This lecture intends to introduce students to the language of Organic Chemistry through an in depth discussion of ring structures. Specifically, we will explore pericyclic reactions (sigmatrophic rearrangement, electrocyclic., and cycloaddition) and aromatic compounds. Students should expect to gain a deeper understanding of molecular orbital theory, and the principles and methods used in organic synthesis.

Does there exist a function which is continuous exactly on the irrational numbers? It turns out the answer is yes: https://en.wikipedia.org/wiki/Thomae%27s_function. The natural next question is: "What about he complement? Is there a function exactly continuous on the rationals?" The answer to this is, somewhat surprisingly, no! To answer this question we will need to develop some powerful tools in analysis such as the Baire Category Theorem and Borel Hierarchy which tell us about what happens when we take repeated intersections, unions, and complements of infinitely many open sets.

Elliptic curves are a class of cubic curves (defined by a cubic polynomial in two variables) with deeply surprising and beautiful properties. The most surprising fact about elliptic curves is that their points naturally form a group structure under some geometric operation. Elliptic curves show up everywhere in mathematics from torii defined over the complex numbers to generating abelian extensions of number fields. Elliptic curves also feature prominently in the Birch and Swinnerton-Dyer conjecture, one of the millenium prize problems, and Andrew Wiles' proof of Fermat's last theorem. In this class we will develop the fundamental results about elliptic curves such as the Mordell-Weil theorem and Weierstrass function theory before diving head-first into some more advanced territory mentioned above. This class will be a meandering relentless rollercoaster through some of the most beautiful connections in modern mathematics rather than a reasonably-paced, well-structured, and rigorous development of a topic. Expect proofs to be "sketched", definitions to be hand waved, and lots of inane terminology to be used. That said, expect to come away with an broader awareness of open problems in mathematics, a much richer appreciation of the interplay between geometry and algebra, and a healthy respect for cubic polynomials in two variables.

This course will be given as an introduction to superconductors, via the marvelous effects its unique properties induce AND the beautiful mathematics that underlies it. Why in the heck do they have zero resistance? By introducing certain tools from quantum mechanics, I will attempt to describe to you why the field of superconductivity is awesome.