Learning

What I learned today:

1/4

• Briefly looked at Spectral theory (spectral theorem for compact operators) to prepare for my functional analysis class. By brief I mean I skimmed through the wiki page.

1/1/2024

• Hahn-Banach Theorem for normed spaces: for a subspace U of X and a continuous linear functional u’: U -> F, there exists an extension linear functional x’: X -> F. This tells us that the dual space is “somehow large” (for a Hilbert space the space is isometrically isomorphic to its dual space, but this might not be generally the case.)

12/20

• Classification of Compact Riemann surfaces (Artin 11.9) (vanishing locus of f: C^2 -> C^2) as a n-sheeted branch covering space (every fibre consists of n points, and every neighborhood on the points of fibre homeomorphic to open neighborhood of the image).

12/19

• Hilbert’s Nullstellensatz (Artin Theorem 11.8.6): Maximal ideals in C[x_1,x_2,…,x_n] are in bijective correspondence with points in C^n. This is the starting point of Algebraic Geometry: algebraic property of C[x]/I is closely related to the geometric properties of V (algebraic variety).
• Notes on Representation Theory by Max Steinberg (UCLA Math ‘24) , had a taste on what quivers and McKay correspondence are. Quivers look very alike to a category from my understanding, and the McKay correspondence classifies all the finite subgroups (symmetries) of SU(2) (In bijection with ADE quivers).