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 Metric Spaces!

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
  1. Optimization Models: modeling optimization problems as linear, quadratic, and semidenite programs
  2. Symmetric Matrices: using spectral decomposition of symmetric matrices for positive denite matrices and their role in optimization
  3. Singular Value Decomposition: computing SVDs in the context of linear equations and optimization
  4. Least Squares: solving systems of linear equations and least squares problems
  5. Convexity: identifying key properties of convex sets and convex functions for optimization
  6. 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,
  7. 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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