Diff-Geom-2023

Differential Geometry MAT432/MAT733 Fall 2023

Professor Sormani (About Myself)

MAT733: Differential Geometry Curves in E3, curvature, torsion, fundamental existence theorem for space curves, geometry of a surface, inverse and implicit function theorems, Gauss curvature, and Minimal Surfaces. 4 credits

MAT432 Differential Geometry: Metric Spaces, Curves in Euclidian 3 space (E3): Curvature, torsion, fundamental theorem of ordinary differential equations, fundamental existence theorem for space curves. Surfaces in E3: geometry on a surface, Inverse Function Theorem, Implicit Function Theorem, and Gauss curvature. Coordinate charts, Fubini's Theorem, orientation and an introduction to Riemannian Geometry. 4 credits

Prerequisite: Linear Algebra MAT313 and Vector Calculus MAT226. Also Analysis MAT320 is recommended but not required.

Course Webpage: https://sites.google.com/site/professorsormani/teaching/Diff-Geom-2023

Professor Sormani: google "Sormani Math Lehman” for my page

Contact: sormanic@gmail.com (do not call the office and leave messages)

Welcome to the Course Playlist

Differential Geometry combines everything you learned in Vector Calculus with everything you learned in Linear Algebra and applies it to understand smooth geometric shapes. It has applications in Pure Mathematics, Applied Mathematics, Geometric Analysis, Astronomy, Physics, Computer Graphics, Computational Geometry, Data Science and many other subjects. This is just the first course in a subject which you can continue to study for many more years in different ways. I have a Doctorate in Mathematics with a specialization in Geometric Analysis which is a type of Differential Geometry. You can see more about my research on my webpage and you can also see the research completed by undergraduates and masters students that have worked with me here and here. You can see more about my doctoral students here. - Prof Sormani

Class Meets: Online Asynchronously (see below for links to weekly lessons and videos which can be completed at any time during each week). Students will submit class notes and homework weekly by sharing google docs filled with photos of your handwritten work, questions, and a selfie holding that work. Everyone will be given the opportunity to resubmit incorrect work after receiving feedback from the professor.

Office Hours: Asynchronous: put your questions in your googledoc for the week, email the professor there is a question there, and the professor will insert photos with answers to your questions next to the questions.

Materials, Resources and Accommodating Disabilities

Recommended Textbook: All definitions and theorems and examples will be provided in the notes for this course. Those who wish may purchase Bloch, A First Course in Geometric Topology and Differential Geometry, ISBN 0-8176-3840-7, Chapters I, IV, V, and VI but YOU DO NOT NEED TO PURCHASE THIS TEXTBOOK. Students should have access to a copy of the textbooks that they used for Calculus and for Linear Algebra to consult as needed throughout this course.

MATLAB: MATLAB will be used for computations and graphing in this course. Do not use AI software like ChatGPT because that software has not been developed to provide correct mathematicsw. MATLAB, on the other hand, does mathematics correctly and is often used in the workplace. YOU DO NOT NEED TO PURCHASE MATLAB AS YOU CAN ACCESS IT ONLINE THROUGH LEHMAN. Mathworks for CUNY Students has the info

Googledocs: Everyone will have to create a google account with a professional name that they are willing to share with the professor and classmates. All work will be submitted by sharing your googledoc for that week with the professor using the correct title on that doc stating the course number, the week number and your name:

MAT432F23-WeekX-YourNameHere

MAT733F23-WeekX-YourNameHere

Google automatically sorts work submitted with the correct name so be sure to write it correctly. You may use a preferred name. Note googledocs are not pdf files: they are documents which allow gluing in of photos so that you may glue in your work on a photo and I can glue in feedback on a photo. A video about googledocs.

Laptops: If you do not have a computer or laptop, it is recommended to contact the math department for information about how to borrow one for the semester. For those who have NYSaves College Accounts, the purchase of a laptop is an allowable expense. A tablet or even smart phone suffices for watching lessons and submitting homework but the laptop becomes helpful later when using MATLAB.

Tutoring: There is no tutoring available for this advanced a course, but the professor gives students a chance to try and try again so that you can learn. Students should not use tutoring services for this course as it will be considered to be cheating. Students may work together on joint projects, and should always cite each other by full name when working together.

Online Resources: Wikipedia is a fairly reliable source for advanced mathematics. Also useful is conducting a "google scholar" search rather than just a "google" search. Please note that there are many websites with completely incorrect information posted by students and tutors. A google scholar search will take you to published work which has been checked by mathematicians and editors.

Stress: Student Services can help with Many Concerns both Personal and Academic!

For students that become overwhelmed with the course, one solution is to do less work and aim for a lower grade. Communicate with the professor about your career goals so that the professor can let you know which assignments may be skipped. For example, you might be told that you can skip a particular week (which would lower your grade by only 5 points), but some weeks are too important to skip and are needed throughout the course or might be needed for your career.
For students that are worried they are failing, communicate with the professor about your career goals so that the professor can point out extra credit work that has been assigned which matches your skill set and is appropriate for your goals. Do not wait until the last minute. Sometimes it is impossible to catch up if you wait too long.

Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disabipty Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.

Accommodating Holidays: If you have a holiday during a lesson or extra lesson, let the professor know, and something will be arranged for you.

Pronouns and Preferred Names: I will use the gender neutral they/their for all students in the class as I know it can put pressure on some students to have to choose pronouns. It is easiest for me to just use last names for everyone, and you may also address me by last name: Sormani. It is alright if you mispronounce my name or do not spell it correctly. We all come from different places and it can be hard to pronounce every name correctly.

Feedback: At times you may be asked to correct another students' work and you should do so kindly, explaining that you "believe something might be incorrect" and you should be prepared for the possibility that the original solution might correct and your suggestion might be incorrect at any time because you are peers. In my experience feedback by peers is ony correct about half the time, and that often both the original solution and the correction are both correct. My own feedback will be more aggressive because I am a professor with a doctorate in mathematics, but you may always ask what exactly is wrong if my feedback is unclear. Note that often my feedback will recommend that you read certain notes or watch a video again. Occasionally I will provide more details in a photograph inserted into your doc.

Respect: Everyone will treat each other with respect and dignity. Let me know if you have concerns. Talk to an advisor or another professor if you have concerns about me.

Course Objectives

At the end of the course students should be able to:

1. Write Proofs that sets in metric spaces are open or closed

2. Compute the curvature and torsion of a space curve

3. Compute the change of coordinates, the Jacobian, normal, and area of a parametrized surface

4. Apply the Implicit and Inverse Function Theorems

5. Find the First and Second Fundamental Forms

6. Compute Gauss Curvature

7. Compute Mean Curvature and Verify that a Surface is a Minimal Surface

8. Plot surfaces using Maple or MATLAB

Grading Policy

Classwork and Homework: 70% Each week students will submit their classwork and homework in a googledoc for that week: MAT432F23-WeekX-YourNameHere or MAT732F23-WeekX-YourNameHere. There are 14 weeks in this course and each weekly doc will earn a score of 0-5. There are two deadlines for each week's work.

Submission deadline: If the doc is submitted before its submission deadline with complete classnotes copied from the videos that is 2 points. The rest of the 3 points will be earned by doing the homework correcty. Try all the problems and ask questions inside the doc before the submission deadline.

Resubmission deadline: You will have the opportunity to resubmit incorrect work and improve your score to a perfect score if it is fixed by the resubmission deadline. A video about googledocs submission ans resubmission of homework.

Group Projects I-II: 20% There will be an asynchronous online group project where you all contribute work one step at a time uploaded as a photo into a common googledoc, you will give feedback on your classmates work, and provide revisions. You will contribute at most once per day to allow other students the opportunity to work as well. You will not be required to be online at the same time as your classmates and you will only be scored on your personal contributions.

Final Project: 10% In the place of a final exam there is a final project due at the end of finals week. Each student will study a different geometric problem using the methods taught in the course. Students will start working on these projects halfway through the course. The will be similar to the homework assigned in the second half of the course but there will be no assistance on aspects of the project. One resubmission after personalized feedback will be allowed. It will be possible to earn extra credit on this project if the work is exceptional and includes extra work.

Extra Credit: up to 10% All semester long there will be extra credit problems assigned inside the lessons. Graduate students taking MAT732 should be doing this extra credit if they hope to earn an A.

Scores to Letter Grades:

MAT432: This is an undergraduate course:

A 95-110%, A- 90-95%, B+ 85-90%, B 80-85%, B- 75-80%, C+ 70-75%, C 65-70%, C- 60-65%, F 0-60%

MAT733: This is a graduate course so students have to complete better work:

A 95-110%, B 85-95%, C 75-85%, F below 75%

At some point in the course you may realize that you have already earned enough points to earn a C. Do not disappear even if a C is all you need for your degree. Communicate with your professor to ensure that you have learned everything you need to meet your career goals.

Note about options at Lehman College:

At Lehman College, student may withdraw from a course they are afraid of failing to avoid an F. The grade of W will appear on the transcript and the course will not affect your GPA. Always consult with the professor before withdrawing as accomodations may be possible.

At Lehman College, students who have at least passing work before the final exam and suffer hardship may request an incomplete in the course and complete the course in January. At the end of the semester, if a student has enough points so that 10 more points is passing, then they may request an incomplete from me. It is acceptable to request an incomplete if you already have a B or C average but need more time for the final project. At the end of January students awarded a temporary grade of INC will have their grade changed to the grade they have earned over the extra month. I will not allow incompletes to run past January.

Differential Geometry

Differential Geometry combines everything you learned in Vector Calculus with everything you learned in Linear Algebra and applies it to understand smooth geometric shapes. It has applications in Pure Mathematics, Applied Mathematics, Geometric Analysis, Astronomy, Physics, Computer Graphics, Computational Geometry, Data Science and many other subjects. This is just the first course in the subject which you can continue to study for many more years in different ways. I have a Doctorate in Mathematics with a specialization in Geometric Analysis which is a type of Differential Geometry. You can see more about my research on my webpage and the you can also see the research completed by undergraduates and masters students that have worked with me here and here. You can see more about my doctoral students here. - Prof Sormani (About Myself Playlist)

Schedule:

Weekly lessons will be posted here, which must be completed on time. The lessons will include notes and videos, as well as homework assignments.

Welcome to the Course Playlist (just goes over this webpage/syllabus)

Week 1 Introduction to Metric Spaces due 10pm Sept 3, fix by noon Sept 9

Week 2 Curves and MATLAB due 10pm Sept 10, fix by noon Sept 16

Start Group Project I on Sept 12: Group Project I Team A or later: Group Project I Team B.

Week 3 Open and Closed Sets due 10pm Sept 17, fix by noon Sept 23.

Week 4 Frenet-Serrat Formulas and Fundamental Theorem of Curves due 10pm Sept 24, fix by noon Dec 20.

All students start Week 5 on Sept 25 seriously aiming for the Oct 1 deadline.

Week 5 Maps between Metric Spaces due 10pm Oct 1 for all students, fix by noon Oct 7

Week 6 Differentiation due 10pm Oct 8, fix by noon Oct 14

End of Group Project I this week. It will close on October 8 at midnight.

Week 7 Diffeomorphisms, Isomorphisms, and Homeoorphisms due 10pm Oct 15, fix by noon Oct 21

Start Group Project II begins this week. Everyone will work together. Request access by clicking on the link after completing corrections to your Week 6 homework.

Week 8 Inverse Function Theorem and Atlases of Coordinate Charts due 10pm Oct 22, fix by noon Oct 28

Week 9 Implicit Function Theorem due 10pm Oct 29, fix by noon Nov 4

Students who are behind schedule see this checklist of priority work for Weeks 6-9.

Week 10 Surfaces and Submanifolds due 10pm Nov 5, fix by noon Nov 11 -> Nov 15

Your Final Project should be started alone.

Week 11 Vectors, Normals, and First Fundamental Form due 10pm Nov 12, fix by noon Nov 18

I have allowed students to fall behind schedule but I need everyone to start getting back on track to see how they are progressing towards their final grades.

Nov 18 is the absolute final deadline for fixes to Week 7 Part I, Week 8 Part I, Week 9 Part I, and Week 9 Part II even for those who suffered hardship.

Nov 25 is the absolute final deadline for fixes to Week 10 Part I, Week 10 Part II, and Week 11 Part I.

Week 12 Differentiation of Vector Fields due 10pm Nov 26, fix by noon Dec 2

Group Project II should be finished this week by Dec 2.

Week 13 Areas and Second Fundamental Forms due 10pm Dec 3, fix by noon Dec 9

Week 14 Minimal Surfaces due 10pm Dec 10, fix by noon Dec 16

Dec 16 is the absolute final deadline for Part I of Weeks 12-14.

Create a status doc here to tally your grade.

Final Project due 10pm Dec 17, fix by noon Dec 23

Useful Topic for Data Science and Probability:

Extra Credit:

Hessians and Laplacians Playlist

Heat Equations and Diffusion Maps Playlist

Extra Credit is to add these last two topics to your final project.