PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Stephen Simpson, Jason Rute, Jan Reimann.

Title:Reverse mathematics, Young diagrams, and the ACC
Seminar:Logic Seminar
Speaker:Stephen G. Simpson, Pennsylvania State University
Abstract:
In abstract algebra, a ring is said to satisfy the ACC (ascending chain condition) if it has no infinite ascending sequence of ideals. According to a famous and controversial theorem of Hilbert, 1890, polynomial rings with finitely many indeterminates satisfy the ACC. There is also a similar theorem for noncommuting indeterminates, due to J. C. Robson, 1978. In 1988 I performed a reverse-mathematical analysis of the theorems of Hilbert and Robson, proving that they are equivalent over RCA_0 to the well-orderedness of (the standard notation systems for) the ordinal numbers omega^omega and omega^{omega^omega} respectively. Now I perform a similar analysis of a theorem of Formanek and Lawrence, 1976. Let S be the group of finitely supported permutations of the natural numbers. Let K[S] be the group ring of S over a countable field K of characteristic 0. Formanek and Lawrence proved that K[S] satisfies the ACC. All of these results concerning the ACC involve well partial ordering theory. I now prove that the Formanek/Lawrence theorem is equivalent over RCA_0 to the well-orderedness of omega^omega. The proof involves an apparently new, combinatorial lemma concerning Young diagrams. I also show that, in all of these reverse-mathematical results, RCA_0 can be weakened to RCA*_0. This recent work was done jointly with Kostas Hatzikiriakou.

Room Reservation Information

Room Number:MB315
Date:03 / 04 / 2014
Time:02:30pm - 03:45pm