## Fall 2019 Schedule

**Cool Mathematics**

I will in fact talk about some unsolved, or recently solved, problems in Mathematics. The main purpose of the meeting, however, will be to organize the Undergraduate Colloquium for those interested in taking it for credit (1 hour credit/no credit). For many participants this will be the first class in which they have to write a technical report. This is a complicated yet gratifying task. I will discuss some of the issues involved and also give a first introduction to the use of LaTeX.

**Penrose Tiling**

*Abstract:*A given set of tiles admits a tiling of the plane if the plane can be covered without overlaps by congruent copies of tiles from the set. Sometimes, the tiling is periodic: the whole pattern can be moved without rotation to a copy of itself. The usual tilings of the plane by squares or hexagons have this translational symmetry. In 1974, Penrose discovered that there are some sets of tiles that can tile the plane but must do so non-periodically. The simplest Penrose set consists of thick and thin rhombs with matching rules that dictate which edges can be adjacent. The up-down generation method and the pentagrid method for existence of the Penrose tiling will be discussed. Slides of the lecture are available: http://www.math.utah.edu/~treiberg/PenroseSlides.pdf

**Classifying Pythagorean Triples**

*Abstract:* Number theory begins as the study of the natural numbers: 1, 2, 3, etc. When studying
the natural numbers, one quickly arrives at the fascinating mysteries surrounding
prime numbers. You might wonder: How many primes are there? How can we tell if a given
number is prime? Is there a pattern to the primes? We know some things about prime
numbers, for instance that there are infinitely many of them. However, there are
many things which are still not understood, largely because the "pattern" of the prime
numbers is unpredictable. There are many other interesting objects in mathematics
that behave as erratically as the prime numbers do, and the techniques of number theory
can often be used to study these objects as well. As an example, consider right triangles
whose side lengths are all natural numbers. We will classify all such triangles,
using techniques from arithmetic geometry and the theory of algebraic numbers. The
only prerequisite for this talk is high school algebra.

** The Fast Fourier Transform**

*Abstract:* In 1965 Cooley and Tukey discovered a revolutionary algorithm: the Fast Fourier Transform
(FFT). It reduces the number of operations required to do frequency analysis of a
signal of length N from N^{2}to about N log(N). We will look under the hood of this ubiquitous algorithm, and explore
a few applications including noise reduction, image compression (JPEG) and sound compression
(MP3).

**Isoperimetric problem and the Calculus of Variations**

The isoperimetric problem is one of the oldest problems in mathematics. On a plane, the problem asks for the plane region with a given perimeter that encloses the largest area. The fact that the circle is the solution to this problem appears to be intuitive and was known already in Ancient Greece, but it was not until 1879 that Karl Weierstrass gave the first rigorous proof of this statement. This leads to the development of an emerging field in modern mathematics known as the Calculus of Variations. Besides its links with other branches of mathematics such as differential equations and functional analysis, it finds its diverse applications in physics, engineering, economics, and biology. In this talk, we will introduce several classical problems in the field and discuss the classical techniques for solving these problems explicitly.

**University of Utah Mathematics Alumni**

Please join us for the September 25 Undergraduate Colloquium where four University of Utah alumni will share the career paths taken from a mathematics degree to their current jobs, followed by Q&A. This is a great opportunity to engage with professionals in a range of STEM careers.

**The Department of Mathematics Undergraduate Colloquium welcomes; **

** Evan Dudley, VP Credit Risk Technology, Goldman Sachs**** Jeremiah Perry, VP Liquidity Risk, Goldman Sachs Dan Eardley, Director, Mathnasium Jonathan Bown, **

**Quantitative Modeling Analyst, Zions Bank**

## Math 3000 (Receive Credit for Attending)

The Undergraduate Colloquium is open to anyone to attend; however, if students would
like to receive credit, you may register for **Math 3000**.

This is a 1 credit hour CR/NC course. To receive credit:

- You may not miss more than 2 of the colloquia
- You will need to write a short paper on one of the topics presented during the semester. (Report Specifications)

Course Coordinators

Peter Alfeld | Lisa Penfold |

Course Instructor | Administrative Coordinator |

ugrad_services@math.utah.edu |