The focus of this project is investigating a variety of questions in low dimensional topology, which is the study of shapes of spaces with dimension at most four. Central in this subject is the study of knotting and linking of circles in three-dimensional spaces, called knot theory. For instance, knots and links can be used to construct three- and four-dimensional spaces. Low dimensional topology and knot theory have various connections to cosmology and physics (the shape of the universe and string theory), and biochemistry (the knotting behavior of DNA molecules). Unintuitively, many classification questions in topology are harder in dimensions three and four, and over the past three decades, topologists have developed modern tools (with roots in physics) for studying these questions. Two examples of such tools are Heegaard Floer invariants, which grew out of gauge theory, and Khovanov homology, which has roots in representation theory. This project will further develop these invariants and investigate their similarities and relations. Moreover, it will harness their power to study symmetries of surfaces (two-dimensional spaces), in connection with hyperbolic geometry and dynamics, and will investigate several fundamental questions in low-dimensional topology such as finding the minimum number of times a knot must cross itself to become unknotted.
In parallel, this project aims to make mathematics and in particular topology accessible to a broad audience, through educational activities at all levels with an emphasis on diversity and inclusion. These activities include, establishing a Math Circle program in the Athens-Clarke County public library and organizing summer Math Camps for high school and middle school students, running a summer research experience project for undergraduate students and a topology summer school for graduate students and postdocs. The major goals of this research program are organized around four areas. First, developing new invariants for studying spatial graphs and graph concordance. The main tool will be a generalization of (minus) Heegaard Floer homology called, tangle Floer homology. Studying spatial graphs up to concordance has applications in studying equivariant concordance between knots, and strong concordance between links. Second, studying mapping class group and extensions of surface diffeomorphisms over handlebodies using another generalization of Heegaard Floer homology, called bordered Heegaard Floer homology. Third, further developing the bordered Heegaard Floer homology tools by generalizing and refining the contact invariant defined by the PI and her co-authors and use it to address open questions in contact topology. Lastly, the PI focuses on connections and similarities between Khovanov homology and Heegaard Floer invariants, with the three major goals of defining new concordance invariants to study smooth 4D Poincare conjecture, finding new lower bounds for the unknotting number and developing new invariants for transverse knots.
- Funder: NSF
- Amount: $549,999
- PI: Akram Alishahi