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MAA Invited Paper Session: Concrete Computations in Algebra and Algebraic Geometry Abstracts

Friday, August 7, 1:00 PM - 3:20 PM, Marriott Wardman Park, Delaware B

This session will bring together researchers in computational or combinatorial algebra and algebraic geometry whose research is concrete and accessible.

Sarah Mayes-Tang, Quest University
Karen Smith, University of Michigan

Continued Fractions Can Resolve Singularities?!

1:00 PM - 1:20 PM
Robert Walker, University of Michigan

For the lion's share of number theorists, continued fractions are not a tool for solving not-so-trivial problems. They're not hip enough perhaps. In birational geometry, however, they can be used to obtain minimal resolutions of several classes of singular algebraic varieties—a not-so-trivial problem to be sure. For coprime integers $$a>b>1$$, the affine plane curve $$x^b = y^a$$ $$\in$$ $$k^2$$ ($$k$$ any field) has a cusp at the origin. Say we want to resolve this isolated cuspidal singularity. The goal of the talk is to show how the continued fraction expansion of $$a/b$$ can serve as a "cheat code" for completing this task.

The Search for Indecomposable Modules

1:30 PM - 1:50 PM
Courtney Gibbons, Hamilton College

Short Gorenstein rings have many properties that make them interesting and easy to work with. For instance, they are finite dimensional vector spaces, and so are their finitely generated modules. In this talk, I will discuss how finding the answer to a simple question about indecomposable modules over a short Gorenstein ring led through very interesting mathematics, including the study of continued fractions.

The Importance of $$\alpha$$

2:00 PM - 2:20 PM
Mike Janssen, Dordt College

The classical algebra-geometry dictionary relates an ideal $$I$$ in a polynomial ring to its corresponding zero locus $$Z$$ at which all polynomials in the ideal vanish. A recent object of study in projective algebraic geometry is the initial sequence $$(\alpha(mZ))_{m\geq 1}$$, where $$\alpha(mZ)$$ is the degree of a polynomial of least degree vanishing to order at least $$m$$ on $$Z$$. In 2010, Bocci and Chiantini used classical algebraic geometric methods to classify all finite sets of points in $$\mathbb{P}^2$$ for which the first difference $$\alpha(2Z) - \alpha(Z)$$ is small. We will discuss their result and recent generalizations.

Pictures of Syzygies

2:30 PM - 2:50 PM
Timothy Clark, Loyola University

We describe a diversity of pictorial, combinatorial, and topological objects whose structure can be used to encode and understand the notion of $$syzygy$$ in commutative algebra. A $${syzygy}$$ is a tuple $$(a_1,\ldots,a_n)\in R^n$$ such that $$a_1g_1+\cdots a_ng_n=0$$, where $$g_1\ldots g_n$$ is a set of generators for a module over a commutative ring $$R$$.

When Do 10 Points Lie on a Cubic Curve?

3:00 PM - 3:20 PM
Will Traves, US Naval Academy

David Wehlau and I found a ruler and compass construction to check when 10 points in the plane lie on a cubic curve. I'll explain our construction and how it relates to previous work by Pappus, Pascal, Cayley and Bacharach. I'll also describe a new class of problems raised by our work.