## 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 $\mathbb{R}$ 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.

## 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.