A Heegaard Floer theoretic approach to 4-dimensional genus questions

Mathematical breakthroughs in the 20th century revealed that four-dimensional spaces are more mysterious than spaces in any other dimension. One of the best ways to study these spaces is to break them into smaller pieces and examine the knots and surfaces within. One way of distinguishing spaces is to determine surfaces of least complexity, for example, “minimal genus”. This project focuses on minimal genus questions in a variety of contexts. The PI brings expertise in Heegaard Floer theory, which has proved effective at addressing these kinds of questions. As part of this project, the PI plans to organize a yearly colloquium and special lecture for the Association for Women in Mathematics Student Chapter at the University of Arkansas. Additionally, the PI will co-organize a regional conference in topology and geometry that serves the EPSCoR regions of Arkansas, Oklahoma, and beyond, and continue to lead the Math Olympiads for Elementary and Middle Schools (MOEMS) team at the Fayetteville Public Library. Minimal genus problems are central to the study of low dimensional manifolds. The project addresses variations of the minimal genus problem and its broad implications. The PI will study the relationship between the knot concordance group and the homology cobordism group. The PI will address the existence of deep slice knots in contractible 4-manifolds. The PI will study an analog of the Thurston norm for knots in rational homology spheres. The PI will develop bordered Floer techniques to study knots that are not freely equivariantly slice. Finally, the PI will study minimal genus problems in the context of contact topology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.