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 reading# Topology

## Measurable 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.

## Geometry & 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.

## What Is a Delta Complex?

It’s time to get our hands dirty with some topology! Instead of studying spaces directly, we’re going to study a way of building topological spaces. In particular, we’re going to view a space as a collection of analogous subspaces appropriately glued together satisfying a few restrictions.

Continue reading## An Introduction to Topological Groups

We’re going to talk about an interesting merger of abstract algebra and topology, namely topological groups.

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