## Notes: Modern Algebra I

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

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

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.

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.