Mathematics Colloquium

WHEN:
Friday April 29, 2:30-3:30PM

WHERE:
Durham Science Center, Room 255

WHAT:

ABSTRACT:

The factorial function, n!, is a ubiquitous thing in mathematics, as it appears in many different disciplines, like number theory, calculus, abstract algebra, combinatorics, etc.. Here's one such occurrence: Prop. Consider n+1 arbitrary integers. Then the product, P, of their pairwise differences is divisible by the product 0!1!2!...n! (and this is sharp, i.e. there is no larger divisor that will work universally). Now, rather than considering n+1 arbitrary integers, one could consider n+1 arbitrary elements of some specific subset S of the integers (like the even numbers, or the set of primes). In this new setting, one could seek a similar result to the above, i.e. a divisor of P that is sharp, but what should replace 0!1!...n! ? In a 2000 article, Manjul Bhargava constructed a "generalized" factorial function, n!_S, defined for a subset, S, of the integers. It bears many of the properties we would expect of a factorial function, and it also answers the above question (the divisor should be 0!_S 1!_S...n!_S) and many more. In this colloquium talk, I will introduce and define this factorial function, explain how it generalizes a number of classical number theory and algebra problems, and discuss some of its properties and some open questions surrounding it. The talk should be accessible to everyone.

    Back to the Mathematics Colloquium Page