Notes: Real Analysis I & II

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!

Notes for Real Analysis I

Notecards for Real Analysis II

Tool Sheet for Real Analysis I Finals

 

Tool Sheet for Real Analysis II Finals

 

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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.

Reflection: Planning/Time Management

As a Teacher

  • Pre-Calculus
    • Lesson Planning is taking a lot less time this time around, since I spent a lot of time documenting the lessons from last year. Over-planning was actually an issue last semeseter; I would never actually complete everything I planned, and that was mildly frustrating. Also, there’s an element of improvisation that happens depending on the mood of the classroom, which means that scripting every sentence is simply not possible anyway.
  • Calculus II
    • The first two weeks I only prepared by rereading the text on the sections that the lead professor went over, but on the second week students asked me questions that definitely stomped me because I had not seen the problem before hand. Now, I ask students to email me their questions before the TA session, so that I can prepare with care. Planning short lectures on Calculus II material has been easier after that change in preparation, especially with access to great tools.

As a Student

  • Algebra
    • I’m really glad to be practicing with SAGE, but I’ve honed in on specific problems and end up spending significant time cleaning up code instead of doing proofs. I need to find a balance in what I’m focusing on when learning.
  • Real Analysis II
    • I should really be spending more time on this course. The lectures in this cource has been a blessing for preparing for Calculus II TA sessions, since it helps get me into the “mood” to do Calculus. 🙂
  • Combinatorics
    • Obsessing over the details in this class takes too long. I’m in a similar situation with Algebra, but there’s this one is more like I spend 4-5 hours on one proof and run out of time for the other problems.

There’s a lot of juggling, and I have to get better at this soon before the midterm season.

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:

    1. Create an account (New users only):
      1. Go to user.wolfram.com and click “Create Account”
      2. Fill out form using a @sfsu.edu email, and click “Create Wolfram ID”
      3. Check your email and click the link to validate your Wolfram ID
    2. Request the download and key:
      1. Fill out this form to request an Activation Key
      2. Click the “Product Summary page” link to access your license
      3. Click “Get Downloads” and select “Download” next to your platform
      4. 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.

Precalculus Resources: Intro & Review of Functions

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: 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,

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

Website Infrastructure

The notes page is set up, next I aim to finish up organizing the precalculus resources and general non-math resources.

Now, the site contains links all my notes starting from Fall 2015, which send visitors directly to my google drive, but I wonder if that is secure/wise.

Maybe there is a better way to both host the files and allow for instant updates as I compile my LaTeX files. Right now, this blog links to individual files within my google drive which is synced to a directory on my laptop. When I compile on my computer, I work directly in that directory so that the file linked in google drive is automatically synced. I want the same functionality and convenience but I’m not sure if linking to the drive is secure, since it’s a personal account.

Time to do some non-math research~

Update: I hear that wordpress has great support, so I made a post at the forums. I wonder if there’ll be an easy solution?

Why a Blog?

So, why start a blog when there is approximately… 1.5 \times 10^{6} posts created per day? (Checked @ 8 AM 1/8/17)

For me, this website is a bit more than a collection of math notes and lists of resources. I am aiming to chronicle the challenges of being a student and a teacher at the same time. On the other hand, I want to also document the most fun parts of being in an intensive graduate program!

So this blog will be part survival guide, part chicken-soup-for-the-soul, and part scrapbook.

Oh. By the way, I survived the first semester of graduate school!