# Mathematics

## Mathematics Main Menu

# Hudson Colloquium Series - 2011

## Fall 2011

**November 16
Sungkon Chang**

I

*ntroduction to Differential Geometry*

Abstract: This talk is the ?rst part of my presentation for Differential Geometry and General Relativity, and the second part will be given in Spring 2012. Differential Geometry was rapidly developed around 1800 by great mathematicians such as Gauss, Riemann, and Poincaré, and became the source of modern mathematical subjects such as Lie/Representation theory, Algebraic Topology, and Algebraic Geometry. As the ideas in all these purely abstract mathematical subjects were used in modern physics, Differential Geometry serves as a subject in the undergraduate curriculum by which both math and non-math major undergraduate students may appreciate pure mathematics and the practicality of the art of mathematical thinking. Fully understanding this talk requires Calculus 3 and linear algebra, but as the subject directly appeals to our geometric intuition, I would like to invite all students who are interested.

**November 2
Lorrie Hoffman**

*Best estimators in broken-line regression*

Abstract: Assuming cause and effect for physical phenomena leads scientists to expect that all data sets are generated by some underlying model. In simple models the effect (outcome, say y(t), called the dependent, predicted or response variable) is linearly related to the cause (perhaps, time t, called the independent or predictor variable). Regression analysis is a statistical technique used to compute parameters of underlying models. Depending on the definition of “best”, we can generate different estimators (formulas/functions) by conducting a regression analysis. Parameters of the model are estimated by these statistics/estimators as functions of the data. Certain properties such as unbiasedness and minimum variance are considered strong indicators of the “goodness” of the estimators. We will define and explore some of these properties when assessing various estimators in the single line model case. We will briefly discuss extending the investigation to estimators under the broken-line regression scenario, the case where there are two lines, one consisting of a trend line and the other a horizontal ray that meet at a joinpoint. This work is theoretical in nature and mirrors the assessment efforts via simulation undertaken in the paper by Hoffman, Knofczynski and Clark (2010).

**October 14
Mark Budden, Western Carolina University**

*Paley graphs*

Inspired by Paley's use of the Legendre symbol to construct Hadamard matrices, Sachs first defined Paley graphs in 1962. Such graphs have vertices identified with the elements of a finite field $\mathbb{F}_q$, where $q\equiv 1\pmod{4}$ is a power of a prime, and an edge connects vertices $a$ and $b$ if and only if $b-a\in \mathbb{F}_q ^{\times 2}$, the subgroup of squares in the multiplicative group $\mathbb{F}_q ^{\times}$. In this talk, we will consider a generalization of Paley graphs known as character difference (di)graphs and will investigate some of their basic properties. In particular, we find that many problems involving adjacency in such (di)graphs may be solved through the tedious evaluation of character sums in number theory. Furthermore, many classical number theoretic problems (such as finding the size of maximal difference sets of quadratic residues) may be approached by studying the properties of such (di)graphs.

**October 4
Tricia Muldoon Brown**

*The Robinson-Schensted Algorithm*

Abstract: We introduce two familiar combinatorial objects, partitions and permutations, and give a bijection between sets of these objects. An algorithm due to Robinson and Schensted that makes use of another combinatorial object, Young diagrams, defines this bijection. The talk concludes with some questions.

**September 21
Travis Trentham**

*The Axiom of Choice*

Abstract: In this discussion we will be looking at the Axiom of Choice and some of the statements which are equivalent to it. In particular, we will be looking at Zorn's Lemma and the Well-Ordering Principle as well as some applications of all three. These applications will highlight not only the utility of the Axiom but also how it tends to keep popping up in a surprisingly broad spectrum of mathematical disciplines. Additionally, we will look at how the Axiom arises in a way that makes its usage seem both inevitable and necessary. However, there are a number of remarkable (some would say "disturbing") consequences of its usage, which we will be looking at that might make you want to completely reject its veracity. This discussion will be example-heavy and should quite accessible to an undergraduate audience.

**September 7
Jim Brawner**

*Folding Graphs into Polyhedra*

Abstract: In the branch of mathematics called graph theory, a graph is a collection of dots, called vertices, some of which are connected by line segments or curves, called edges. In this talk we will explore the question of which graphs can be folded, by gluing together various vertices, and perhaps stretching edges as needed, into three-dimensional frames of particular polyhedra. To whet your appetite, can you see how you might fold the bug-shaped graph on the left into a cube? How about the “hexastar” graph on the right? Can you also fold the hexastar graph into an octahedron? The talk will focus on finding graphs that can be folded into two different polyhedra.

## Spring 2011

**April 20
Lorrie Hoffman**

*A Non-Least-Squares Method For Estimation of Broken-line Regression to Determine Plateau Time of Outputs In Some Animal Science Studies*

Abstract: Broken-line regression has been used to estimate parameters in increasing or decreasing linear trends that later plateau. Most approaches employ least squares methods that under usual assumptions lead to maximum likelihood estimates (Gill, 2004). In Hoffman, Knofczynski and Clark (2010) an estimation procedure called MMNPR is derived and simulation studies show that parameter estimates of the broken-line regression perform better than the MLEs in small sample situations when data is from distributions where error terms are ill-behaved. A review of applied science literature revealed many studies using broken-line regression. We note three animal studies that report parameters based on least squares approaches: 1) determining times of steady milk production in cows (Sahinler, 2009), 2) transport of urea in mice (Marini, Lee and Garlick, 2006) and 3) the maximum sustained effect of dietary copper on chicks (Robbins, Norton and Baker, 1979). We investigate the appropriateness of underlying assumptions concerning correlation and homoscedacity. We then revisit the analysis of the animal data using the MMPNR technique, discussing differences between parameter estimates based on least squares and those based on the MMNPR technique.

**April 6
Robin Wilke**

*Generating Lattice Polygons With Specified Number of Interior Points*

Abstract: How many unique lattice polygons are there with three interior points? The answer is... God knows! Finding the number of nonisomorphic two-dimensional convex lattice polygons with a specified number of interior points i is surprisingly hard: the number of inequivalent polygons with i ≥ 2 is still an open problem. We propose a new way to construct a complete set of lattice inequivalent polygons for any i, based on the relative minimality or maximality of polygons instead of the number of sides, and demonstrate it for i = 1 and i = 2. The algorithm presented allows insight into which polygons can be generated from which other polygons, inducing a directed graph structure into any set of inequivalent polygons for a value of i. Some combinatorial properties of these graphs will also be discussed. Finally, we discuss applications to cryptography, the theory of modular forms, and several problems in computational geometry.

**March 23
Sungkon Chang and others**

*Reflections on the 71th Annual Putnam Mathematical Competition*

Abstract: In December 4, 2010, an intrepid team of Armstrong students spent most of their Saturday working on a dozen frighteningly challenging mathematical problems. Why? Just another installment of the notoriously difficult William Lowell Putnam Mathematical Competition. Members of the team and faculty members will discuss solutions of their favorite problems from the most recent competition.

**March 2
Michael Tiemeyer**

*Four-Cycle Frames of the Complete Multipartite Graph*

Abstract: PDF version

**February 9 (postponed from January 26)
Joshua Lambert**

*Exploring Rational Residue Graphs*

Abstract: During the early 1960's, Horst Sachs introduced Paley graphs to the world of mathematics. Paley graphs tie together graph theory and number theory. We shall explore the symmetry of these rational residue graphs alongside the calculation of the number of triangles and the number of edge-disjoint hamiltonian cycles.