A good practice in math, especially when studying a type of mathematical objects, is by considering what doesn’t fit the definition.

# mathematics

# Notes: Real Analysis I, II and Graduate Analysis

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

# Notes: Graduate Algebra

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.

# Precalculus Resources: Spring 2017 Midterm I Review (Section 0.1 – 2.2)

I have gathered quizzes that relate directly to the upcoming midterm, as well as included a sample midterm. Please feel free to use these for practice! If you have trouble with the material, visit this page for additional resources. Don’t forget to grab a free version of wolfram alpha pro to help you if you’re enrolled at San Francisco State University!

## Quizzes #1 – #4:

## Practice Midterm:

# Notes: Modern Algebra II

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.

# San Francisco State University Math Resources: Mathematica & Wolfram Alpha Pro

SF State has a site license for Mathematica. Under this site license, Mathematica can be installed without further cost on any computer owned by San Francisco State University, by SF State faculty, or by SF State students. This software works on Windows, Macintosh, and Unix.

Under the terms of the site license negotiated by CSU, free WolframAlpha Pro accounts are also now available to our faculty, students, and staff.

If you are an SFSU student, click here for a free WolframAlpha Pro account.

### Mathematica

If you are a SFSU student and you want to download the *Mathematica* software to your computers, then follow the instructions below:

- Create an account
*(New users only)*:- Go to user.wolfram.com and click “Create Account”
- Fill out form using a @sfsu.edu email, and click “Create Wolfram ID”
- Check your email and click the link to validate your Wolfram ID

- Request the download and key:
- Fill out this form to request an Activation Key
- Click the “Product Summary page” link to access your license
- Click “Get Downloads” and select “Download” next to your platform
- Run the installer on your machine, and enter Activation Key at prompt

For both *Mathematica* and WolframAlpha Pro requests on the links above, **it is important that only sfsu.edu email addresses be used.** Requests from other email addresses will be rejected.

# Reflection- Group Work: Size & Clarity

I’m wondering how group activities can be done with 40+ students; I often give short, paired activities during lecture, which (hopefully) helps with engagement, but if I want to give a longer “exploration” activity in class, there are challenges.

### Group Size

Checking in with each group is takes a significant amount of time, scaling linearly. I tried to keep group size to 3-4 students, but dividing students into groups of 4-5 lead to having about 8-10 groups. The down time caused some groups to finish faster than others. There must be an optimal ratio for the number of students to square foot of classroom. I currently have 40 students, but the rooms are a lot smaller than the previous semester, I must alter the group activity plans. I also had 80 minutes sessions on Tues/Thurs instead of 50 and a much larger classroom for discussions/group work in the 4th hour. My lesson plans will require more adjustments, I suspect.

Some groups had members forging ahead before everyone understood, and that’s always a challenge as well. I wonder if larger groups are a good idea, because conversations between more than 5 students often turn to a few pairs and few solo working in parallel…

### Clarity: Instruction and Feedback

This time around, when I lead the group activity I focused on the clarity of my instructions, and I’m trying a different approach this semester compared to last. Previously, I gave handouts with specific procedure, but it was confusing for some students. This time, I tried a different approach, where I verbally and visually give instructions on the board and forgo all printed handouts in order to allow for students to make their own notes instead of using data sheets. I think I might bring back the data sheet for the later activities, so that the students can know what I specifically expect from them. I should specify that students should use technology to assist in graphing and calculations.

In terms of feedback, I tried to ask questions and check for understanding, but the number of groups is high, which means I must spend less time per group, or make the group size larger. If I maintain that the groups have no more than 5 people, then I will have at least 8 groups – which can cause my feedback to the students to be less precise and more brief. Perhaps I can take a vote with my students, to see if they prefer trying larger groups, given the challenges above?

Looking back, I admire my high school teachers who managed to deal with 30 students at a time, and were able to conduct experiments in labs, with open flames, too.

# Notes: Number Theory and 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,

- Al Cuoco & Joseph J. Rotman, Learning Modern Algebra, MAA Textbooks, 2013.
- Frederick M. Goodman, Algebra: Abstract and Concrete, edition 2.6.

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

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.

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.

**Syllabus:**

- 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,
- 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.

# Reflection: Following My Passions

When I was in my late teens to early twenties, I knew I liked math but I wanted to try everything that would require application of math: I jumped from applied math to physics to engineering. I loved the process of learning, and I took a lot of different classes, and by the time I was halfway through the second semester of Mechanical Engineering courses, I finally realized that I was only really interested in the math, and talking about the math.

Then today, I went digging through my time-capsules on the internet. I have blogs scattered across a lot of different platforms, and I found this post over on Hubpages that I wrote in 2009. I’m pretty sure this reaffirms that I’ve always wanted to teach math.

Looking back, I’m glad that I took a long, winding path. I needed to grow a lot spiritually and emotionally before I was ready to take on teaching. Hopefully I’ll maintain my capacity for growth in the upcoming years.