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.

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:

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.

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!