Math 4997: Invariants in Low-Dimensional Topology

Classroom:  Department of Mathematics Lounge, Lockett Hall
Time:  MWF, noon – 12:50


The Classroom
is a space for me to post lectures, videos and other materials related to this and other classes I’m teaching.

A Communication-Intensive (CxC) Course

This is a certified Communication-Intensive (C-I) course, which means it meets all of the requirements set forth by LSU’s Communication across the Curriculum (CxC) program, including:

  • instruction and assignments emphasizing informal and formal writing and speaking;
  • teaching of discipline-specific communication techniques;
  • use of draft-feedback-revision process for learning;
  • practice of ethical and professional work standards;
  • 40% of the course grade rooted in communication-based work;
  • a student/faculty ratio no greater than 35:1.

Students interested in pursuing the LSU Distinguished Communicators certification may use this C-I course for credit.  For more information about this student recongnition program, visit CxC.

Textbook: The official and non-required text for the course will be Grid Homology for Knots and Links, by Peter Ozsváth and Zoltan Szabó.  This is a fantastic textbook that will serve as a lasting resource for anyone interested in the subject.

Prerequisite: The prerequisite for this course is Math 2057 or 2058.

Course Content

This is a Vertically Integrated Research (VIR) course focusing on low-dimensional topology. The central aim of a VIR course is to of  to bring together undergraduates, graduate students, post-docs and senior faculty to learn about and work on problems at the boundaries of current mathematical research. This semester, we’ll be learning about and exploring a package of incredibly powerful invariants of knot, links and 3–manifolds called Heegaard Floer theory. In addition to learning about some amazing mathematics, we’ll (hopefully) get a chance to try our hand at solving some open problems in the field.

Communication-Intensive Activities:

A good mathematician is one who knows the statements to many theorems.  A great mathematician is one who also understands the proofs to most of the theorems they know.  An exceptional mathematician is one who, in addition to knowing a great many theorems and proofs, also knows examples illustrating why each hypothesis in a given theorem are necessary.

This simple statement illustrates a broad distinction between basic mathematical competency and true mathematical fluency.  By this point, I expect that all of you are mathematically competent.  That is, I expect that you understand what a mathematical statement (Theorem, Lemma, Proposition, etc.) is, and roughly what it means to prove a mathematical statement.  By this point in your mathematical careers, you probably all have a decent library of (true) mathematical statements floating around in your head.  Mathematical fluency is an entirely different beast.  It’s not just about memorizing lots of mathematical statements or even their proofs – mathematical fluency is about developing gut-level understandings of what certain mathematical statements are saying, why they are true, and how they fit into the broader mathematical tapestry.

Hand-in-hand with mathematical fluency is the ability to identify and speak appropriately to one’s audience. Who is trying to understand the mathematical statement or proof? Are you trying to convince yourself? A fellow classmate? An anonymous grader? Me? What may look and feel like a complete proof to someone with lots of mathematical training may be utterly unconvincing to a novice. By working hard to distill mathematical statements to as child-like a form as possible and to understand these statements and proofs at a gut-level, we position ourselves to be both better mathematicians and communicators.

Individual roles

This course brings together undergraduates, graduate students, postdocs and senior faculty to learn about and study current problems in mathematics. Members of each of these groups will play a unique and important role in our progression. Each undergraduate will be paired with two graduate student to form a “mentorship group”. The graduate mentors will help guide the undergraduates (and one-another!) as we all learn, process, and present the material. The postdocs and senior faculty will keep track of everyone’s progress, ensuring that everyone understands their roles and that each of the students are appropriately up-to-speed.

Type 1: (writing) Mathematical fluency, thinking and writing

Your task in a Type 1 assignment is to learn the material, distill it in your own mind until you have a clear understanding of why each statement is true.  Your task is to then to discuss and write a class lecture with your mentorship group. Everyone will be required to participate in this activity, including postdocs and senior faculty. Constant feedback will be given throughout the preparation process to ensure that the final prepared lecture is both well-understood by the speaker and appropriate for the audience.

Type 2: (speaking) Professor for a day (or two)

Type 2 (speaking) assignments will focus on building the skills necessary for lecturing on mathematics. Though this might sound a bit daunting at first, the exercises should be lots of fun! Your task here is to deliver the lectures your prepared and honed within your mentorship group. Here, though, the emphasis will be on clarity of presentation, board technique, and ability to address audience member’s questions.

Homework and Grades:

There will be several flavors of homework problems assigned throughout the course.

C-I Assignments: There will be (BLANK) many Communication-Intensive assignments

  • Two Assignments of Type 1
  • Two assignments of Type 2

Final Course Grades: Your grade will be determined entirely by your performance on the communication-intensive activities discussed above.

Grading Scale: A – at least 90%; B – at least 80%; C – at least 70%; D – at least 60%.

Ethical Conduct: You are strongly encouraged to talk to others about the material you are learning — this includes fellow students, me and anyone else in the department.  The more engaged you are with the material the better. That said, the work you hand in must be your own. The best way to ensure that the work you turn in accurately represents your own understanding of the material (and to avoid any suspicion of copying) is to write out your homework solutions on your own, not whilst working with a friend. Cheating on exams is unacceptable. Any cheating on homework assignments or during midterms or finals will result in you receiving no credit for the assignment/exam.  Students must abide by LSU’s Code of Student Conduct.

Disability Support: Students who may need accommodations because of a documented disability should meet with me privately within the first week of classes. In addition, students with disabilities should also contact the Office of Disability Services.