Syllabus

Introduction #

Analysis is the mathematical language of convergence and approximation. Its ideas are ubiquitous in modern mathematics appearing in fields as diverse as machine learning, probability, dynamical systems, differential equations, as well as geometry and topology, and number theory. At its heart is the following question:

Given a mathematical object such as a function or a matrix, it is possible to approximate it using simpler objects? And can we answer important questions about the original object by studying just the simpler ones?

Rather than studying convergence individually in each of the many contexts within which it appears in mathematics, we unify the course around the notion of a metric space; in this way, our work can be applied universally. Two courses naturally follow: Math 2805: Mathematical principles of machine learning and Math 3603: Advanced analysis. Both explore important applications of this class.

Course description: Building on the theoretical underpinnings of calculus, develops the rudiments of mathematical analysis. Concepts such as limits and convergence from calculus are made rigorous and extended to other contexts, such as spaces of functions. Specific topics include metric spaces, point-set topology, sequences and series, continuity, differentiability, the theory of Riemann integration, and functional approximation and convergence.

Course Objectives: At the end of the course, a student should be able to:

  • describe the real numbers as a complete, ordered field;
  • use the definitions of convergence in the context of a general metric space;
  • determine the continuity, differentiability, and integrability of functions ;
  • apply the Mean Value Theorem and the Fundamental Theorems of Calculus;
  • compose rigorous proofs as evidenced by organization, argument, and style; and
  • describe the use of analysis in modern mathematics, including its use in the study of dynamical systems and probability.

Books and Materials #

We will very loosely follow three texts:

All three are wonderful references. My personal favorite is the second, but the course is closest to the outline of the first book. We will not use any of the texts explicitly so neither is explicitly required, but if you can, procure a copy of at least one of the three to serve as a reference. Often you will get an alternate and useful perspective on what we are working on in class.

Reading and watching mathematics: There may be times this semester where you will be asked to read a section or watch a short video about a topic related to the course. Learning mathematics is not a spectator sport. Reading mathematics is not like reading a novel; watching mathematics is not like watching an action thriller. Some paragraphs are easy to digest, but you may find yourself looking at one line of text for five or more minutes trying to understand what the author is trying to say. Use the pause button when watching a video. As you read or watch, take notes, just as you do in class. This is crucial! If questions arise, write them down and ask during office hours.

Class meetings and office hours #

We will meet in Searles 217 every Tuesday and Thursday at 2:50pm. Whenever possible and appropriate, lectures will be recorded for your enjoyment and to help you review your notes. If it becomes necessary this semester, we will meet over Zoom. The link is available on the menu to your left.

I will hold formal office hours in my office Searles 205 at:

4:15pm on Mondays, Tuesdays, and Wednesdays.

There may be times in the semester where I will have to reschedule office hours due to a conflict. I will let you know by email beforehand. I am available throughout the week for additional meetings. To schedule either individually or as a group, please send me an email with a couple of times that will work. And please don’t hesitate to set these up, I am very happy to see you.

Homework #

Homework problems will be assigned after every class, and will be due once a week. Each submission will consist of three parts:

  • Cover sheet: In addition to your name, I will also ask you to recognize individuals you worked with and sources you used to complete your work. There will also be a space to briefly discuss your weekly group meeting.

  • Class notes: As you participate in class, I expect you to actively take notes. You will submit them as part of the homework each week. They will be graded generously, but should be complete and legible.

  • Homework problems: Problems will be assigned during every lecture and formally posted on this website.

Your homework will be graded using GradeScope. To begin, you will need to set up an account.  I will send you the code for our class by email.  Each homework assignment will need to be submitted as a .pdf file.  If you edit your homework electronically, make sure you can save or export your work in this format.  If you write-up your homework the old-fashioned way using pencil and paper, use a scanner or a phone scanner app.  See the help document for a list of suggestions. Please let me know if this does not work for you; I will come up with an alternative.

Collaboration and groups: Throughout the course of the class, you will be a part of a group. While your work will be written-up individually, I would like all of you to check in with each other and discuss the course material and homework at least once a week outside of class.

Exams #

There will be three exams during the semester, tentatively scheduled for:

  • Wednesday, October 5;
  • Thursday, November 10;
  • and a take-home final exam.

I expect that the first exam will be a take-home and the second an in-class affair. More details will be announced in class.

Course Assessment #

Your grade will be based on three components: homework (30%), exams, (60%), and class engagement (10%). The latter will be assessed based on participation during my lectures, your contributions to your group as well as other students’ success in the course, as well as the discussion board. There are many ways to earn a perfect score in this category; find a way to contribute to the class and I will be happy to recognize your effort in this category.

  • contribute during lectures: questions, answers, or perspective;
  • participate actively in your group;
  • post on out discussion board;
  • if you wrote good notes for a lecture, post them on the discussion board; students missing class will truly appreciate this resource,
  • and make sure to recognize the contributions of others on the homework cover sheets.

At the end of course, I will ask you to fill out a brief form enunciating your contributions in this area. In this category, my goal is to recognize you for work that you already do.

Password #

A password will be required to access some features of the course. I will announce it in class, but if you forget it, send me a quick email.

Some axioms #

Federico Ardila enunciated the following axioms in his Todos Cuendan. They form a lens through with I view both teaching and doing mathematics.

  • Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
  • Everyone can have joyful, meaningful, and empowering mathematical experiences.
  • Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
  • Every student deserves to be treated with dignity and respect.