing.  This talk, too, is meant to be understandable by good undergraduates or by mathematicians in all fields.