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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.

Precalculus Resources: Spring 2017 Midterm II Review

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!

Topics Covered:

  • 2.4: Polynomials
    • Vertex of a parabola
    • Determine the behavior of a polynomial near - \infty or \infty
  • 2.5: Rational Functions
    • Determine the behavior of rational functions near - \infty to \infty
    • Determine the vertical and horizontal asymptotes of a rational function
    • Determine the holes of a rational function, if they exist.
  • 3.1: Exponential and Logarithmic Functions
    • Their definitions as inverses
    • Practice using #23 – #32 in Precalculus – Prelude to Calculus, 3rd Ed. 
  • 3.2: Power Rule
    • Change of Base
  • 3.3: Product and Quotient Rule
    • Read p.249-p.252 for applications in scientific settings
  • 3.4: Exponential Growth
    • Compound Interest (n times per year)
  • 3.5: e and the Natural Logarithm
    • Understanding the definition of e will enhance your understanding of proofs later on. For now, we know it’s a real number that is approximately equal to 2.71, and it’s associated with Continuous growth!
  • 3.7: Exponential Growth Revisited
    • Read p.304-p.306 for an application of the approximation methods discussed in 3.6 used in financial estimations.
    • Focus on Continuous Compound Interest
  • 4.1: Unit Circle
  • 4.2: Radians
    • Make sure you can convert between degrees and radians.
  • 4.3: Sine and Cosine
    • Definition using unit Circle
    • Domain and Range of Sine & Cosine
    • If you want a challenge, try answering #45 from this section.

Quiz 5:

Review for 2.4 & 2.5

Khan Academy Videos:

Institute of Mathematics and its Applications at University of Minnesota: SAGE Coding Sprint Journal

April 5th, 2018

  • Met with the team of SAGE & Normaliz Developers at Institute of Mathematics and its Applications at University of Minnesota.
  • Discovered the issues I had connecting PyNormaliz to SAGE was due to installing SAGE using the Binary, which limits the use of custom packages. While making from source allows for the use of most recent version,
  • While waiting for the fresh install of SAGE, read an article about unit testing by Jeff Knupp here.
  • Learned about using parallel compiling option for SAGE install, and also learned about the System Monitor (was cool to see all the cores get used at once)
  • Begin writing Unit Tests

April 6th, 2018

  • Shifted to the developer branch of SAGE
  • Installed the newest developer version of PyNormaliz, which installs Normaliz 3.5.3
  • Rewrote Top Down algorithm using Polyhedron(rays=[[vector1],…,[vectorN]],backend=’normaliz’)
    • Allows for the use of common operations like verifying if a vector is contained in the Cone.
    • Resolved multiple issues (that all arose basically due to me being a newb.)

April 8th, 2018

  • Added multiple githubs to record my work besides my thesis
  • Begin redesign of Bottom Up using the ideas above.

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

 

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.

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:

    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.