## Thoughts About the NSF GRFP Application

Having received a 2021 NSF GRFP Fellowship, I thought it would be useful to share a little about my application process, some methods I used to strengthen my application, and tips that other NSF fellows past down to me. A fellow forewarning to take all of the following with a healthy grain of salt; the competition pool changes year-to-year and there are aspects of my application that may have made me a weaker or stronger applicant if this was a different year. The following statistics should also be taken into account: I received the fellowship as a senior undergraduate from an R1 state school.

## Interpreting the Size of the Cantor Set

The Cantor set provides some of the most pathological examples in real analysis. Introduced by G. Cantor in 1883, the Cantor set (or Cantor dust) can be thought of as the remainder of the unit interval after removing open middle thirds ad infinitum. In the following, we discuss the pathology relating to the “size” of the Cantor set where, depending on how you define it, the Cantor set has the size of a point, the entire real line, or somewhere in-between.

## The Contraction Mapping Theorem and Your New Favorite Math Tale

The contraction mapping theorem, also known as the Banach-Caccioppoli theorem, guarantees the existence of a fixed point for maps which “shrink”. In the following we’ll discuss the theorem, provide a sketch of the proof, and describe a neat application that will be sure to impress your math and non-math friends.

## Kashiwara Crystals and Bases of Representations

The theorem of highest weight tells us that for each dominant weight there exists a unique irreducible finite dimensional representation with the dominant weight as the highest weight. We may further decompose these representations into a sum of weight spaces. It turns out that bases of these representations admit beautiful combinatorial structure with respect to the weight space decomposition. In the following we describe this structure via Kashiwara crystals.

## My Summer UMN REU Expierence: Researching During A Global Pandemic

Participating in an REU is a joy, but doing it during a pandemic? Here I outline what my REU experience was like during COVID-19.

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.

## 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!

## 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 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.