My current research is in trisections of smooth 4-manifolds. Trisections are a 4-dimensional analog of Heegaard splittings of 3-manifolds. A trisection of a smooth, closed 4-manifold is a decomposition three 4-dimensional 1-handlebodies whose pairwise intersections are 3-dimensional handlebodies and triple intersection is a surface.
There are many analogies between the theories of trisections and Heegaard splittings. For example, every smooth closed 4-manifold admits a trisection which is unique up to the appropriate notion of stabilization. There are also trisection diagrams where is a closed, genus surface and each of are -tuples of non-separating, simple, closed curves in such that each triple is a Heegaard diagram for
Publications and Preprints:
- Obstructing Relative De-stabilizations of Trisections, with Thomas Kindred. In preparation.
- Classifying low-genus and Isotopic Relative Trisections, with Patrick Naylor. In preparation.
- Relative Group Trisections, with Jason Joseph. In preparation.
- The Relative L–invariant of a Compact 4-manifold with Boundary, with Gabriel Islambouli, Maggie Miller, and Maggy Tomova. arXiv:1908.05371, 2019. Accepted, Pacific Journal of Mathematics.
- Camp Euclid: A Research Experience for Youth – in a Virtual Environment, with Juliette Benitez and David T. Gay. To appear in AMS Notices, December 2020.
- Trisections of 4-manifolds via Lefschetz fibrations, with Burak Ozbagci. Mathematical Research Letters 26(2): 383-420, 2019.
- On Dividing a Rectangle, with Robert Dumitru*, Quinn Perian*, Alexander Nealey*, Mohammed Mannan*, Eddie Beck, David Gay, Dipen Mehta*, Anish Pandya*, Ejay Cho*, 2018. (This paper was the result of a research group of Camp Euclid 2016. (*) indicates high school co-author.)
- Trisecting 4-manifolds with Boundary, with David T. Gay and Juanita Pinzón Caicedo. Proceedings of the National Academy of Sciences Oct 2018, 115 (43) 10861-10868
- Trisecting Smooth 4-dimensional Cobordisms, arXiv:1703.05846, 2017
- Diagrams for Relative Trisections, with David T. Gay and Juanita Pinzón Caicedo, Pacific Journal of Mathematics 294 (2):275-305, 2018
- Relative Trisections of Smooth 4-manifolds with Boundary, Ph.D. Thesis, The University of Georgia, 2016