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


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.