You might have a quite extensive collection of mathematical texts on one of you shelves, but have you completely read through any of them? While still in lockdown and with the spring semester over, I’ve had a lot of time to read through my shelf. When reading textbooks I primarily use two strategies, and I’d like to share these in the following.

Continue reading# Author: Henry Twiss

## A Tea-Time Paper Post

During our (now extended) time cooped up at home I thought it would be fun to put together a small post consisting of links to some very interesting relatively short papers that can be read with your favorite cup of tea. Descriptions are given before each link with the necessary background. Stay safe everyone!

Continue reading## Understanding Varieties

Varieties are a basic structure in algebraic geometry. They were the central objects of study before Grothendieck reinvented the entire theory in his treatise *Éléments de géométrie algébrique* by introducing schemes. In the following we will introduce varieties and define algebraic curves.

## Tangent Vectors and Differentials of Smooth Maps

Tangent vectors of functions are discussed early on in a standard calculus course. They are described either as directional derivatives or as velocities as curves. In manifold theory, we would like to generalize these ideas of *calculus on * to *calculus on manifolds*. While there is an algebraic and geometric viewpoint of tangent vectors on manifolds, the algebraic realization is often faster to devlope the theory with. However, the geometric realization can be incredibly useful for computations. In the following, we’ll discuss both the algebraic and geometric viewpoints of tangent vectors, prove the equivalence between then, and see how both are useful by discussing the differential of a map.

## The Connection Between Hopf Algebras and Groups

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 reading## 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## The Arithmetic of Polynomial Rings over Finite Fields

We’re going to discuss the arithmetic of polynomial rings over finite fields and its similarity to the arithmetic of the integers. We’ll first run though some preliminary observations about the arithmetic of polynomial rings which will suggest that it behaves similar to the arithmetic of the integers.

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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