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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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.

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