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Notes: Real Analysis I, II and Graduate Analysis

/ T Chan / 1 Comment

I took Real Analysis I & II with Professor Alex Schuster in 2016-2017. These comprehensive notes were compiled using lecture notes and the textbook, An Introduction to Analysis (Fourth Edition), by William Wade.

Please feel free to download and print these notes for your convenience.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Real Analysis I

Real Analysis II

Graduate Analysis

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Notes: Graduate Algebra

/ T Chan / Leave a comment

I took Graduate Algebra with Professor Matthias Beck in Spring 2017. These comprehensive notes were compiled using lecture notes and the textbook, David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Please feel free to download and print these notes for your convenience.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

  • Polynomial rings,
  • irreducibility criteria,
  • Gröbner bases & Buchberger’s algorithm,
  • field extensions,
  • splitting fields,
  • Galois groups,
  • fundamental theorem of Galois theory,
  • applications of Galois extensions,
  • introduction to the polynomial method with applications in graph theory and incidence geometry.

Notes: Modern Algebra II

/ T Chan / Leave a comment

I took Modern Algebra II with Professor Matthias Beck in Spring 2016. These comprehensive notes were compiled using lecture notes and the textbook, David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Please feel free to download and print these notes for your convenience.

Featured Image: Icosahedron and dodecahedron Duality
Credit: Images from Algebra: Abstract and Concrete by Frederick M. Goodman

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

  • 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,
  • field extensions,
  • primitive roots,
  • finite fields.

Precalculus Resources: Intro & Review of Functions

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In the first couple of chapters, Dr. Axler covers the properties of real numbers, and gives a throughout exploration of functions in his textbook for Math 199 (Precalculus at SFSU).

Flashcards

You can take the file to a print shop to have directly printed and cut for you, or print it double sided on regular paper and cut + paste onto index cards. Get creative!

Helpful Khan Academy Videos:

More resources to come!

Notes: Number Theory and Modern Algebra I

/ T Chan / Leave a comment

I took Modern Algebra I with Professor Matthias Beck in Fall 2015. These comprehensive notes were compiled using lecture notes and the textbooks,

Please feel free to download and print these notes for your convenience.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

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

Featured Image: Dodecahedron-Icosahedron Duality
Credit: Images from Algebra: Abstract and Concrete by Frederick M. Goodman

Notes: Mathematics of Optimization

/ T Chan / Leave a comment

I took this course with Professor Serkan Hosten in Fall 2016. These comprehensive notes were compiled using lecture notes and the textbook, Optimization Models by Giuseppe Calaore and Laurent El Ghaoui, Cambridge University Press.

Please feel free to download and print these notes for your convenience.

Featured Image: Analysis explanation of why when optimization a linear objective function over a convex shape will lead to a optimal solution on the boundary of the feasible region.
Image Credit: Figure 08-19 in Optimization Models by Giuseppe Calaore and Laurent El Ghaoui, Cambridge University Press.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Syllabus:

  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. Semidefinite programs: modeling certain convex optimization problems as semidefinite programs

Fun fact: The subtitle for the site, “#teamnosleep” originated from my study group in this class. Homework assignments were intensely hard.