My name is Sherwood Hachtman, and I am an undergraduate at Rutgers University, where I study mathematics. I am presently investigating ultraproducts of the collection of finite symmetric groups, as part of the 2008 Rutgers Math REU. My mentors in this project are Paul Ellis and Scott Schneider.

When working with an infinite collection of models, one can use the ultraproduct construction to obtain a new model whose behavior "averages" (in a certain sense) the behavior of those in the collection. For example, let us denote the ring of integers with arithmetic modulo n by Zn. We may take an ultraproduct on the collection {Zn : n is a natural number} and thereby obtain a new, bigger, ring; call it ∏(Zn). What are the properties of this new structure? Łoś's Theorem tells us that a first-order sentence is true in this ultraproduct if and only if it is true in "almost all" of the models used to construct it. The meaning here of "almost all" is determined by the ultrafilter used in the construction. It is well-known that Zn is a field if and only if n is prime. Thus by Łoś's Theorem, ∏(Zn) is a field if and only if {n : n is prime} is in the ultrafilter used.

I am studying ultraproducts on the collection of finite symmetric groups Sn, and in particular, how the properties of the group obtained are determined by the ultrafilter used. As a starting point, I will try to ascertain how many nonisomorphic groups arise in this fashion. (There are at most 2c (where c is the cardinality of the real numbers), since there are precisely 2c ultrafilters on N, the set of natural numbers.) The answer to this question may depend on whether the Continuum Hypothesis (CH) is assumed, and therefore may be independent of the usual axioms of set theory (ZFC, for those in the know).

Forthcoming is an explanation of the terms used above, with background and facts of interest. In the meantime, please refer to the excellent Wikipedia articles on ultraproducts, ultrafilters, symmetric groups, and the Continuum Hypothesis.

Update on progress as of July 16, 2008: We have completed proof of the existence of exactly c nonisomorphic ultraproducts ∏(Sn) under the assumption of CH; the same holds for the ultraproduct over the collection of alternating groups, ∏(An).

More interesting is the observation that the normal subgroups of ∏(An) are in fact linearly ordered by the inclusion relation, ⊂! A nice isomorphism and a previous result then allow us to prove the consistency with ZFC of the existence of 2c nonisomorphic ultraproducts. I'm currently composing a hopefully cogent and pretty document explicating these results. That will in all likelihood be posted here upon its completion.

Update of August 5, 2008: A paper inexplicably bearing my name has been completed and submitted for publication. It is quite good, and I am excited.

Also, for those interested, here are some hastily prepared slides from a 10-odd minute presentation I delivered on July 17.

First Second Thard Fourth

Disclaimer: My primary purpose in this talk was to thoroughly motivate and define the basic notions of ultrafilter and ultraproduct. I avoided delving deeply into our results. As originally presented, these slides were supplemented by the use of a whiteboard and verbal communication. Without these things, they make much less sense. To best simulate the genuine experience, you can imagine them read aloud in a hurried, nervous monotone.

Please feel free to contact me with questions, corrections, praise, or polite criticism at hachtman[at]eden[dot]rutgers[dot]edu.

ul•tra•prod•uct [ul-truh-produkt] -noun. Like a product, but better.