I am interested in low-dimensional topology and, in particular, interactions between 3-dimensional and 4-dimension manifolds, constructions of exotic 4-manifolds, and knot theory. Recently, I have been exploring which negative definite plumbings of disk bundles over the 2-sphere with associated graphs containing exactly one cycle have boundaries that also bound rational homology circles. Such plumbings can be used to construct rational homology 3-spheres that bound rational homology 4-balls as well as exotic 4-manifolds via cut-and-paste. In a related direction, I have also been exploring so-called χ-slice links, which are links in the 3-sphere that bound properly and smoothly embedded surfaces with no closed components and Euler characteristic 1 in the 4-ball. One motivation behind studying such links is that the double covers of the 3-sphere branched along χ-slice links with nonzero determinant are rational homology 3-spheres that bound rational homology 4-balls.




Publications and Preprints
Geography of symplectic Lefschetz fibrations and rational blowdowns.
Abstract We produce simply connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic 4-manifolds. We present an example of a construction of a minimal symplectic exotic CP^2#-5CP^2 through this procedure applied to a genus-3 fibration.

  with R. Inanc Baykur and Mustafa Korkmaz
On the nonorientable 4-ball genus of torus knots.
Abstract The nonorientable 4-ball genus of a knot K in the 3-sphere is the minimal genus of nonorientable surfaces in the 4-ball bounded by K. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable 4-ball genus of any knot. This bound is sharp for several families of torus knots, including T(4n,(2n±1)2) for all positive even integers n, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever p is an even positive integer and p/2 is not a perfect square, then the torus knot T(p,q) does not bound a locally flat Möbius band for almost all integers q relatively prime to p.

  with Fraser Binns, Sungkyung Kang, and Paula Truöl
Classification of torus bundles that bound rational homology circles.
Abstract In this article, we completely classify torus bundles over the circle that bound 4-manifolds with the rational homology of the circle. Along the way, we classify certain integral surgeries along chain links that bound rational homology balls and explore a connection to 3-braid closures whose double branched covers bound rational homology 4-balls.

  to appear in Algebraic and Geometric Topology
Using rational homology circles to construct rational homology balls.
Abstract Motivated by Akbulut-Larson's construction of Brieskorn spheres bounding rational homology 4-balls, we explore plumbed 3-manifolds that bound rational homology circles and use them to construct infinite families of rational homology 3-spheres that bound rational homology 4-balls. In particular, we find infinite families of torus bundles over the circle that bound rational homology circles and provide a simple method for constructing more general plumbed 3-manifolds that bound rational homology circles. We then use these rational homology circles to show that, for example, -1-surgery along any twisted positively-clasped Whitehead double of any knot bounds a rational homology 4-ball and 1-surgery along any unknotting number one knot K with a positive crossing that can be switched to unknot K bounds a rational homology 4-ball.

  Topology and its Applications, Vol. 291 (2021)
Tight contact structures on some plumbed 3-manifolds.
Abstract In this article, we prove a generalization of a theorem of Lisca-Matic to Stein cobordisms and develop a method for distinguishing certain Stein cobordisms using rotation numbers. Using these results along with standard techniques from convex surface theory and classifications of tight contact structures on certain 3-manifolds due to Honda, we classify the tight contact structures on a certain class of plumbed 3-manifolds that bound non-simply connected 4-manifolds. Moreover, we give descriptions of the Stein fillings of the Stein fillable contact structures.
Symplectically replacing plumbings with Euler characteristic 2 4-manifolds.
Abstract We introduce new symplectic cut-and-paste operations that generalize the rational blowdown. In particular, we will define k-replaceable plumbings to be those that, heuristically, can be symplectically replaced by Euler characteristic k 4-manifolds. We will then classify 2-replaceable linear plumbings, construct 2-replaceable plumbing trees, and use one such tree to construct a symplectic exotic ℂP2#6(-ℂP2).

   Journal of Symplectic Geometry, Vol. 18, No. 5 (2020), pp. 1285-1318

PhD Dissertation
Cut-and-paste operations and exotic 4-manifolds.

Research Talks
Math Seminar
Spelman College
October 2022
Math Club
Agnes Scott College
September 2022
Topology Seminar
University of Georgia
April 2022
AMS Southeastern Sectional Meeting
University of Southern Alabama
November 2021
Research Horizons Seminar
Georgia Tech
November 2021
Geometry Topology Working Seminar
Georgia Tech
October 2021
Algebra-Topology Seminar
University of Alabama
October 2021
Geometry Topology Seminar
Georgia Tech
September 2021
Low-dimensional topology and symplectic geometry weekend
Remote
April 2021
Geometry Seminar
University of Virginia
December 2020
Topology Seminar
Brandeis University
October 2020
AMS Fall Western Sectional Meeting
UC Riverside
November 2019
Geometry and Topology Seminar
University of Massachusetts Amherst
October 2019
Moab Topology Conference
Moab, UT
May 2019
Symplectic Geometry, Gauge Theory, and Categorification Seminar
Columbia University
April 2019
Geometry and Topology Seminar
University of Massachusetts Amherst
September 2018
Geometric structures on 3- and 4-manifolds
Inter-University Centre, Dubrovnik, Croatia
June 2018
Joint Mathematics Meetings
San Diego, CA
January 2018
Tech Topology Conference
Georgia Tech
December 2017
Joint Mathematics Meetings
Atlanta, GA
January 2017
Perspectives in Topology and Geometry of 4-manifolds
Inter-University Centre, Dubrovnik, Croatia
June 2016
AMS Southeastern Sectional Meeting
University of Georgia
March 2016
Geometry Seminar
University of Virginia
November 2015

Past Expository Talks
A history of 4-manifolds
Graduate Student Seminar
University of Virginia
March 2018
Building exotic 4-manifolds
Graduate Student Seminar
University of Virginia
October 2015
Intro to Kirby Calculus
Topology Student Seminar
University of Virginia
February 2015

Conferences/Workshops
AMS Western Section Meetings, UC Riverside, CA November 2019
Knot concordance and low-dimensional manifolds, Le Croisic, France June 2019
Moab Topology Conference, USU Moab May 2019
Virginia Topology Conference, University of Virginia December 2018
Knotted Surfaces in 4-manifolds, UMass Amherst October 2018
The topology and geometry of low-dimensional manifolds, UT-Austin July 2018
Geometric Structures on 3- and 4-manifolds, Dubrovnik, Croatia June 2018
Joint Mathematics Meetings, San Diego, CA January 2018
Low-Dimensional Topology Conference, UCLA January 2018
Tech Topology Conference, Georgia Tech December 2017
58th Annual Texas Geometry and Topology Conference, UT - Austin November 2017
2017 Virginia Topology Conference, University of Virginia November 2017
CMO-BIRS: Low Dimensional Topology and Gauge Theory, Oaxaca, MX August 2017
Kylerec Workshop, Truckee, CA May 2017
Joint Mathematics Meetings, Atlanta, GA January 2017
Virginia Topology Conference, University of Virginia November 2016
Low Dimensional Topology Workshop, Central European University, Budapest, Hungary July 2016
Perspectives in Topology and Geometry of 4-manifolds, Dubrovnik, Croatia June 2016
Knots in the Triangle (Knots in Washington XLII), North Carolina State University April 2016
Graduate Student Topology & Geometry Conference, Indiana University April 2016
AMS Southeastern Sectional Meeting, University of Georgia March 2016
Joint Mathematics Meetings, Seattle, WA January 2016
William Rowan Hamilton Geometry and Topology Workshop, Trinity College, Dublin, Ireland August 2015
Princeton Low Dimensional Topology Workshop, Princeton University June 2015
Redbud Topology Conference, Oklahoma State University April 2015
Topology Student Workshop, Georgia Tech June 2014
Graduate Student Topology & Geometry Conference, UT - Austin April 2014
Tech Topology Conference, Georgia Tech December 2013