Below are some notes written for presentations, mostly given at the graduate/research level. Notes for talks given at the undergraduate level (with the exception of the talk given May 17, 2019) are provided on the “Additional resources/links” page.
My notes on Haar measure that were for a presentation given in Dr. Falkner’s 6212 class during spring semester 2014.
My notes on completions, the I-adic topology, and integer-valued polynomials that were for two presentations given for the Commutative Algebra seminar on 1/26/15 and 2/9/15.
Notes on ultrafilters, ultraproducts, and some of their applications to commutative algebra presented to the UCR Commutative Algebra seminar on 5/22/15.
Talk on Ultrafilters in Commutative Algebra for the Graduate Student seminar 11/3/2015. This is a modestly improved version of the notes for the talk given in May 2015, adjusted so that the Commutative Algebra prerequisites were not assumed.
Dissertation proposal. This is the written portion for my candidacy exam (taken February and March 2016), which describes the motivation behind my research and the directions that I would like to go with it.
Talk on the historical origins of ideal theory for the Graduate Student Seminar on 11/28/2017. This explores how some historical questions on Diophantine equations like Fermat’s Last Theorem and Bachet’s equation, as well as quadratic forms and reciprocity laws, naturally led to the definition of an “ideal” in the sense of modern abstract algebra and ring theory. The talk mostly follows a paper of Harold Edwards, and it is meant to be accessible to a beginning math graduate student, not necessarily a specialist in algebra or number theory.
This is, in my opinion, a significantly improved version of the talk given in the OSU Graduate Student Seminar in 2017. Some information and topics were omitted from the 2017 talk, while other topics (specifically the examples) were done in greater detail so that an undergraduate audience could follow. Some of the content (namely Kummer’s first definition of “ideal primes” by means of constructing a “g_q” and the related discussion on norms) was briefly glossed over in the actual talk. [Note: The image on slide 13 is taken from here.]
Slides for my talk on MacLane’s, Inoue’s, Loper and Tartarone’s, and my own research on the construction of valuations defined on fields of rational functions in 1 and 2 variables over a field K that when restricted to K define a discrete valuation on K. This is a talk I gave for the graduate student seminar at OSU on January 21, 2020. It begins by summarizing the known results classifying all valuations on K(x) extending a discrete valuation on K as either inductively defined (potentially via an infinite sequence) or via pullbacks of inductively defined valuations found by Saunders MacLane, K. Alan Loper, and Francesca Tartarone. Then, after giving a few results on the utility of this classification toward the classification of certain integrally closed domains inside K[x] that can be represented as an intersection of certain classes of valuation rings defined by valuations in the ways mentioned in the preceding sentence, the talk concludes by summarizing the great deal of care that is needed to try to find analogues to these results on constructing valuations and classifying integrally closed rings to the 2 variable case. Some original (and heretofore unpublished as of January 2020) results are given as well in this conclusion, as this topic is the subject of my dissertation. It is my hope that these slides may be followed by someone with the background in algebra obtained in a typical graduate algebra course, not necessarily a specialist in commutative algebra or valuation theory.