Fall 2019 Schedule
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.
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 N2to 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
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.
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 18th 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 P3 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 19th 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)
|Peter Alfeld||Lisa Penfold|
|Course Instructor||Administrative Coordinator|