# Division of Mathematics & Natural Sciences

The Division of Mathematics and Natural Sciences at Penn State Altoona offers courses in agriculture, biology, biochemistry, molecular biology, chemistry, geography, geoscience, mathematics, meteorology, physics, and statistics. The Division has approximately 35 tenured and tenure-track postitions, 15 full-time instructors, 40 part-time instructors, 3 staff assistants, and 3 lab technicians. The Division offers bachelor's degrees in Biology, Environmental Studies, Mathematics, and Science and minors in Biology, Chemistry, Environmental Studies, Mathematics, Mathematics Applications, and Natural Sciences.

Friday, November 9, 2012
3:00 p.m., 143 Hawthorn Building

## Barry Minemyer

Binghamton University

"Simplicial Isometric Embeddings of a finite Simplicial Complex into R^p_q."

Description: "After a brief history of how this research came about we will prove that every finite metric simplicial complex $\mathcal{X}$ admits a simplicial isometric embedding into $\mathbb{R}^{p}_{q}$ where $p = q = \text{max} \{ d, 2n + 1 \}$ and $d = \text{max} \{\text{deg}(v) \, | \, v \text{ is a vertex of } X \}$. Here, $\mathbb{R}^{p}_{q}$ is $\mathbb{R}^{p + q}$ endowed with the Lorentzian inner product that has $p$ eigenvalues of 1 and $q$ eigenvalues of -1. The proof of this theorem is existential, but we will give a second proof which is (somewhat) constructive (but increases dimension) which will allow for an example. We will then use these results to give a description of a flat metric (interior to a fixed simplex) in terms of only the edge lengths of the simplex and barycentric coordinates."

"This talk will cover the first part of my thesis. If time permits I will give an overview of the results that will be included in the second part of my thesis. This talk should be mostly self-contained so that any of the faculty who do not study topology or geometry should be able to follow. We will (quickly) review all necessary definitions and facts."