Research

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 \bf{X} is a decomposition \bf{X\boldsymbol{=} X_1 \boldsymbol{\cup} X_2\boldsymbol{\cup} X_3} three 4-dimensional 1-handlebodies \bf{X_i \cong \natural^k S^1 \times B^3} whose pairwise intersections are 3-dimensional handlebodies \bf{X_i\boldsymbol{\cap} X_j\boldsymbol{\cong}\boldsymbol{\#}^g S^1\boldsymbol{\times} S^2} and triple intersection \bf{X_1 \cap X_2 \cap X_3} 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 \bf{\boldsymbol{(}F,\boldsymbol{\alpha, \beta, \gamma)}}, where \bf{F} is a closed, genus \bf{g} surface and each of \boldsymbol{\alpha, \beta, \gamma} are \bf{g}-tuples of non-separating, simple, closed curves in \bf{F} such that each triple \bf{\boldsymbol{(}F, \boldsymbol{\alpha, \beta)}},\bf{\boldsymbol{(}F, \boldsymbol{\beta, \gamma)}},\bf{\boldsymbol{(}F, \boldsymbol{\alpha, \gamma)}} is a Heegaard diagram for \bf{\#^g S^1 \times S^2.}

Untitled
A Genus 3 Trisection of S^4

Publications and Preprints:

  1. Classifying low-genus and Isotopic Relative Trisections, with Patrick Naylor. In preparation.
  2. Relative Group Trisections, with Jason Joseph. In preparation.
  3. The Relative L–invariant of a Compact 4-manifold with Boundary, with Gabriel Islambouli, Maggie Miller, and Maggy Tomova. arXiv:1908.05371, 2019. Submitted, under revision.
  4. 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.
  5. Trisections of 4-manifolds via Lefschetz fibrations, with Burak Ozbagci. Mathematical Research Letters 26(2): 383-420, 2019.
  6. 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.)
  7. 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
  8. Trisecting Smooth 4-dimensional Cobordisms, arXiv:1703.05846, 2017
  9. Diagrams for Relative Trisections, with David T. Gay and Juanita Pinzón Caicedo, Pacific Journal of Mathematics 294 (2):275-305, 2018
  10. Relative Trisections of Smooth 4-manifolds with Boundary, Ph.D. Thesis, The University of Georgia, 2016