Math Talks

University of Utah

Patrick Dylan Zwick

This is a collection of files for talks I've given in or for the math department or at other departments.

Undergraduate Colloquium Fall 2007 - How Far Can You Go With Calculus?

Abstract - So there's no question calculus is useful. In fact, it's probably the most useful idea in the history of mankind. But, what you might not be aware is that by using some pretty basic ideas from calculus you can actually prove some amazing things. Things you wouldn't expect, and that hint at much deeper mathematics. In this talk I will give three examples of this where we will use basic calculus ideas, especially sequences and series, to prove some astonishing results. We will, for example, derive a closed form formula for the Fibonacci sequence and figure out a proof of the infinitude of primes, discovered by Euler, that is completely different than Euclid's classic proof. It should be fun, and while the results will probably be new, the math used will not go beyond first year calculus.

Note - I really only did two examples, and only the first really involved calculus. However, the first result, which was the first real result in the field of analytic number theory, is so cool that it makes up for all the rest. At least, I think so.

Lecture Notes
Notes without Drawing
Notes with Drawing


There's a silly little diagram representing the ancient construction of the golden ratio. I wanted to draw it in Latex, but couldn't figure it out before the talk, so I just drew it by hand and scanned it. I've posted both the scanned version with the drawing, and the nicer version without the drawing.

GSAC Spring 2008 - NP or Not NP. That's a million dollar question.

Abstract - In this talk I introduce the concept of algorithmic complexity and give some examples. We'll then examine two classes of problems, P and NP, and discuss some ideas about them such as NP completeness. We'll end with a statement of one of the seven millennium problems, that by the end of the talk you should understand.

Lecture Notes
Notes without Drawing
Notes with Drawing


Again, the notes with drawing contain three drawings on the notes that I drew by hand and then converted to a .pdf file with a scanner.
A link to the algorithms class I plugged at the end of the talk is here

Undergraduate Colloquium Spring 2008 - Knots, Links, and Invariants

Abstract - This talk is a basic introduction to knot theory. I introduce the concept of a knot, and the more general concept of a link, and look at some of the questions mathematicians and knot theorists ask about these objects. I then look at invariants, which are constructions on knots that help to answer some of these questions, and I develop three invariants that can be useful in distinguishing knots.

Lecture Notes
Notes


Undegraduate Colloquium Fall 2009 - Groebner Bases

Abstract - Grobner bases are an exciting tool in computational commutative algebra that have been applied in a wide variety of problems since their invention about forty years ago. Given their relatively recent development and widespread use, you might expect they're complicated and difficult to understand, but they're not. Basically, all you need to understand is polynomial division! In this talk I'll discuss the problems inherent in dividing polynomials with more than one variable (more precisely, in determining if a given polynomial is in an ideal, which is a term I'll define), and then I'll explain what a Grobner basis is and how it addresses these problems.

Lecture Notes
Notes


GSAC Fall 2009 - Voting and Group Choice

Abstract - In this talk, we'll discuss different voting systems that seem, at first glance, to be fair, and then explore the limitations and problems of each of them. We'll then discuss what kind of properties we'd expect a fair voting system to have, and then see how restrictive these seemingly simple properties are on our possible voting systems. We'll end with a proof of Arrow's Impossibility Theorem, which is a surprising result about the possible voting systems allowed given a few simple criteria. The talk assumes no background in the subject, and should be understandable by everybody.

Lecture Notes
Notes


Graduate Student Combinatorics Conferences 2013 - Tropical Quadrics

Abstract - This is a quick 20 minute talk I gave in 2013 at the graduate student combinatorics conference at the University of Minnesota in April, 2013. The talk introduces the basic ideas behind tropical geometry and its particular application to the study of tropical quadrics. Then, some combinatorics questions that arise in the study of tropical quadrics are discussed.

Lecture Notes
Notes


Ph.D. Dissertation Defense on Tuesday, May 20th, 2013 - Symmetric Tropical Matrices

Abstract - These are the Beamer slides for my dissertation defense talk. They've been modified slightly, removing some slides specific to the day. This is my talk about my dissertation. The main theme is tropical and Kapranov ranks of symmetric tropical matrices. A much more detailed descrption, along with the dissertation itself and some related files, can be found here.

Beamer Slides
Slides