Extracurriculars
The lecturers of some of the graduate courses were kind enough to let me join their courses before enrolling in a postgrad degree (or maybe I just didn't correct their assumption...). Here is an overview.
Algebra II: 1,0
Algebraic Topology: 1,0
Functional Analysis: 1,0
Harmonic Analysis: 1,0
Hilbert's Nullstellen- & Basissatz, Noetherian & Artinian Rings, Zariski Topology, Conrete & Abstract Spectra, Krull Dimension & Transcendence Degree, Localization, Nakayama's Principal Ideal Theorem, Integral Extensions & Noether Normalization, Hilbert Polynomials & Dimension, Associated Graded Algebra, Regular Local Rings & Smoothness, Tensor Products & Basic Homological Algebra
Homotopy Category & Fundamental Groupoid, Covering Spaces, Monodromy Functor & Galois Correspondence Theorems, Classification of Surfaces, Simplicial - & Axiomatic Homology + Computation & Application, Relative & Reduced Homology
General Topology (Compactness, Separability, Countability, Nets, Tychonoff, ...), Normed Spaces (Examples, Separability, Riesz's Lemma, ...), Operators & Functionals (Hahn-Banach, Strong/Weak/Weak* Topologies, Extremal Points & Krein - Milman Theorem, ...), Operators on Banach Spaces (Baire's Theorem, Open/Inverse Mapping Theorem, Closed Graph Theorem, Banach-Steinhaus Theorem, ...), Hilbert Spaces (Orthogonality & Separability, Riesz's Representation Theorem, ...), Spectral Theory (Compact Operators, Spectral Theorem for Compact Normal Operators, ...)
p-adic Numbers & Integers, Ostrowski's Theorem, Riesz-Markov Representation Theorem, Haar Measure: Existence & Uniqueness via Izzo's Fixed Point Argument, Topological Vector Spaces, Markov-Kakutani Theorem, Modular Function, Unitary Representation Theory, Schur's Lemma, Induced *-Representation on L1, Measure Algebra, Abstract Fourier Transformation, Riemann-Lebesgue Theorem, Ascoli's Theorem, Stone-Weierstraß, Bochner's Theorem, Plancherel's Theorem, Fourier Inversion, Pontrjagin Duality
Advanced Topics in Algebraic Topology: 1,7
Commutative Differential Graded Algebras, Stoke's Theorem, Poincare Lemma & Homotopy Invariance, Poincare Duality, Künneth Formula, Bundles & G-Torsors, Poincare-Hopf Index Theorem & Characteristic Classes, Thom Isomorphism
Operator Theory: 1,0
C*-Algebras & Abstract Spectral Theory, Analytical Functional Calculus, Gelfand Theory & Application, Spectra of Operators: Properties & Examples, Courant-Fischer Min-Max-Principle, Trace Class Operators & Hilbert-Schmidt, Fredholm Operators & Indices + Applications, Spectral Theory of Bounded Self-Adjoint Operators, Unbounded Operators, Quardratic Forms & Friedrich Extensions, Spectral Families, Application: Moment Problems, Unitary Groups on Hilbert Spaces & Stone's Theorem, Uncertainty
Probability Theory: 1,7
Probability Spaces, Independence, Jensen's & Chebyshev's Inequality, Weak Law of Large Numbers, Borel-Cantelli Lemmas, Strong Law of Large Numbers, Kolmogorov's 0-1-Law, Weak Convergence of Distributions, Charactersitic Functions & Inversion Theorem, Levy's Continuity Theorem, Conditional Expectation, Martingales & Concrete Applications, Stopping Times, Polya's Urn
Seminar: From Homotopical Algebra Towards Higher Categories (Visiting)
Chain Homotopies & Quasi Isomorphisms, Model Categories from Localization, Quillen equivalence of Top and sSet, Projective/Injective Model Structure on Functors, Quasi-Categories/(infinity, 1)-categories, (Co)Limits in Quasi-Categories, Derived Algebraic Geometry/Derived Deformation Theory, Dold-Kan Correspondence