Lecture Notes

Convergence rates for quadratic cost, M-smooth functions, gradient descent for smooth functions, Euclidean projection, minimum principle, Frank-Wolfe method, proximal gradient descent, sub-gradients, Danskin's theorem, step-size tuning, stochastic coordinate descent, stochastic gradient descent, Nesterov accelerated gradient descent, and interior point method.

Ordered sets, Archimedean property, Completeness, Metric spaces, compactness, connectedness. Continuity and uniform continuity.Monotonic functions, Functions of bounded variation, Absolutely continuous functions. Derivatives of functions and Taylor`s theorem. Riemann integral and its properties, characterization of Riemann integrable functions. Improper integrals, Gamma functions, Sequences and series of functions, uniform convergence, and its relation to continuity, differentiation, and integration.

Topological groups, linear Lie groups, one-parameter groups, Lie algebras, nilpotent, solvable, and semi-simple Lie algebras, Haar measure, modularity, unitary representations, compact self-adjoint operators, Schur orthogonality relation, Peter-Weyl theorem, representations of SU(2) and SL(2), character functions, central functions, and Fourier series.

Vitali set construction, sigma-algebra, measure spaces, continuity, Carathéodory extension, Lebesgue measure, measurable functions, integration, monotone and dominated convergence theorem, product measure spaces, Fubini`s theorem, and introduction to Lp-spaces.

Topology and basis, sub-basis, order topology and product topology, subspace, Hausdorff space, continuous function and homeomorphism, continuous functions, product topology and box topology, metric topology, metrizable space and uniform topology, sequence lemma, first countable, quotient map, connectedness, linear continuum, intermediate value theorem, path-connectedness, local connectedness, compact spaces, min-max value theorem, Ureshyon lemma, homotopy and fundamental groups.

Review of quantum mechanics and relativity, Lagragian viewpoint of covariant fields, Klien-Gordan equation, energy-momentum tensor, scattering, cross section, differential cross section, spin, Dirac equation, covariant form of Dirac equation and solutions of Dirac equation. 

Introduction, Recap of normed linear spaces, linear operators, continuity, derivatives, quotient space, Banach space, bounded linear functionals and dual spaces, Hahn-Banach theorem, Minkowski function, Hahn-Banach separation, separable Banach spaces, ongoing.... 

Introduction, embedded submanifolds in Euclidean space, tangent space, retraction maps, first-order optimality condition, Riemannian gradient descent, ongoing.....

Introduction, classification theorem, topological manifold, smooth structure on manifolds, maximal atlas, tangent vector, germs, functors, tangent space, ongoing.... 

Equivalence relation, partition of a set, group, cancellation laws, permutation, ongoing.....

Axiom of quantum mechanics, Hilbert space, trace-class linear maps, Banach space, linear operators, dual space, BLT theorem, strong and weak convergence, ongoing....

Coming soon... (After Mid-semesters)