Rouben Rostamian

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MATH 423: Differential Geometry

Spring 2022 Course information

Class Time/Place:	MoWe 5:30pm–6:45pm, ILSB 201
Office:	MP 402
Phone:	410–455–2458
Email:	rostamian@umbc.edu
Office hours:	MoWe 4:00–5:30, or by appointment

Textbook and course content

Textbook

L. M. Woodward and John Bolton, A First Course in Differential Geometry: Surfaces in Euclidean Space Cambridge University Press, 2019.

Both printed and e-book versions are available. Frankly, I don't see how one can ever seriously study mathematics from an e-book. Do yourself a favor and buy the printed version. Not a good idea to cut corners here.

Coverage

We will cover much of

Chapter 1:

Curves in Rⁿ

Chapter 2:

Surfaces in Rⁿ

Chapter 3:

Tangent planes and the first fundamental form

Chapter 4:

Smooth Maps

Chapter 5:

Measuring how surfaces curve

Chapter 6:

The Theorema Egregium

Software

Maple from https://www.maplesoft.com/ is an all-purpose Computer Algebra System (CAS).

The main (indeed, the only) point of this course is to understand the shapes and properties of curves and surfaces in 3D. That can be done all in one's head, as it has been done for generations, but that understanding can be greatly enhanced with the proper use of graphical capabilities of the modern computer. In this course we will use Maple mostly for graphics, and to a lesser extent for computational purposes.

If you have a reasonably fast internet connection, you may be able to access Maple remotely by logging into UMBC's GL Linux machines. If your local machine is Windows, you will need a good deal of know-how to make it talk to Linux. If your local machine is a Mac, it should be easier. If your local machine is Linux, then it's quite straightforward.

If you can afford it, a better option is to buy Maple. It will cost you $\$$75 after discounts with student ID, but it's a terrific software and well worth it, considering that its normal retail price is about $\$$2400.

To purchase Maple, go to https://webstore.maplesoft.com. Select "Student" under "Choose a Category". The normal student price is $\$$99 but you get a 25% discount if you enter the "Promotion Code" associated with this course. I cannot put the promotion code on this public website; e-mail me to ask for the code.

Prerequisites:

Linear Algebra (Math 221) and Multivariable Calculus (Math 251)

Recommended preparation:

Math 301 and a grade of A in Math 251

Course Goals/Objectives

Differential geometry has its origins in Karl Friedrich Gauss's paper, Disquisitiones generales circa superficies curvas (General Investigations of Curves Surfaces) which he presented to the Royal Society of Göttingen in 1827. The paper remains quite fresh and readable (in the English translation, anyway) after the passage of almost 200 years. There have been great developments in the subject since then, and it has moved far away from its origins, but in this introductory course we will limit ourselves pretty much to what Gauss did in 1827. You may download the PDF of the English translation of Gauss's paper from https://www.gutenberg.org/files/36856/36856-pdf.pdf.

The course has the following specific learning goals:

1. Master the definitions, theorems, and proofs presented in the book, and in lectures in class.
2. Develop a geometric intuition and analytical capability to solve questions posed as examples and exercises.
3. Learn how to translate intricate theoretical constructions into static or dynamic computer graphics.

Workshop approach

This course will run mostly as a workshop. At each meeting, after a brief (15 minutes or so) lecture that introduces new material, the students will split into groups of 3 each and work together on the implications and extensions of the new idea. The work will involve paper-and-pencil calculation, proofs, and programming in Maple. There will be occasional homework with problems that require extended thinking and calculations. The solutions of the homework problems will be brought in to share and examine with the group, or for oral presentations in class.

Exams and grades

Due to the interactive nature of the workshops, homework will not be collected/graded. There will be one take-home exam based on the course's prerequisite materials at the beginning of the semester, two exams during the semester, and a final exam at the end. The course grade will be based entirely on the exam scores, weighted as 10%, 30%, 30% and 35%. (I know, this adds up to 105%.) An overall score of 85% or more will earn a grade of "A". The cut-offs for the grades of B, C, D will be 75%, 65%, and 55%.

Cool demos (made in Maple)

Class notes

 

Safety rules during the pandemic

UMBC requires you to wear a KN95 mask that covers your nose and mouth in all classrooms regardless of your vaccination status. This is to protect your own health as well to assure others around you that you respect their health and safety concerns.

Anyone attending class without a mask or wearing one improperly will be asked by the instructor to put on a mask or fix their mask in the appropriate position. A student that refuses to comply with this directive will be asked to leave the classroom. The failure to do so will result in the instructor requesting the assistance of the University Police.

UMBC’s on-campus safety protocols, including masking requirements, are subject to change in response to the evolving situation with Covid-19.

UMBC Honors Code

By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal.

The PDF document UMBC Policy for Undergraduate Student Academic Conduct spells out the official academic integrity policies for undergraduates.

Student Disability Services (SDS)

Services for students with disabilities are provided for all students qualified under the Americans with Disabilities Act of 1990, the ADAA of 2009, and Section 504 of the Rehabilitation Act who request and are eligible for accommodations. The Office of Student Disability Services is the UMBC department designated to coordinate accommodations that would allow for students to have equal access and inclusion in their courses.