Research
Classification of Staircases for Hirzebruch Surfaces
Joint with Dusa McDuff, Ana Rita Pires, and Morgan Weiler, I have written a series of papers about the elliposid embedding function of Hirzebruch surfaces. Given the eccentricity of an elliposid, this function measures the minimum scalling of the symplectic form needed to symplectically embed the elipsoid into a Hirzebruch surface. A property we can classify about this function is called an infinite staircase. Having an infinite staircase implies their are infinitely many relevant obstructions necessary to compute the ellipsoid embedding function. In this series of papers, we classify this property for Hirzebruch surfaces. Hirzebruch surfaces can be parameterized by a b-value in the interval (0,1). The below image, made by Ana Rita Pires, is a number line from (0,1) which gives a visualization of the classification.
Staircase symmetries in Hirzebruch surfaces, with Dusa McDuff. To appear in Algebr. Gom. Topol.
This work explores the symmetries of infinite staircases occuring in the embedding problems. This paper enables us to reduce the study of staircases in Hirzebruch surfaces to just the red intervals in the above image. Iterating the symmetries brings the red intervals to the various blue and orange intervals in an interweaving fashion seen in the figure.
Staircase patterns in Hirzebruch surfaces, with Dusa McDuff and Morgan Weiler. To appear in Comm. Math. Helv.
This paper gives an almost complete classification for which Hirzebruch surfaces have infinite staircases. Starting with the first infinite layer of red intervals shown on the number line in the figure above, we construct a cantor set of intervals in between each pair of red intervals in the first layer. The symmetries then bring this same structure to the orange and blue intervals. We show:
for each b-value contained in an open interval in the figure, there are no infinite staircases,
for each b-value at the endpoint of one of the intervals in the figure, there is an infinite staircase,
for each b-value which is the limit point of the endpoints coming from the cantor set, there is an infinite staircases.
In summary, we showed there is a countable number of Cantor sets worth of Hirzebruch surfaces with infinite staircases, and a dense set of Hirzebruch surfaces that do not have infinite staircases.
A classification of infinite staircases for Hirzebruch surfaces, with Ana Rita Pires and Morgan Weiler.
We complete the open questions left in Staircase patters in Hirzebruch surfaces to finish the classification of which Hirzebruch surfaces have infinite staircases. This involves showing that the b-values at the limiting point of the symmetries (i.e. where the blue and orange intervals glue together) do not have infinite staircases. We also prove a speical case of a conjecutre of Cristofaro-Gardiner, Holm, Mandini, and Pires, which states there is only one Hirzebruch surface with rational blow up size that has an infinite staircase. In the picture, this special rational Hirzebruch surface is the limit point of the interweaving circles given by the symmetries.
Unobstructed embedding in Hirzebruch surfaces. To appear in J. Symp. Geo.
We use almost toric fibratons to consturct full fillings at the accumulation point for a certain family of Hirzebruch surfaces. This work explores how the constructions of symplectic embedding are relating to the obstruction given by symplectic spheres of self intersection -1. The work was also necessary in proving various b-values do have infinite staircases.
Staircases in Other Convex Toric Domains
Four-periodic infinite staircases for four-dimensional polydisks, with Caden Farley, Tara Holm, Jemma Schroder, Morgan Weiler, Zichen Wang, and Elizaveta Zabelina. To appear in Involve. This is work from the SPUR REU run by Tara Holm, Morgan Weiler, and myself in Summer 2022. Here, we follow sequences of ATFs I constructed for my research in the 2-fold blow up of the complex projective plane to compute inner corners for the embedding function for a specific polydisk. This works gives evidence for the connection between infinite stiarcases for Hirzebruch surfaces and polydisks.
Staircases in convex toric domains, with Dusa McDuff and Dan Cristofaro-Gardiner. In this upcoming work, we show many different convex toric domains do not have infinite staircases and generalize work of Cristofaro-Gardiner, Holm, Mandini, and Pires to the irrational case.
For my thesis work, I am considering embeddings of ellipsoids into other toric targets. Currently, I'm focused on the two-fold blow of the complex projective plane and generalizing these results of the Hirzebruch surface into other targets.
Undergraduate Research:
With Julianna Tymoczko, we formed a bijection between maximal springer fiber cells and combinatorial objects represented by webs.
Jaeho Choi, Nitin Krishna, and I participated in the SMALL REU under the guidance of Alejandro Sarria. We worked on understanding the global regularity of solutions to a family of partial differential equations. On the L^p regularity of solutions to the generalized Hunter-Saxton system