Mathematics Colloquium Series

The Mathematics Colloquium series continues on Thursday, October 20 at 3:30  in VH1224  with a presentation by two Truman students on their summer research. Laipeng Zheng will be speaking on “A Novel Matrix Decomposition”. Miguel Fernandez will be speaking on his REU research “Minimal Pentagonal Tilings and Isoperimetry on the Plane with Density”. Come learn about opportunities for summer research, how to apply, and how to get accepted!

Laipeng’s Abstract:

In the summer, I was fortunate to work with Professor Michael Adams using research grants provided by Truman State University TruScholar Program to conduct a mathematics research project. We study novel matrix decomposition, related to loop analysis. Loop analysis provides an analytical technique of interest to ecologists to study population dynamics.  Prior quantitative studies have focused on development of transitional matrix and understanding of eigenvalues to explain population dynamics. Our study focuses on a new way to decompose the population projection matrix directly into cyclic summands. Computer simulation of different projection matrixes provides calculation of eigenvalues of newly decomposed cycles. Our research has found some interesting mathematical properties with regard to these new matrix decomposition methods. Dimension of space generated by cyclic summands is equal to dimension of cycle space generated by adjacency matrixes. This is an important step to understand loop decompositions and to quantify eigenvalues of associated loops with respect to eigenvalues of the projection matrix.

Miguel’s Abstract:

In 2001, Thomas Hales proved that regular hexagons provide a least-perimeter unit-area tiling of the plane, better than squares and equilateral triangles. We seek the least-perimeter unit-area tiling of the plane by pentagons. Work by my advisor Frank Morgan and his students resulted in a proof that two other pentagons, called Cairo and Prismatic, yield least-perimeter unit-area tilings by convex pentagons. Over the summer, we found uncountably many mixtures and classified the doubly periodic ones by their wallpaper symmetry groups. We also consider tilings by mixtures of convex and nonconvex pentagons and perimeter-minimizing tilings on various flat tori.

Our second summer project considers isoperimetric curves on the plane with density. It is well known that on R^2 the least perimeter curve that encloses a given area is a circle. What happens when we give the plane a density that weights both area and perimeter? The log convex density conjecture says that if the density is radial and its log is convex, circles about the origin minimize weighted perimeter for given area. We present our results so far regarding the borderline case of the plane with density e^r, and offer numerical evidence hinting that circles about the origin are indeed isoperimetric.