I began typing up my notes in LaTeX during the last couple of semesters in my undergrad at SFSU. Thus, I have maintained a collection of condensed notes from those times.
These notes are specifically the theorems, remarks and corollaries used in proofs within the homework, and are written without justification.
I’ve organized my notes from these classes into four major topics:
Please contact me if you find any mistakes in the notes!
Algebra
Class 
Professor 
Syllabus/Topics Covered 
Elementary Number Theory 
Dr. Matthias Beck 
 Divisibility
 Primes
 Congruences
 Arithmetic functions
 Primitive roots
 Quadratic reciprocity
 Continued fractions

Modern Algebra I 
 Integers & the Euclidean algorithm
 Complex numbers, roots of unity & Cardano’s formula
 Modular arithmetic & commutative rings
 Polynomials, power series & integral domains
 Permutations & groups

Modern Algebra II 
Review of basic properties of groups and rings and their quotient structures and homomorphisms, group actions, Sylow’s theorems, principal ideal domains, unique factorization, Euclidean domains, polynomial rings, modules, tensor products, field extensions, primitive roots, finite fields. 
Graduate Algebra 
[Coming Soon] 
Analysis
Class 
Professor 
Syllabus/Topics Covered 
Real Analysis I 
Dr. Alex Schuster 
In this course we will prove many of the results from
Calculus. We will examine in detail the concepts of limits, continuity, differentiation and
integration. 
Real Analysis II 
[Coming Soon] 
Geometry
Class 
Professor 
Syllabus/Topics Covered 
Geometry 
Dr. Joseph Gubeladze 
 Prove theorems in incidence geometry;Discuss the strengths and weaknesses of Euclid’s Elements as an axiomatic system;
 Compare various approaches to Euclidean geometry;
 Prove theorems in Euclidean and projective geometries;
 Classify Euclidean motions and apply transformational methods to prove theorems and analyze symmetry patterns in the plane;
 Discuss the significance of Euclid’s fifth postulate in the development of nonEuclidean geometry;
 Define axiomatic system and discuss the importance of models for an axiomatic system.

Applied Mathematics
Class 
Professor 
Syllabus/Topics Covered 
Mathematics of Optimization 
Dr, Serkan Hosten 
 Optimization Models: modeling optimization problems as linear, quadratic, and semidenite programs
 Symmetric Matrices: using spectral decomposition of symmetric matrices for positive denite matrices and their role in optimization
 Singular Value Decomposition: computing SVDs in the context of linear equations and optimization
 Least Squares: solving systems of linear equations and least squares problems
 Convexity: identifying key properties of convex sets and convex functions for optimization
 Optimality Conditions and Duality: developing criteria to identify optimal solutions and using dual problem, in particular, in the context of linear models, Linear and Quadratic Models: employing the geometry of linear and quadratic models for solution algorithms,
 Semidenite programs: modeling certain convex optimization problems as semidenite programs

Probabilty and Statistics with Computing 
Dr. Mohammad Kafai 
 Discrete Distributions
 Bernoulli
 Binomial
 Geometric
 Negative Binomial
 Hypergeometric
 Multinomial
 Poisson
 Continuous Distributions
 Uniform
 Normal
 Gamma
 Exponential

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