Notes on Learning Trajectories and Progressions

To pay homage to mathematician peers and mentors, these notes will maintain the same familiar formatting structure of definitions, theorems, and corollaries similar to mathematical text. These notes contain a few original ideas in the form of conjectures and remarks. Outside of these comments, the words arranged below and in subsequent posts are summaries of readings from the STEM education journal club at LSRI.

Miscellaneous Updates

The very most recent news is that I have decided to join University of Illinois at Chicago to work on a PhD in Learning Sciences, with a focus on Mathematics Education. I’m very excited to join the Learning Sciences Research Institute (LSRI) in continuing my research!

I’m moving this July, and this site will see significantly more activity; especially WRT notes on math education! I’ll be compiling notes and some definitions here as a resource

Some Highlights of 2018-2019:

  • Spring 2018:
    • Received the Sally Casanova Scholarship for the academic year 2018-19 (See page 14 of this booklet).
    • Participated in the California Forum for Diversity in Graduate Education
    • Finalized all the code for my thesis
    • Teaching:
      • Math 199 – Precalculus
      • Math 227 (x2) – Calculus II (as TA)
  • Summer 2018
    • Generalized code to any dimension, final verification of code with Dr. Gubeladze
    • Worked on implementing California Executive Order 1110 – part of a team with tenured faculty working on creating the curriculum for stretch courses.
    • Created a gradebook template and a lesson plan template for general use.
  • Fall 2018:
    • Applied to twelve PhD programs, and had a bit of an existential crisis because PhD applications can be grueling.
    • Worked on writing out my thesis, as most of my work was coding up to this point.
    • Defended my master’s thesis, “Computational Verification of the Cone Conjecture”
    • Teaching:
      • Math 107 – Math for Business Calculus I
      • Math 197 – Prelude to Calculus I
  • Spring 2019
    • Out of 12 I was waitlisted at one and accepted into two.
    • Finalized edits for my master’s thesis and submitted to archives
    • Updated LaTeX template for Masters Thesis for STEM majors at SFSU
    • Graduated from the Masters program!
    • First semester to finish grading finals not on the day that grades are due 😀
    • Teaching:
      • Math 108 – Math for Business Calculus II
      • Math 198 – Prelude to Calculus II

Masters Thesis

I defended my thesis, “Computational Verification of the Cone Conjecture”, in December 2018, and submitted all final edits in May 2019.

My advising committee:

  • Dr. Joseph Gubeladze
  • Dr. Matthias Beck
  • Dr. Serkan Hosten

Abstract:

The set of polyhedral pointed rational cones form a partially ordered set with respect to elementary additive extensions of certain type. This poset captures a global picture of the interaction of all possible rational cones with the integer lattice and, also, provides an alternative approach to another important poset, the poset of normal polytopes. One of the central conjectures in the field is the so called Cone Conjecture: the order on cones is the inclusion order. The conjecture has been proved only in dimensions up to 3. In this work we develop an algorithmic approach to the conjecture in higher dimensions. Namely, we study how often two specific types of cone extensions generate the chains of cones in dimensions 4 and 5, whose existence follows from the Cone Conjecture.

A naive expectation, explicitly expressed in a recently published paper by Dr. Gubeldaze, is that these special extensions suffice to generate the desired chains. This would prove the conjecture in general and was the basis of the proof of the 3-dimesional case. Our extensive computational experiments show that in many cases the desired chains are in fact generated, but there are cases when the chain generation process does not terminate in reasonable time. Moreover, the fast generation of the desired chains fails in an interesting way—the complexity of the involved cones, measured by the size of their Hilbert bases, grows roughly linearly in time, making it less and less likely that we have a terminating process. This phenomenon is not observed in dimension 3. Our computations can be done in arbitrary high dimensions. We make a heavy use of SAGE, an open-source mathematics software system, and Normaliz, a C++ package designed to compute the Hilbert bases of cones.

Full Text is below:

The actual latex code is hosted on github.

Updates on GitHub

Finally set up a github to house all the custom LaTeX files that I’ve created!

Will be updating this over time – this particular github is mostly meant for personal use, though I’m happy to share what I have and continue to update this github.

As I continue to create new tools, I’ll be hosting them here.

You’ll see currently the following repositories:

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

Tool Sheet for Real Analysis II

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: