Mathematics

Hudson Colloquium Series - 2012

2012

November 28
Sungkon Chang
The Geometry of Einstein's Theory of Relativity, Part II

Abstract: This talk is the second part of the series Geometry of Einstein's Theory of Relativity, and the main goal of the talks is to introduce how abstract geometry was used to realize Einstein's vision.

Around 1800 differential geometry for Euclidean spaces was developed by mathematicians and mathematical physicists, and in 1828 Gauss published his famous Theorema Egregium (Remarkable Theorem), which concerns curvature of surfaces.  In 1854, inspired by Gauss' theorem, Riemann expressed in his work so-called Riemannian Geometry where he introduced an intrinsic property of a differential manifold, so-called Riemannian curvature tensor.  Although Riemann's published work was received with great enthusiasm in the community, the development was slow, and in 1900 when the development of tensor calculus culminated it was not received with enthusiasm.  However, in 1916 Einstein successfully used Riemannian curvature tensor in his general theory of relativity, and it made a great impact.   Although a solid mathematical foundation of Einstein's theories came  later, the abstract geometry used in Theory of Relativity is a supreme example of the practicality of the art of mathematical thinking.

Introduced in the earlier talk was the special relativity.  In this talk we shall review the basic elements of Riemannian Geometry and the special relativity, and focus on the geometry of general relativity.  Calculus III and Linear Algebra are the minimum pre-requisites for following the contents of the talk, but all those who are interested are invited.

November 14
Dr. Tim McMillan
Two Events from the History of Mathematics

Abstract:  In this colloquium we will present two vignettes from math history. First, Euler solves the Basel Problem, evaluating  \sum_{n=1}^\infty 1/n^2.  Mathematicians puzzled over this problem for several years successively improving decimal approximations until Euler brought it to its surprising solution. The second episode will give an illustration of what is called Archimedes Method. This procedure is alluded to in some old Greek correspondences, but until a late 19th Century discovery was not at all understood. Archimedes used his method to determine certain geometric measurements. Knowing the solutions he then proved them axiomatically.

October 24
Dr. Paul Hadavas
Mathematics in Politics

Abstract:  The field of Operations Research covers many topics in mathematics.  Two of those topics, Game Theory and Analytics, have direct applications to politics.  In this talk, we will cover an overview of these two areas, how they relate to politics, and what they say about the upcoming election. 

October 10
Dr. Michael Tiemeyer
On Four-Cycle Factorizations

Abstract:  A talk about the ups and downs of the STEP experience during Summer 2012.

September 12
Dr. Michael Tiemeyer
"Clicker Classroom Response System"

Abstract:  What is it and why might you be interested in using it?

April 18
Dr. Sungkon Chang and Robert Fenney
The Geometry of Einstein's Theory of Relativity, Part I

Abstract: Around 1800 the differential geometry for Euclidean spaces was developed by mathematicians and mathematical physicists, and on top of these accomplishments Riemann defined Riemannian Geometry which embraced sporadic examples of non-Euclidean geometry discovered during that time.  By birth of Riemannian Geometry a wealth of non-Euclidean geometries were created, and it is an excellent example of the art of mathematical thinking that realized one's imagination and vision.

Albert Einstein knew that Riemannian Geometry is the right framework to realize his vision, and in 1905 Special Relativity was published, and the final form of General Relativity was published in 1916.   Although a solid mathematical foundation of these theories came later, the abstract geometry used in these theories is a supreme example of the practicality of the art of mathematical thinking.

In this talk we shall introduce the basic elements of Riemannian Geometry and the geometry of a spacetime.  We shall end this talk with an introduction to General Relativity, and in a sequel which will be given in the fall we will discuss the General Relativity and its application to Cosmology.  Having taken Calculus III and Linear Algebra will help to follow the contents of the talk, but we would like to invite all current Calculus-III students, and all those who are interested.

Spring 2012

April 6
Jim Coykendall, North Dakota State University
A Tour of Factorization

Abstract: Factorization is one of the first concepts that any person encounters in their mathematical journey. Although factorization is a familiar concept, it can be quite deep and mysterious; sometimes it baffles the intuition. 

Throughout the past 20 years or so, the concept of factorization in integral domains has become a rich and vibrant area of study in algebra. The aim of this talk is to give an overview of modern factorization theory from an example-oriented point of view. We will not be heavy-handed with respect to rigor, but rather, try to give the audience a general intuitive feel for the questions and results that motivate this area of research. And many examples will be given to shine light on the beauty and elegance of the structure of factorization.

March 21
James Brawner, Sungkon Chang, Joshua Ferrerra
Reflections on the 72th Annual Putnam Mathematical Competition

Abstract: In December 3, 2011, 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 7
Dr. Joshua Cooper, University of South Carolina

The Minimum Number of Givens in a Fair Sudoku Puzzle is 17

Abstract: Add one more reason to love the number 17*.  McGuire, Tugeman, and Civario surprised many in the mathematics-of-Sudoku community on January 1, 2012 by posting the result of a huge computation apparently confirming the long-held suspicion that the fewest number of givens in a fair Sudoku puzzle is 17.  The proof is a combination of clever programming techniques, mathematical analysis of so-called "hitting sets", and a massive supercomputer computation.  We discuss how they did it, what it means, and where to go from here.  * Just for some purely mathematical examples: 17 is a twin prime, a Mersenne prime exponent, an Eisenstein prime, a Fermat prime, the number of wallpaper (plane isometry) groups, the length of the longest Berlekamp-Graham "perfectly distributed" sequence in [0,1], the least nontrivial hexadecimal repunit prime, the number of orthogonal curvilinear coordinate systems up to conformal symmetry for which the 3-variable Laplace equation can be solved using separation of variables, and the least number that can be written as the sum of a positive cube and a positive square in two different ways.

February 22
Dr. Michael Tiemeyer
Cycle Frames of Multipartite Graphs

Abstract:
Let M(b,n) be the complete multipartite graph with b parts of size n.  A z-cycle system of M(b,n) is said to be a cycle-frame if the z-cycles can be partitioned into sets such that each set induces a 2-factor of M(b,n) minus some part.  The existence of a z-cycle-frame of M(b,n) has been settled when z = {3,4}.  Here, we consider z-cycle-frames when z>4 is even.

February 8
Dr. Travis Trentham
Purgatory Domains 
 

Abstract: We begin by recalling that an irreducible x in an integral domain D is any nonzero nonunit of D that cannot be expressed as a product of more than one nonunit of D.  For example, every prime integer is irreducible in the ring of integers.  Sometimes irreducibles are also called atoms.  If every nonzero nonunit of our domain D can be expressed as a product of atoms, then we say D is atomic.  Every UFD and every Noetherian domain is atomic.  Examples abound of (non-field) domains that contain no atoms at all.  These rings are called antimatter domains and it can be argued that they constitute the worst of all possible worlds from a factorization point of view.  However, there is a class of domains which form a middle ground in that they contain atoms but also have elements which do not admit atomic factorizations.  Such a ring is called a Purgatory domain.  Hence, every domain is either atomic, an antimatter domain, or lies in Purgatory.  We will consider examples of these rings and demonstrate how the isomorphism class of any domain can be uniquely associated to the isomorphism class of a Purgatory domain.  Hence, if there is a largest class among these three types of domains, it must be the Purgatory class.

January 20
Dr. Yan Wu, Georgia Southern University

Stability Analysis of Adaptive Control of SIngle-Loop Thermosyphon via Wavelet Network

Abstract
Compactly supported orthogonal wavelets have certain properties that are useful for controller design.  A wavelet network serves as a universal approximant for functions in L^2(R^n) in  such as a nonlinear state feedback controller.  The mechanism of a wavelet controller is revealed via integration with linear time-invariant systems.  We obtain closed-form bounds on the design parameters of a wavelet controller which guarantee asymptotic stability of the wavelet-controlled LTI systems.  We further investigate global stability of wavelet-controlled nonlinear dynamical systems, such as a loop thermosyphon, in which the nonlinearity of the system is approximated so that a classical proportional state feedback control is sufficient for stabilizing the chaotic flow in the thermosyphon.