Contents
Fall 2023 Teaching
I am teaching Math 274 (Calculus II) and Math 477 (Topology) this semester.
Courses Taught
Math 273 (Calculus I)
James Stewart rightly wrote that calculus is “considered to be one of the greatest achievements of the human intellect.” Not only do the ideas of calculus have a beauty of their own, but it would be hard to overstate the influence the subject has had on scientific innovation (see the quote here that begins “Without calculus,…”). In calculus, you study change in the context of functions, and functions are the way that we model the world with mathematics.
This is a first semester calculus course, covering what has become the standard curriculum for university calculus courses in the US. The majority of the semester focuses on limits, derivatives, and a few applications of derivatives. Integration is introduced towards the latter part of the semester and the fundamental theorem of calculus is covered.
Book. We will use OpenStax Calculus, Volume 1, by Herman and Strang. The book is available for free online — both in the browser and in PDF format (for the PDF, follow the browser link, and click on “Download a PDF”). If you wish, the linked website also has the option to order a print copy.
Coursework. In addition to tests and a final exam, the students will complete regular quizzes and homework from the web-based system WebWork. In the course there are also several lab assignments, in which the students get an introductory experience with SageMath software. See the course site on Blackboard for more details.
Math 274 (Calculus II)
Calculus I focused on the main ideas of calculus – limits of functions, derivatives and their uses, and integrals. However, only so much time is available in one semester. In that first semester, integrals are introduced near the end and the Fundamental Theorem of Calculus is presented. In Calculus II, we pick up from that point. The class discusses uses of integration and techniques for integration (when an antiderivative is less easy to determine). In addition, approaches are presented for approximating integrals when an elementary antiderivative does not exist. Instead of approximating the integral, one could try to approximate the function being integrated. This leads to an important technique for approximating functions with a power series.
Book. We will use OpenStax Calculus, Volume 2, by Herman and Strang. The book is available for free online — both in the browser and in PDF format (for the PDF, follow the browser link, and click on “Download a PDF”). If you wish, the linked website also has the option to order a print copy.
Coursework. In addition to tests and a final exam, the students will complete regular written homework and quizzes. In the course there are also some lab assignments that reinforce class ideas from another viewpoint and, in a few cases, introduce a new related idea. 1 In this course, students will use SageMath software for these labs. See the course site on Blackboard for more details.
Math 267 (Introduction to Abstract Mathematics)
When I am asked what this course is about, I answer that the objective is to help students transition from simply doing math procedures to using math ideas in order to create new insights. It involves truly disciplined, critical thinking. It also requires creativity. Such skills are among the most desirable to have in a competitive job market.
Often described as a “bridge to higher mathematics,” one of the main purposes of courses like this one is to help students gain the (math specific) reading and writing skills necessary to succeed in many upper-division courses and beyond. A lot of this effort centers around learning how to read and write proofs in mathematics. As a part of this, the students are expected to develop their ability to communicate in the mathematical realm, using correct quantifiers, clear definitions, etc.
Book. We will use the book Reading, Writing, and Proving by Daepp and Gorkin. Rather than going in a linear fashion through the book, class readings jump around to different sections.
Team Homework. Students work together in rotating teams of 4. In order to make the experience as effective as possible, I suggest the following.
| Do | Don’t |
| Individually work on all the exercises before team meetings | Think about just 1/4 of the exercises. |
| Write multiple drafts of a solution | Have last-minute team meetings |
| Throw away ideas that go nowhere | Ditch an idea without first explaining it to a teammate |
| Ask questions, of yourself & teammates
what is this \(A\) that you wrote? |
Let a team homework solution be submitted without trying to understand it |
Math 457 (Differential Geometry)
In this course, we will build up the ideas of differential geometry on curves and surfaces in \(\mathbb R^3\). The “geometry” part of the title is there because the subject works with angles and lengths in a fundamental way. The “differential” part is there because you don’t restrict yourself to straight lines and planes, but you parameterize a given curve (or surface) and use derivatives to understand its relationship to its tangent lines (or planes). A big part of the course involves the study of the curvature of both curves and surfaces.
The textbook is Elements of Differential Geometry by Millman and Parker.
Differential geometry (in a more general setting, at least) has many important applications. It is an essential foundation for Einstein’s general relativity — the currently best description of gravity in modern physics. Aspects of it are used for the study of conservation laws in physics and for thermodynamics. It also has a wide array of other applications: in geology, in chemistry, even in computer science (in image processing, for example).
Math 477 (Topology)
In this course, we will use the textbook Principles of Topology by F.H. Croom. Class materials can be found on the class Blackboard site.
On a few occasions throughout the semester we will discuss some applications. These will be drawn from several resources, including Topology and its Applications by W. Basener and Elementary Applied Topology by R. Ghrist.
The video below gives a taste of how topologists solve problems. The videos on the YouTube channel 3Blue1Brown are generally pretty great.
Update: In 2020, after the video above was made, Joshua Greene and Andrew Lobb solved the inscribed square problem (in fact, they solved a more general problem). They did so by using topology and symplectic geometry techniques.
Here is a link to their paper, the preprint, and a link to an article in Quanta Magazine that was written for a more general audience.
Math 490 (Senior Seminar)
This semester the topic for my senior seminar course will be Mathematics and Machine Learning (ML). We will discuss some of the kinds of problems that ML attempts to solve, some of the algorithms used, and the mathematics that is necessary to justify the use of techniques, to understand and interpret the results, and to explore new approaches.
In place of a textbook students should study from their class notes. Class materials will be made available on Blackboard and this website. Course material is being adapted from a few sources, including the book Mathematics for Machine Learning by Deisenroth, Faisal, and Ong. The entire text is available here.
Giving presentations, and typing your work. I strongly endorse the use of \(\LaTeX\). The easiest way to begin is to create a free account on Overleaf, where you can create and store your \(\LaTeX\) documents in the cloud. This platform allows you to share a project with others (allowing multiple people to edit). If you wish to download the \(\LaTeX\) software to your personal device, there are a few options. Some, for Windows, are recommended by Dr. Kolesnikov here. If you use a Mac, then I believe that TeXShop is a popular option.
The way to create presentations using \(\LaTeX\) is by having the document class be beamer. Here is a template file for making such a presentation. Individual slides are made by putting them in a \begin{frame}...\end{frame} environment. To make content on a slide appear after clicking/hitting forward arrow, you can use the \onslide<2-> command (content after this command appears on the slide after going forward once). If you want to have content appear only after clicking twice, change the 2 to 3, and so on. Alternatively, you can make text transition from nearly transparent to regular (on a click) by using the \pause command.
Overleaf has resources for learning LaTeX as well as tutorials. For an encyclopedic resource for using \(\LaTeX\), go to the Wikibooks LaTeX site.
- Working with computational software gives students the ability to work on techniques that would be overly cumbersome to complete by hand. At the same time, such software tools should always be viewed as merely a supplement to thinking through math problems on one’s own (with pen and paper). It is impossible for you to make computational software be the powerful tool that you want it to be without spending time in “analog mode,” thinking through the ideas on your own. ↩