Failure of Shapiro Conjecture for Fl(2,3;4)

Math 648: Computational Algebraic Geometry

Web for a maximally inflected quintic plane curve

For this class, I want students to complete a smallish project, which should involve some mathematics and some computation. This will be done in pairs.
Each group will make a presentation to the rest of us of their project. This will be about 40 minutes and would include a brief discussion of the mathematics, as well as a brief demonstration of some software that was written for the project. You would also hand in a short typed (LaTeX) description of your work. It is not intended to be onerous, but to stimulate each of you to go a little further than the lectures. Different topics may have different emphasis on how much to write up and how much to code.

Schedule:

Tuesday, 28 April Blocker 302
2:30–3:10	Aaishah Ale-Rasool and Pooja Joshi	The Hilbert Polynomial and Two Invariants of a Variety
3:15–3:55	David Cates and Jacob Guerreso	Canonical Heights in Arithmetic Dynamics
4:00–4:40	Yunpeng Meng and Ruzho Sagayaraj	Computing geometric intersection numbers and self‐intersection numbers for loops on surfaces using Gröbner‐Shirshov bases
4:45–5:25	Nipun Amarasinghe and Phillip Speegle	Secant Loci and Partial Elimination Ideals
Wednesday, 29 April Blocker 302
1:00–1:40	Daniel Dale and Somak Dutta	Certified Homotopy Continuation using interval arithmetic
1:45–2:25	John Ajayi and Kai Neshat	Truncated Gröbner Bases for Integer Programming
2:30–3:10	Andrew Biehl and William Taylor	Implicitizing curves in three ways	Write-Up
3:15–3:55	Quentin Holmes and Joshua Im	Rational Normal Scrolls

Location:
We have Blocker Room 302 for Tuesday, April 28 2:30--5:30 PM and Wednesday, April 29 1:00--4:00 PM If needed, we will also have some in Blocker 121 1:00--3:00 PM on Monday, May 4.

Presentations:

Aaishah Ale-Rasool and Pooja Joshi, The Hilbert Polynomial and Two Invariants of a Variety.
We can use the Hilbert polynomial to determine the dimension and the degree of a variety. In this presentation, we will talk about this relation and compute a few examples for the same.
David Cates and Jacob Guerreso, Canonical Heights in Arithmetic Dynamics.
We will give an introduction to the notion of height functions in arithmetic geometry and number theory, with a focus on dynamical settings. Then we will show how one can compute canonical heights numerically using an algorithm.
Yunpeng Meng and Ruzho Sagayaraj, Computing geometric intersection numbers and self‐intersection numbers for loops on surfaces using Gröbner‐Shirshov bases.
We discuss Gröbner‐ bases which are a generalisation of Gröbner bases to Lie algebras and associative Algebras. Using the natural association between ideal membership and word problem for groups, we use a Gröbner‐Shirshov basis corresponding to a specific presentation of the surface group of a surface of genus g≥2 to compute geometric intersection numbers and self intersection numbers of loops on that surface.
Nipun Amarasinghe and Phillip Speegle, Secant Loci and Partial Elimination Ideals.
Let X be a projective variety and let q be a point not on X. The collection of points p of X for which the line passing through p and q intersects X with multiplicity at least k is called the kth secant locus of X. We define the kth partial elimination ideal of a homogeneous ideal and use it to get an algorithm to compute the ideal of the kth-secant locus of X.
Daniel Dale and Somak Dutta, Certified Homotopy Continuation using interval arithmetic.
Homotopy continuation is a standard method for numerically solving systems of polynomials, however in some applications such as for monodromy actions, standard implementations are poorly suited. We describe such a method for certifying path calculations using interval arithmetic and Mooreâ's criterion as implemented in the package Algpath.
John Ajayi and Kai Neshat, Truncated Gröbner Bases for Integer Programming.
We will discuss what is Integer Programming, the Buchberger Algorithm for Integer Programming, multivariate grading of the Toric ideal associated with the integer programming problem, and a truncated Buchberger Algorithm to solve the problem.
Andrew Biehl and William Taylor, Implicitizing curves in three ways.
Implicitization is the act of finding a system of implicit equations for a variety corresponding (as closely as possible) to a given collection of parametric equations. We remind the audience of two methods (namely, via Gröbner bases and resultants) for implicitizing rational curves and then present a third, new method (via moving line bases).
Quentin Holmes and Joshua Im, Rational Normal Scrolls.
We will give the definition of rational normal scrolls and prove some of their basic properties. In order to do this, we will go over the necessary prerequisite definitions and basic results. Many pictures will aid us along the way as these objects are very visual in nature.

Last modified: Sun Mar 29 16:31:47 CDT 2026