Hopf algebras are known to have a copious ammounts of structure which makes them useful in studying representations while, on the other hand, groups come equipt with little structure. In the following we will “realize” Hopf algebras as groups and comment on why we call quasitriangular Hopf algebras quantum groups. Hopefully, this will help demistify the confusion that comes along when studying Hopf algebras. For those of you who don’t know Hopf algebras, don’t worry! We will review them as well.
Continue readingAuthor: Henry Twiss
Presentation: Using Holonomy to Understand Riemannian Manifolds
This is the script for a presentation I gave for a class in Riemannian Geometry at the University of Minnesota in the Fall of 2019. In it, I dicuss the basics of holonomy, history of the theory, de Rham’s decomposition theroem, the Ambrose-Singer theorem, and Berger’s classifiction.
Continue readingPresentation: The Root Datum of Special Linear and Projective Linear Groups
This is the script for a presentation I gave for the Directed Reading Program at the University of Minnesota in the Fall of 2019. In it, I dicuss how the root datum is an extension of the root system, and how we can use it to distinguish between linear algebraic groups.
Continue readingPresentation: Asymptotics of Moments of Dirichlet L-series and Infinite Kac-Moody Lie Algebras
This is a presentation I gave for the Student Number Theory Seminar at the University of Minnesota in the Fall of 2019. In it, I discuss current research I am pursuing with Adrian Diaconu at the University of Minnesota.
Continue readingMeasurable Spaces & Topological Spaces, an Analogy
While I was in Indiana a few weeks ago at Notre Dame for their Geometry & Topology conference I had a very enlightening conversation with my roommate at the time (and now good friend). He had come across a small section in one of Rudin’s analysis textbooks which highlighted an analogy between measurable spaces and topological spaces. I’d like to dive into that analogy in what follows.
Continue readingGeometry & Topology RTG
Previously this week (week of August 3rd) I was able to attend the Geometry & Topology RTG workshop at the University of Notre Dame. The workshop was a week long event consisting of two parts, I and II, the first being an introduction into geometry & topology, and the latter being lectures on more advanced topics including student presentations. I’ve included their website link here. I attended part II and thought I’d speak about my experiences.
Continue readingPresentation: The Functional Equation for Riemann Zeta Over Function Fields
This is a presentation I gave for my Algebraic Number Theory class during the Spring 2019 semester. In it I prove the functional equation for the Riemann zeta function for global function fields of transcendence degree one.
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