*The Undergraduate Mathematics Research Colloquium Series features public lectures by mathematics researchers and educators about significant mathematics research that is accessible to undergraduate students.*

*This activity is sponsored by the RTG in Logic and Dynamics in the Department of Mathematics at UNT. Contact the UMRC series organizer, Professor John Krueger, at jkrueger@unt.edu for more information.*

**Link to archive for UMRC events occurring after Fall 2011 **

**Organizer: Professor John Krueger**

December 6, 2011**Speaker:****William Cherry **(University of North Texas)* Title:* Analogies between Numbers and Functions

**When we first begin studying mathematics, we think of numbers and functions as very different sorts of objects. We also tend to think of numbers as relatively simple compared with functions. As we start to study abstract algebra, we also start to study similarities between numbers and functions. I will illustrate various instances of this similarity between numbers and functions. I will discuss some basic examples, such as long division and partial fractions. I will then move to the Stothers/Mason theorem for polynomials and the ABC conjecture for integers. Here I will make a connection to Fermat's Last Theorem. In my lecture I will also introduce some exotic absolute values that are different from our familiar notion of absolute value. By the end of my talk, I will try to convince you that although numbers feel much more intuitive to us, questions about numbers are, in fact, much more subtle that questions about functions.**

*Abstract:*poster

October 18, 2011**Speaker:****Patick McDonald **(New College of Florida)* Title: *Constructing Undergraduate Research Problems: Ideas and Examples

*Constructing research projects in mathematics for undergraduates is much more of an art than a science; recognizing latent ability and shaping experiences to develop it is a subtle problem requiring a variety of resources. In this talk I will discuss a number of important parameters involved in the process of creating a successful undergraduate research project and I will describe strategies which have proven successful over a range of topics. Examples of successful projects will be presented. These examples will include results involving spectral geometry, harmonic analysis, probability, mathematical modeling, and genetic algorithms. The talk is intended for a general math audience.*

**Abstract:**poster

September 16, 2011**Speaker: ****Solomon Friedberg **(Boston College)* Title:* Packing Primes

Abstract: Prime numbers are the building blocks of all whole numbers and the basis for modern applications such as sending credit card numbers over the Internet securely. In this talk I will describe the ways that the prime numbers, though individually distributed erratically, accumulate into regular patterns when packed together. In looking at this, we will find out about links betwen prime numbers and the numbers "e" and "pi," which seem to have nothing to do with primes whatsoever!

poster

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# Spring 2011

**Organizer: Professor Steve Jackson**

April 26, 2011**Speaker:****Joseph A. Gallian **(University of Minnesota Duluth)* Title:* Using mathematics to create symmetry patterns

*We use video animations to illustrate how mathematics can be used to create computer generated symmetry patterns. Polynomials, exponential functions, logarithms and modular arithmetic are used to transform basic images into symmetry patterns. These methods were used to create the image for the 2003 Mathematics Awareness Month Poster.*

**Abstract:**poster

March 29, 2011**Speaker:****Peter Doyle **(Dartmouth College)* Title:* Surfing backwards

*If you could surf backwards on the web, how would you decide where to go?*

**Abstract:**poster

March 2, 2011*Speaker:** ***John Quintanilla **(University of North Texas)* Title: *The mathematics of music and language

*Trigonometry has plenty of applications beyond solving for the missing parts of a triangle. In this talk, we will discuss the application of trigonometry to music, phonetics, and voice recognition software. While foreshadowing sophisticated mathematical applications like the Fourier transform, the talk will be accessible to students who have completed a course in Precalculus and Calculus I.*

**Abstract:**poster

January 25, 2011**Speaker:****David Vogan **(Massachusetts Institute of Technology)* Title:* Regular polyhedra and simple groups

*In two dimensions there are regular polygons with any number of sides bigger than two, and all of them are pretty easy to understand. In three dimensions, there are regular polyhedra having 4, 6, 8, 12 or 20 sides; no other numbers are possible, and some of these polyhedra are amazingly subtle geometric gadgets. I'll talk about how Felix Klein rephrased this fact in terms of "group theory" (and what that means). Building on Klein's work, John McKay about twenty years ago related regular polyhedra to a special kind of "graph." Here's a picture of the graph for the regular icosahedron :*

**Abstract:**3

l

1 - 2 - 3 - 4 - 5 - 6 - 4 - 2

Each of the numbers is one half the sum of the adjacent numbers. I'll give at least some hints about where this graph comes from.

poster

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# Fall 2010

**Organizer: Professor Steve Jackson**

December 7, 2010**Speaker:****Yiannis Moschovakis **(UCLA and University of Athens)* Title:* Solving equations in algebra and arithmetic

*Do the equations 2x+3=0, x5-x+1=0, 17x-37y=1, x5+y5=z5 have solutions or integer (whole number) solutions, and how many? These simple problems are introduced in Elementary School arithmetic and High School algebra, but their serious examination leads to deep and difficult problems in many areas of mathematics, including number theory, geometry, computer science---and somewhat surprisingly, mathematical logic. The purpose of this talk is to trace how these problems arise and to discuss some of the fundamental results to which they have led. The lecture is elementary, requiring little background beyond High School algebra and the basics of analytic geometry.*

**Abstract:**poster

November 2, 2010**Speaker:****Michal Misiurewicz **(Indiana University Purdue University Indianapolis)* Title:* Periodic orbits of simple dynamical systems

*Let*

**Abstract:***f*be a continuous map of a closed interval into itself. In the theory of Dynamical Systems we iterate such a map (repeat it again and again). If we start with some point

*x*and after n iterates get back to

*x*, such

*x*is called a periodic point. The smallest

*n*for which this happens is the period of

*x*. Sharkovsky's Theorem tells us that existence of periodic orbits of some periods implies existence of periodic orbits of some other periods. It can be generalized in several ways. One is to consider not only the period of a point, but also the pattern of its movement under the iterates of the map. The other is to replace the interval by a more complicated space.

poster

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