625.687.3VL - Applied Topology
Description
The course is both an introduction to topology and an investigation of various applications of topology in science and engineering. Topology, simply put, is a mathematical study of shapes, and it often turns out that just knowing a rough shape of an object (whether that object is as concrete as platonic solids or as abstract as the space of all paths in large complex networks) can enhance one’s understanding of the object. We will start with a few key theoretical concepts from point-set topology with proofs, while letting breadth take precedence over depth, and then introduce key concepts from algebraic topology, which attempts to use algebraic concepts, mostly group theory, to develop ideas of homotopy, homology, and cohomology, which render topology “computable.” Finally, we discuss a few key examples of real-world applications of computational topology, an emerging field devoted to the study of efficient algorithms for topological problems, especially those arising in science and engineering, which builds on classical results from algebraic topology as well as algorithmic tools from computational geometry and other areas of computer science. The questions we like to ask are: Do I know the topology of my network? What is a rough shape of the large data set that I am working with (is there a logical gap?)? Will the local picture of a part of the sensor field I am looking at give rise to a consistent global common picture?
Expanded Course Description
Prerequisites
Multivariate calculus, linear algebra and matrix theory (e.g., 625.409), and an undergraduate-level course in probability and statistics.
Instructors
Tatyana Sorokina
Christopher Boswell
Course Structure
This course uses the Virtual Live format. It combines the face-to-face class held at APL (section .3VL) with remote students participating online through live web-conferencing. You may choose to attend in-person/on-site or remotely or both. Students who wish to enroll and attend in Virtual Live courses in-person/on-site at the Applied Physics Lab, COVID-19 and Flu vaccination compliance is mandatory for enrollment in this course. All students must upload proof of vaccination or request and receive an exemption in the Vaccination Management System (https://vms.jh.edu/my-vaccines) prior to the start of class.
Course Topics
A tentative course outline may include these elements:
- Topology, topological spaces, open and closed sets. / The notion of abstract and geometric simplicial complexes.
- Bases for topology. Examples. / Construction of simplicial complexes.
- Metric Spaces./ Boundary maps for simplicial complexes.
- Position of a point with respect to a set./ Chain Complexes.
- Finite topological spaces/ Poset Topology.
- Limit points and Compactness./ Groups and Rings overview.
- Continuous maps and connectedness./ More algebra overview.
- Homeomorphisms/ Homotopies
- Classification of surfaces./ Chain complexes and computing simple homologies
- Homologies/ Project Proposals
- Midterm/ Persistent homologies
- Chain maps and exact sequences
- Betti numbers and bar codes
- Project presentations
Course Goals
The number of useful applications of topology continues to grow, including in areas of optimal assignment, simulation (in engineering and biology), and Topological Data Analysis. This introduction to the use of topology will include a theoretical framework to enable students to help design new applications of topological concepts. This will include set-theoretic and algebraic topological concepts and examples that demonstrate how topological ideas and techniques are applied to study problems in science and engineering. A particular focus is on using methods of algebraic topology to understand large high- dimensional data sets.
Course Learning Outcomes (CLOs)
- understanding of basic topological concepts and their relevance to other branches of mathematics
- ability to translate characteristics of natural phenomena and various data into topological features
- knowledge of algebraic invariants of topological spaces and ability to create algorithms to compute the algebraic invariants or their approximations
- gaining experience in identifying new areas of application of computational topology
Textbooks
There is no required textbook for this course.
We intend to draw from various texts and articles as mentioned in class.
Required Software
Assignments will require some coding that can completed in MATLAB or C++, or another if previously approved by the instructors.
Student Coursework Requirements
Homework (40%). Students are expected to be ready to submit homework or review in teams every week. There will be a choice of a small group or an individual final project. The average number of hours per week that the students should prepare to spend on homework assignments is six to ten hours. Homework will be assigned each class meeting and will be due the next one. It may be graded or discussed in teams. A significant part of the homework will involve some basic programing and will be related to applications.
Midterm (20%). The midterm exam will be in class, closed-book. The content will be mostly of theoretical nature, while the final project will be concerned with applications.
Project (40%). The final project is an application of the Topological Data Analysis methods learned in this class on actual data. The project teams will present their results in a presentation and paper write-up. The goal is to have something than could be presented in a poster session, in a talk at a conference with proceedings, or published in a journal article. As will be discussed in class, sometimes initial analysis of this type does not yield clear insights, while sometimes the initial look does. The project will not be graded on whether it is clearly an important result in the field from which the data is taken, but rather on the proper application of Topological methods, the examination of a number of ways to look at the analysis, and the professionalism of the work, presentations, and documents.
A Project Proposal will be presented early in the process with an outline of the field, the exact data expected to be used, the first planned lines of inquiry. In addition, the Topological metrics and possible means of communicating results will be estimated. Details may change based upon what is found, but if these are substantial should be made in concert with discussions with the Instructors.
The final projects will be shared with the class, with the hope that interesting discussions and ideas for further research will be instructive, interesting, and fun.
Grading Policy
EP uses a +/- grading system (see “Grading System”, Graduate Programs catalog, p. 10).
| Score Range | Letter Grade |
|---|---|
| 100-97 | = A+ |
| 96-93 | = A |
| 92-90 | = A− |
| 89-87 | = B+ |
| 86-83 | = B |
| 82-80 | = B− |
| 79-77 | = C+ |
| 76-73 | = C |
| 72-70 | = C− |
| 69-67 | = D+ |
| 66-63 | = D |
| <63 | = F |
Academic Policies
Deadlines for Adding, Dropping and Withdrawing from Courses
Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.
Academic Misconduct Policy
All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students.
This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.
Students with Disabilities - Accommodations and Accessibility
Johns Hopkins University values diversity and inclusion. We are committed to providing welcoming, equitable, and accessible educational experiences for all students. Students with disabilities (including those with psychological conditions, medical conditions and temporary disabilities) can request accommodations for this course by providing an Accommodation Letter issued by Student Disability Services (SDS). Please request accommodations for this course as early as possible to provide time for effective communication and arrangements.
For further information or to start the process of requesting accommodations, please contact Student Disability Services at Engineering for Professionals, ep-disability-svcs@jhu.edu.
Student Conduct Code
The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically.
For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/
Classroom Climate
JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity.
If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).
Course Auditing
When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.