## 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**

**Root loops and Galois groups**

*Abstract:*In 1829, a French undergraduate named Évariste Galois revolutionized our understanding of polynomials through a careful study of the symmetries of their roots. Though he was killed in a duel just three years later, his ideas pervade modern mathematics. In this talk, we give a pictorial introduction to some of Galois' ideas by exploring the "root loops" of polynomials over the complex numbers.

Prerequisites: It will be helpful, though not strictly necessary, for audience members to know how to add and multiply complex numbers!

**A Fractional Introduction to Fractals **

*Abstract:* How long is the coast of Great Britain? This question, although at first seemingly
simple, has a peculiar answer when approached theoretically from different length
scales due to the fractal-type nature of coastlines. In this talk we will introduce
the notion of fractals, their unique mathematical properties, and their prevalence
in nature, as well as view examples of such strange and fascinating objects.

**Dividing by Three Without the Axiom of Choice**

*Abstract:* Let A and B be infinite sets. If 3 x A is bijective to 3 x B then is A bijective to
B? This question is immediately answered in the affirmative if one evokes the axiom
of choice, but how would one go about "dividing by three" without such a powerful
tool? In this talk we will see just how difficult it can be answer such questions
from the constructivist viewpoint.

**Instant Insanity**

*Abstract:* Fifty years ago, a new puzzle was unleashed upon the ever-patient public. It consists
of four cubes whose faces are colored one of four colors: red, green, blue, and white.
The objective of the puzzle is to arrange the cubes in a row so that along the front,
back, top, and bottom each color appears once among the four cubes. The colors of
the left and right faces of the cubes are of no consequence (indeed, except for the
left of the first cube and the right of the final one, those faces are hidden).

One could call this the Rubik's cube of the sixties because it attracted so much attention and was found to be hard to solve without using some kind of mathematical tool. For this reason, the name under which it was marketed seemed appropriate: Instant Insanity.

In the colloquium we will have a few sets available to allow experiencing the difficulty of solution. Then we will learn about a mathematical technique that leads to a solution.

**Counting Rational Points on Curves and Surfaces**

*Abstract:* In integral calculus, elliptic integrals originally arose in connection with the
problem of giving the arc length of an ellipse. They were first studied by Giulio
Fagnano and Leonhard Euler in the 18^{th} century. Seemingly unrelated, one can also count the number of rational points of
an elliptic curve (roughly a torus) over a finite field – since there are only finitely
many rational points. Hasse's theorem on elliptic curves, also referred to as the
Hasse bound, provides an estimate of the number of points on an elliptic curve over
a finite field, bounding the value both above and below. In my talk, I will explain
how to count the number of rational points on a family of elliptic curves using elliptic
integrals. I will then explain how this can be used to count rational points on certain
K3 surfaces, that is quartic surfaces in P^{3} first studied by Ernst Kummer in the 19th-century.

**Adventures in the Arctic**

Mathematics Ph.D. student, Ryleigh Moore, was one of three American graduate students
invited to participate in the Multidisciplinary drifting Observatory for the Study
of Arctic Climate (MOSAiC) expedition out of Tromsø, Norway from September 20 - October
28, 2019. The flagship German icebreaker,* RV Polarstern*, will be frozen in ice and drift for a full year through the Central Arctic following
in the footsteps of an earlier 19^{th} century expedition under Norwegian explorer, Fridtjof Nansen. MOSAiC will mark the
first time a modern research icebreaker will study near the North Pole throughout
the polar winter; it is hailed as the largest Central Arctic expedition ever, with
19 countries, over 600 people, and a budget exceeding 155 million US dollars.

In this talk, Ryleigh will discuss the science goals of MOSAiC and her experiences
while on the Russian research vessel, *Akademik Fedorov*. She will discuss life on a research vessel, how the expedition identified ice floes
that would be used for instrument deployments, and her role of leading the installation
of three seasonal ice mass balance (SIMB3) buoys in the Central Arctic.

** Proofs without Words**

*Abstract:* A proof without words, or picture proof, is as it sounds: A proof of a mathematical identity or statement which can be demonstrated as self-evident by a diagram without explanatory text. Some argue that proofs without words are not
really proofs, and are thus not acceptable as *real* mathematics. I will instead argue the viewpoint of popular mathematics and science writer Martin Gardner: "In many cases a dull proof can be supplemented by a geometric analogue so simple and
beautiful that the truth of a theorem is almost seen at a glance." We will start with picture proofs that one could use in trigonometry and calculus
courses, and conclude with a picture proof about serial isogons of 90 degrees.

## 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 |