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: