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.

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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.

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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.

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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.

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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.

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