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Students with Disabilities: Students requesting classroom accommodation must first register with the Dean of Students Office. The Dean of Students Office will provide documentation to the student who must then provide this documentation to the Instructor when requesting accommodation.

Tentative List of Topics to be covered

• Real analysis: Basic point set topology, convergence, continuity etc
• Vector spaces and Linear algebra
• Mathematical Probability theory: Sigma algebra, Random variables etc.
• Information Theory: Entropy, Mutual Information, etc.

List of Topics covered --!>

Week	Topic	Additional Reading	Assignment
Aug 19 - Aug 25	Introduction Natural numbers, Rationals Ordered sets, Reals as Dedekind Cuts Countably infinite (proof for rationals)
Aug 26 - Sep 01	Reals are uncountable (Cantor's diagonalization proof) Least upper bound (lub property inplies gub property) Rings and Feilds Convergence, Cauchy sequence	Paper on the Riemann Hypothesis Paper on the Johnson Lindenstrauss lemma	Assignment 1
Sep 02 - Sep 08	Reals are complete Limit Supremum and Limit Infimum Continuity: Limit definition of continuous functions
Sep 09 - Sep 15	Weierstrass's definition of continuous fns, equivalence Metric spaces Basic point set topology epsilon neighborhoods, limit points, interior points, open sets, closed sets Thm: Open iff complement is closed		Assignment 2 Deadline extended to Monday Oct 1.
Sep 16 - Sep 22	Basic point set topology continued; Thm: Arbitrary union, Finite intersection of open sets is open Definition: Topological space, trivial topology, discrete topology Pointwise continuity, Uniform continuity, Absolute continuity
Sep 23 - Sep 29	Linear maps. Vector/Linear spaces. Span, Linear Independence, Basis Matrix representation of a linear map Matrix multiplication with vector and matrix, and relationship to linear maps and composition of linear maps
Sep 30 - Oct 06	Composition of linear maps and Matrix multiplication Gaussian Elimination and LU decomposition
Oct 07 - Oct 13	Midterm I (Monday In-class) Normed (+Complete=Banach) vector spaces, Inner product spaces Inner product (+Complete=Hilbert) spaces Started Gram Schmidt orthogonalization
Oct 14 - Oct 20	Orthonormal basis; advantages, orthonormal vectors are independent. Fourier series		Assignment 3 Deadline extended to Wednesday Oct 31.
Oct 21 - Oct 27	Dual of a vector space Tensors Symmetric and Alternating tensors determinant=signed volume Started eigenvectors/eigenvalues
Oct 28 - Nov 03	Finished eigenvectors and eigenvalues. Unconstrained/constrained optimization The Lagrange Multiplier technique Convex functions and proof of local=global minima
Nov 04 - Nov 10	Mathematical probability theory Sample space, sigma algebra, probability measure. The Borel sigma algebra.
Nov 11 - Nov 17	Random variables, Indicator RV Distribution function, density function. Expected value, Conditional distribution, marginal distribution
Nov 18 - Nov 24	Random variable transform, Independence. Independence of random variables.		Assignment 4 For practice only; No submission required
Nov 25 - Dec 01	Markov and Chebychev's inequalities Weak law of large numbers (and proof) Information theory: Entropy, conditional entropy, mutual information, Kullback Leibler divergence.
Dec 02 - Dec 08	Review Midterm II (Wednesday, In-class)