We invite Dartmouth undergraduate student applications for a Summer Hybrid Undergraduate Research (SHUR) project during the summer of 2024, at the Dartmouth College Mathematics Department, partially funded by the NSF. Our group will pursue a research project in Floer theory with a computer science component.

Floer theory is an area of mathematics that produces invariants of geometric spaces by counting solutions to differential equations. These techniques have had a profound impact on low-dimensional topology, the study of shapes up to bending and stretching in dimensions we can visualize. Additionally, Floer theory provides the language to understand surprising mathematical predictions from string theory in a field known as mirror symmetry. We will work on concrete computational and/or combinatorial problems in these areas. While the machinery of Floer theory and other advanced mathematics inspires and motivates these problems, prior background will not be required to understand or pursue the problems proposed.

The exact problems will be tailored to students' interests and background. We expect that students with many different backgrounds and skill sets can contribute meaningfully and have fun in this research program.The program will also include professional development and learning components. We are committed to providing a positive and welcoming environment for all students, especially those from underrepresented groups. Applications from any Dartmouth undergraduates are welcome.

Participants will receive a small stipend upon satisfactory completion of the program. The tentative dates of the program are from June 20 to August 27.

Applications should be submitted by email to ina.petkova@dartmouth.edu. Applications will be considered on a rolling basis starting immediately, and until April 5, 2024. 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 5, 2024. Decisions will be made by April 19, 2024.

Students from underrepresented groups are strongly encouraged to apply.

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.

Associate Professor

Department of Mathematics

JWY Instructor

Department of Mathematics

Ina Petkova is an Associate 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.

https://math.dartmouth.edu/~ina

Andrew Hanlon is a John Wesley Young Research Instructor at Dartmouth College. His research focuses on homological mirror symmetry, a geometric correspondence explaining observations from theoretical physics. He uses this duality to explore new structures in symplectic topology, algebraic geometry, and commutative algebra. Often, this involves examining diagrams of hypersurfaces on a torus to get a new perspective on geometric spaces called toric varieties.

https://sites.google.com/view/ahanlon-math