Algebraic Geometry Commons^™

Drawing A Triangle On The Thurston Model Of Hyperbolic Space, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan

Blake Mellor

In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.

The Secant Conjecture In The Real Schubert Calculus, Luis D. García-Puente, Nickolas Hein, Christopher Hillar, Abraham Martín Del Campo, James Ruffo, Frank Sottile, Zach Teitler

Zach Teitler

We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

Road Trips In Geodesic Metric Spaces And Groups With Quadratic Isoperimetric Inequalities, Rachel Bishop-Ross, Jon Corson

Rachel E. Bishop-Ross

We introduce a property of geodesic metric spaces, called the road trip property, that generalizes hyperbolic and convex metric spaces. This property is shown to be invariant under quasi-isometry. Thus, it leads to a geometric property of finitely generated groups, also called the road trip property. The main result is that groups with the road trip property are finitely presented and satisfy a quadratic isoperimetric inequality. Examples of groups with the road trip property include hyperbolic, semihyperbolic, automatic and CAT(0) groups. DOI: 10.1142/S0218196712500506

Modular Invariants For Lattice Polarized K3 Surfaces, Adrian Clingher, Charles F. Doran

Adrian Clingher

No abstract provided.

Length-Preserving Transformations On Polygons, Brad Ballinger

Brad Ballinger