Summer 2022: A Computational Approach to Legendrian Knots

About the SHUR

We invite Dartmouth undergraduate student applications for a Summer Hybrid Undergraduate Research (SHUR) project during the summer of 2022, at the Dartmouth College Mathematics Department, partially funded by the NSF. Our group will pursue a research project in low-dimensional topology with a computer science component.

Our research group will investigate phenomena in the intersection between low-dimensional topology, the study of objects which are up-to-4–dimensional, and contact geometry, an active branch of mathematics that originated from Hamiltonian mechanics. In particular, we will be studying knots (strings that are tangled up) that satisfy certain natural contact geometric conditions, as well as interpolating pieces between them called “cobordisms”. We will look into computational and theoretical questions surrounding such knots, with potential projects including 1) further developing a graphical user interface (GUI) to manipulate these planar diagrams, and 2) combining techniques from computer science and theoretical mathematics to search for and obstruct the existence of cobordisms given two knots.

This research has entry points for students with many different backgrounds and skill sets, and it is super fun! We will also provide professional development opportunities, including discussions with mathematicians on research presentation and writing personal statements.

Participants will receive a small stipend upon satisfactory completion of the program. The tentative dates of the program are from June 13 to August 12.

Participants do not need to be on campus for this SHUR. However, in-person activities will also be available if some or all participants are on campus.

How To Apply

Applications should be submitted by email to ina.petkova@dartmouth.edu. Applications will be considered on a rolling basis starting immediately, and until April 29, 2022. The application should include:

  • One reference (list one Dartmouth faculty who can serve as a reference if needed—no letter required at this time).
  • An unofficial transcript.
  • A personal statement listing math courses (with grades), computer science courses (with grades), computer skills, previous research experiences, and explaining why the applicant wishes to participate in the program.
  • The deadline to apply is April 29, 2022. Decisions will be made by May 3, 2022.

Students from underrepresented groups are strongly encouraged to apply.

Program Expectations & Prerequisites

As part of the program, students will attend lectures and discussions led by the faculty on the background of the research topic. After the introduction, students will be presented with concrete, realistic research projects and tasks commensurate with individual interests and strengths, with hands-on guidance by the faculty. Research progress will be assessed throughout the duration of the program, and successful results will be summarized in academic papers and presentations.

This project is suitable for both students interested in research mathematics and students interested in further developing their skills in computer science. Applications from all undergraduate students are welcome, and we especially encourage students from underrepresented groups to apply.


Ina Petkova

Assistant Professor
Department of Mathematics

C.-M. Michael Wong

Lecturer and Research Associate
Department of Mathematics

Ina Petkova is an Assistant Professor of Mathematics at Dartmouth College, working in low-dimensional topology. Her research is in knot theory, with focus on Heegaard Floer homology. She often uses algebraic techniques for studying the variant of Heegaard Floer homology for knots (knot Floer homology), by cutting a knot into pieces called tangles, and studying the individual pieces and their gluing.

C.-M. Michael Wong is a postdoctoral Lecturer and Research Associate at Dartmouth College. His primary research interest is in low-dimensional topology, and specifically cobordisms of 3-manifolds, knots, links, and tangles. The main tools that he uses are various versions of Floer homology and related invariants. He also studies the connections between Floer theory and contact geometry.

SHUR Participants 2022

Mitchell Jubeir

Mitchell is a ’23 at Dartmouth from Maryland triple majoring in math, physics, and economics. He has served as a tutor both through the Tutoring Clearinghouse as well as independently in both math and physics. Outside of academics, he also enjoys playing ultimate frisbee, running, playing chess, and solving various types of puzzles.

Liam Knight

Liam is a rising senior and math major. He has served as a math tutor and grader and performs mathematical outreach with middle school students. Outside of academics, he enjoys trail running, cycling, mountain biking, reading mysteries, and spending time with his dog.

Noah Schwartz

Noah is a Dartmouth ’23 from Massachusetts, majoring in math and physics. At Dartmouth, he has TA’d for the physics department, and worked on research projects in the math department. Outside of academics, Noah is captain of the Frisbee B-team, and he also enjoys hiking, rock climbing, and all sorts of outdoor adventures.

Previous iteration

SHUR 2021