MATH 136 FALL 2024
Introduction to Applied Mathematics

Instructor:  Peter J. Mucha, Kemeny 240, peter.j.mucha@dartmouth.edu, https://mucha.host.dartmouth.edu. 

Lecture Information:  11 (MWF 11:30–12:35), Kemeny 242.

Please hold the 11X hour (Tuesdays 12:15–1:05) open for possible use including one-on-one and small group meetings.

Full class meetings at the 11X time will be held on 11/5, 11/12 and 11/19.

Office Hours:  Tuesdays 1:15–2:15 and Thursdays 9–11 in Kemeny 240, and by appointment (a DM on the Dartmouth Slack will almost always get a faster response than email).

Course Objectives: 

The course aims to introduce a variety of essential methods in applied mathematics that are not covered elsewhere in our core graduate curriculum. Our aim is to develop students' broad conceptual understanding of these methods and the ability to implement them to solve problems. Each student will develop a more detailed expertise on a selected topic through the course project. Depending on specific topics, calculations will be performed by hand, computational symbolic algebra, and/or through numerical packages.

Materials: 

We will organize our explorations of different topics and methods using parts of different textbooks and/or articles that will be listed in References (which you can also find under Pages in the left sidebar). You are not expected to purchase anything.

Assignments, Projects, and Grading: 

Homework assignments will be opportunities to learn some of the material in greater detail. Collaboration on the homeworks is strongly encouraged. 

Your "course project" will be to take one of the topics not selected for our regular lectures and do a deep dive, culminating in a one-hour lecture on the topic.

Attendance Policy: 

Class participation is essential to our exploration of the course material. Please reach out to me in a timely, responsible way to pre-approve any excused absences as needed. 

You are expected to attend class in person unless you have made alternative arrangements due to, e.g., illness, medical reasons, or the need to isolate due to COVID-19 or other illness. For the health and safety of our class community please do not attend class when you are sick. If possible, please let me know in advance in a timely way if you are sick and believe you might miss an upcoming class.

There are many potential reasons to request an excused absence. Please just reach out to me ahead of time and follow up after to discuss how to best catch up on any course material that you miss. In short, be responsible.

Religious Observances: 

Dartmouth has a deep commitment to support students’ religious observances and diverse faith practices. Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible—before the end of the second week of the term at the latest—to discuss appropriate course adjustments. 

Academic Honor Principle: 

The Academic Honor Principle is an essential tenet of the Dartmouth community. Collaboration is strongly encouraged in this course. At the same time, ultimately, all assignments submitted must represent your own understanding of the material. Be generous and honest in your citing assistance and input from your fellow students. You should also be sure to always fairly and completely cite your sources.

Use of Generative Artificial Intelligence: 

As machine learning continues to advance, large language models (LLMs), such as ChatGPT, and other Generative AI (GAI) technologies are becoming more widespread. These models can at times be useful tools to accelerate productivity and understanding. Dartmouth has identified Guidelines on using Generative Artificial Intelligence for Coursework. The use of such technologies is permitted for the assignments in our course, so long as the following guidelines are adhered to:

• When using an LLM or other GAI to aid in completion of an assignment, all prompts and output should be saved and submitted as part of the assignment. This may be in the form of a screenshot, copy and paste, PDF, etc.
• The work that you submit should reflect your own understanding of the assignment.
• Copying the output from an LLM or other GAI and handing it in as your own work is not permitted, similarly to how copying a peer's work and submitting it as your own is not allowed.

Examples of situations where you might find it useful to use GAI in your work include when you know what kind of calculation you want to do but you don't know all of the details or syntax for how to code it, or you forgot an idea or concept from a previous class that is needed for the current item you are working on. Many other reasonable examples are surely possible; you are strongly encouraged to share your experiences using such tools. Please be aware that in many cases these technologies can give answers to a prompt that are completely incorrect (and sometimes wildly so).  As such, you should always be skeptical of any GAI output you see and verify the veracity of the information contained within. If you have any questions about the use of GAI in the class, please reach out to the instructor.

Student Accessibility and Accommodations: 

Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Apply for Services webpage; student.accessibility.services@dartmouth.edu; 1-603-646-9900) and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.

Mental Health: 

The academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. There are a number of resources available to you on campus to support your wellness, including the undergraduate deans, Counseling Center, and Student Wellness Center. I encourage you to use these resources to take care of yourself throughout the term, and to speak to me if you experience any difficulties. 

Diversity and Inclusion: 

Dartmouth is committed to maintaining a diverse and inclusive workplace and welcomes all members of Dartmouth's scholar-educator community to join in cultivating a culture that values and rewards teaching and welcomes diversity in its many aspects. I acknowledge that the distribution of authorship of the books, articles, and original materials referenced therein do not reflect that desired diversity, especially insofar as we consider historical references, but not only as such. I encourage you to talk to me if anything in class or out of class makes you uncomfortable or if you have any suggestions to improve our environment or the quality of the course materials.

Title IX: 

At Dartmouth, we value integrity, responsibility, and respect for the rights and interests of others, all central to our Principles of Community. We are dedicated to establishing and maintaining a safe and inclusive campus where all have equal access to the educational and employment opportunities Dartmouth offers. We strive to promote an environment of sexual respect, safety, and well-being. In its policies and standards, Dartmouth demonstrates unequivocally that sexual assault, gender-based harassment, domestic violence, dating violence, and stalking are not tolerated in our community.

The Sexual Respect Website at Dartmouth provides a wealth of information on your rights with regard to sexual respect and resources that are available to all in our community.

Please note that, as a faculty member, I am a mandatory reporter obligated to share disclosures regarding conduct under Title IX with Dartmouth's Title IX Coordinator. Confidential resources are also available, and include licensed medical or counseling professionals (e.g., a licensed psychologist), staff members of organizations recognized as rape crisis centers under state law (such as WISE), and ordained clergy (see https://dartgo.org/titleix_resources).

Should you have any questions, please feel free to contact Dartmouth's Title IX Coordinator. Their contact information can be found on the sexual respect website at: https://sexual-respect.dartmouth.edu. 

 

Course Schedule: Note future topics listed are tentative and will move around as time permits. A "T" in the schedule means we will have class in the Tuesday X-hour.

Week	Topics
1 MWF	Singular Value Decomposition See Brunton & Kutz chapter 1, Strang (1993), Strang's "... 4 Lines" notes, Moler (2016), Martin & Porter (2012), Kolda & Bader (2009) Monday 9/16: Introductions, "On the Nature of Applied Mathematics" (see Lin & Segel section 1.1), and start SVD Wednesday 9/18: Strang's "4 lines" example of the 4 subspaces,  Friday 9/20: Image Compression, PCA, "eigendigits" example
2 MWF	Fourier Transforms See Lin & Segel chapters 5 and 6, Logan chapter 5, Brunton & Kutz chapter 2, Trefethen chapters 1–4 Monday 9/23: Definitions of the 4 Fourier variants, emphasizing the relationships between the discrete/continuous and bounded/unbounded properties of the 2 spaces Wednesday 9/25: Derivatives, convolutions, Green's function of the heat equation, and FFT Friday 9/27: Smoothness and Spectral Accuracy
3 MWF	Monday 9/30: Simple FFT examples (fftexamples.mlx) Wednesday 10/2: Image Compression examples with FFT and Wavelets "Optimization is the Cornerstone" Friday 10/4: Over- and Under-determined systems, Sparsity and Compressed Sensing (B&K chapters 3–4)
4 MWF	Monday 10/7: CVX Wednesday 10/9: CVX examples with matrices Friday 10/11: Robust PCA I meant for us to also survey some key ideas about clustering and classification from Brunton & Kutz chapter 5, but I can give you some notebooks for that if you want to look at it yourself (just let me know and I can send you the files).
5 WF	No class on Monday 10/14 Essentials of Complex Analysis See Part D of Kreyszig. Wednesday 10/16: Review of complex numbers and functions, limits, derivatives, and analytic functions Friday 10/18: Cauchy-Riemann conditions, complex contour integration, and examples of path independence
6 MW	Monday 10/21: Integral around a pole and Cauchy's Integral Theorem Wednesday 10/23: Cauchy's Integral Formula and Taylor Series No class on Friday 10/25
7 MWF	Monday 10/28: Laurent Series Wednesday 10/30: Residue Theorem Asymptotic Expansion of Integrals See Bender & Orszag chapter 6, Hinch chapter 3 Friday 11/1: Term-by-term integration and integration by parts
8 MTWF	Monday 11/4: Laplace's Method and Watson's Lemma Tuesday 11/5: Method of Steepest Descent Wednesday 11/6: Stirling Series and other examples Course project presentations Friday 11/8: Hayley Coyle on Dynamic Mode Decomposition
9 MTWF	Monday 11/11: Anna Vasenina on SINDy Tuesday 11/12: Xin Jin on Graph Embeddings Methods and Their Applications Wednesday 11/13: Chiyu Wei on Deep Learning Friday 11/15: Patrick Addona on Reduced Order Models
10 MTW	Monday 11/18: Toby Harvey on Symplectic Runge-Kutta Schemes for Adjoint Equations and more Tuesday 11/19: Rohan Kapoor on Wavelets