Browsing by Author "Welch, Amanda Renee"
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- Characterizing Zero Divisors of Group RingsWelch, Amanda Renee (Virginia Tech, 2015-06-15)The Atiyah Conjecture originates from a paper written 40 years ago by Sir Michael Atiyah, a famous mathematician and Fields medalist. Since publication of the paper, mathematicians have been working to solve many questions related to the conjecture, but it is still open. The conjecture is about certain topological invariants attached to a group 𝐺. There are examples showing that the conjecture does not hold in general. These examples involve something like the lamplighter group (the wreath product ℤ/2ℤ ≀ ℤ). We are interested in looking at examples where this is not the case. We are interested in the specific case where 𝐺 is a finitely generated group in which the Prüfer group can be embedded as the center. The Prüfer group is a 𝑝-group for some prime 𝑝 and its finite subgroups have unbounded order, in particular the finite subgroups of G will have unbounded order. To understand whether any form of the Atiyah conjecture is true for 𝐺, it will first help to determine whether the group ring 𝑘𝐺 of the group 𝐺 has a classical ring of quotients for some field 𝑘. To determine this we will need to know the zero divisors for the group ring 𝑘𝐺. Our investigations will be divided into two cases, namely when the characteristic of the field 𝑘 is the same as the prime p for the Prüfer group and when it is different.
- Double Affine Bruhat OrderWelch, Amanda Renee (Virginia Tech, 2019-05-03)Given a finite Weyl group W_fin with root system Phi_fin, one can create the affine Weyl group W_aff by taking the semidirect product of the translation group associated to the coroot lattice for Phi_fin, with W_fin. The double affine Weyl semigroup W can be created by using a similar semidirect product where one replaces W_fin with W_aff and the coroot lattice with the Tits cone of W_aff. We classify cocovers and covers of a given element of W with respect to the Bruhat order, specifically when W is associated to a finite root system that is irreducible and simply laced. We show two approaches: one extending the work of Lam and Shimozono, and its strengthening by Milicevic, where cocovers are characterized in the affine case using the quantum Bruhat graph of W_fin, and another, which takes a more geometrical approach by using the length difference set defined by Muthiah and Orr.