Presentation Topics for Math 204, Spring 2020
Each student in the class will work on a short project and present it to the class at some point during the semester. The presentation can be based on one of the many "Topic" sections from the textbook, or it can be based on some other topic that you select in consultation with me. Some suggested topics are given below.
You will give a ten-to-twenty minute presentation to the class, depending on your topic. You will also turn in a write-up (or possibly PowerPoint slides), which ideally could be published on the course web site. Details will be subject to negotiation. After selecting your topic and getting it approved, and before you give the presentation, you should meet with me to discuss what should go into your project and presentation.
Note that all of the presentation topics must be different, so if you have a preference for some topic, you should claim it soon!
Some Possible Topics
Here are some suggestions for "Topic" sections in the textbook that might make an interesting project, along with a few additional possibilities that I came up with. Note that Topic sections in the book include exercises that might contribute ideas for your presentation and that solutions to those exercises are given in the solution manual for the textbook. However, you are also expected to do some extra research outside of the textbook.
- Topic: Accuracy of Computations (Chapter 1, page 67)— Shows how inaccuracies in numerical calculation can lead to significant errors when solving systems of equations. This is an essential topic that can be done early in the semester. Hopefully someone will decide to do it and get their project out of the way!
- Topic: Analyzing Networks (Chapter 1, page 71) — Best for someone who knows something about the physics of electric circuits. Shows how systems of linear equations can be used to predict current flow in electrical networks, as well as similar things such as traffic flow in networks of roads.
- Topic: Fields (Chapter 2, page 144) — A "field" is an algebraic system with addition and multiplication operations satisfying certain properties. The real numbers are one example, but there are others. The course studies mainly vector spaces over the real numbers, but other fields could also be used. This topic would look at the definition of a field, give some examples, and consider matrices and vector spaces over other fields. It would be good for the person doing this topic to already have some knowledge about the finite fields Zp.
- Topic: Crystals (Chapter 2, page 146) — For someone interested in crystals (maybe a geoscientist), this topic shows how the structure of crystals can be described using vectors.
- Infinite dimensional vector spaces (Not from textbook) — Almost all of the vector spaces in the textbook are "finite dimensional," but infinite dimensional vector spaces are also interesting. This topic would look at examples of infinite dimensional spaces and would consider what a basis for such a space would look like. Could be done after we have covered Chapter 2.
- Topic: Magic Squares (Chapter 3, page 300) — A magic square is a square matrix where the sum of the entries in a row, column, or diagonal is the same for all of the rows, columns, and diagonals of the matrix. This topic discusses magic squares, but it mainly proves something about the vector space of all possible magic squares of a given dimension.
- Topic: Markov Chains (Chapter 3, page 305) — Markov chains are important tool in the study of certain kinds of random processes. The probabilities that determine a Markov chain can be represented by a matrix. This topic has some interesting examples of applying Markov chains to games.
- Topic: Orthonormal Matrices (Chapter 3, page 311) — An orthonormal matrix represents a linear transformation that preserves distance. Any distance-preserving linear transformation of Rn is given by an orthonormal matrix combined with a translation. This is related to the "affine maps" mentioned in the next two topics. (It might be nice to have three people give a set of coordinated presentations on these three topics!)
- Affine maps and computer graphics (Not from textbook) — An affine map is a linear transformation plus a translation. They are an important geometric tool in computer graphics. See http://math.hws.edu/graphicsbook/c2/s3.html, especially Section 2.3.8, and http://math.hws.edu/graphicsbook/c3/s5.html.
- Affine maps, fractals, and the Chaos Game (Not from textbook) — Affine maps can also be used to create fractals, via "iterated function systems." This idea is used in the "Chaos Game," which draws the fractals defined by affine maps using a random process. See http://math.hws.edu/eck/js/chaos-game/CG.html for a web app that implements this idea.
- Dual and double-dual vector spaces (Not from textbook) — The dual vector space, V*, of a vector space V is the space of homomorphisms from V to R. As mentioned in the textbook, for finite-dimensional vector spaces, V* is isomorphic to V, but only because they have the same dimension. You have to choose a basis to get the isomorphism. There is no "natural" way to define the isomorphism, independent of basis. However, the double dual, V**, which is the space of homomorphisms from V* to R, is isomorphic to V in a natural way, even in the infinite-dimensional case.
- Topic: Cramer's Rule (Chapter 4, page 359) — This topic looks at a formula for the solution of a system of linear equations that uses determinants.
- Topic: Stable Populations (Chapter 5, page 452) — This topic discusses a problem in population dynamics that can be represented by a matrix and shows how to solve the problem using eigenvectors and eigenvalues.
- Topic: Page Ranking (Chapter 5, page 252) — "Page rank" is an algorithm (that is, a computational procedure) that was invented by the people who founded Google to rank web pages. It is one of the most famous algorithms ever invented. This section discusses page rank and how it relates to eigenvectors and eigenvalues. (Page rank can also be considered to be a Markov chain; Markov chains are discussed in a previous topic.)
- Topic: Linear Recurrences (Chapter 5, page 456) — A linear recurrence defines a sequence of numbers by defining a term of the sequence as a linear combination of previous terms. An example is the recurrence that defines the Fibonacci sequence: F(n) = F(n-1) + F(n-2). This topic shows how to represent a linear recurrence as a matrix and how to solve the recurrence using eigenvectors and eigenvalues.