Dusa McDuff
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View article: Curvy points, the perimeter, and the complexity of convex toric domains
Curvy points, the perimeter, and the complexity of convex toric domains Open
We study the related notions of curvature and perimeter for toric boundaries and their implications for symplectic packing problems; a natural setting for this is a generalized version of convex toric domain which we also study, where ther…
View article: Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing
Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing Open
We solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded by a piecewise smooth function which exhibits a phase transition from an infinite Fibon…
View article: Polyfold fundamental classes and globally structured multivalued perturbations
Polyfold fundamental classes and globally structured multivalued perturbations Open
Work of Hofer--Wysocki--Zehnder has shown that many spaces of pseudoholomorphic curves that arise when studying symplectic manifolds may be described as the zero set of a polyfold Fredholm section. This framework has many analytic advantag…
View article: Symplectic capacities, unperturbed curves and convex toric domains
Symplectic capacities, unperturbed curves and convex toric domains Open
We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second named author using symplectic field theory. We then compute …
View article: Singular algebraic curves and infinite symplectic staircases
Singular algebraic curves and infinite symplectic staircases Open
We show that the infinite staircases which arise in the ellipsoid embedding functions of rigid del Pezzo surfaces (with their monotone symplectic forms) can be entirely explained in terms of rational sesquicuspidal symplectic curves. Moreo…
View article: Staircase symmetries in Hirzebruch surfaces
Staircase symmetries in Hirzebruch surfaces Open
This paper continues the investigation of staircases in the family of\nHirzebruch surfaces formed by blowing up the projective plane with weight b,\nthat was started in Bertozzi, Holm et al. in arXiv:2010.08567. We explain the\nsymmetries …
View article: Ellipsoidal superpotentials and singular curve counts
Ellipsoidal superpotentials and singular curve counts Open
Given a closed symplectic manifold, we construct invariants which count (a) closed rational pseudoholomorphic curves with prescribed cusp singularities and (b) punctured rational pseudoholomorphic curves with ellipsoidal negative ends. We …
View article: Some Advances in Pure Mathematics Made in the 19th Century [Around the Globe]
Some Advances in Pure Mathematics Made in the 19th Century [Around the Globe] Open
This article is reprinted with permission of the James Clerk Maxwell Foundation (JCMF), which is dedicated to the life and history of Clerk Maxwell. A wealth of information is available at http://www.clerkmaxwellfoundation.org/ . In additi…
View article: Symplectic capacities, unperturbed curves, and convex toric domains
Symplectic capacities, unperturbed curves, and convex toric domains Open
We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second named author using symplectic field theory. We then compute …
View article: Staircase symmetries in Hirzebruch surfaces
Staircase symmetries in Hirzebruch surfaces Open
This paper continues the investigation of staircases in the family of Hirzebruch surfaces formed by blowing up the projective plane with weight b, that was started in Bertozzi, Holm et al. in arXiv:2010.08567. We explain the symmetries und…
View article: Constructing the virtual fundamental class of a Kuranishi atlas
Constructing the virtual fundamental class of a Kuranishi atlas Open
Consider a space $X$, such as a compact space of $J$-holomorphic stable maps,\nthat is the zero set of a Kuranishi atlas. This note explains how to define the\nvirtual fundamental class of $X$ by representing $X$ via the zero set of a map\…
View article: Women's History Month
Women's History Month Open
In the summer of 2017
View article: Smooth Kuranishi atlases with isotropy
Smooth Kuranishi atlases with isotropy Open
Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build suc…
View article: On the stabilized symplectic embedding problem for ellipsoids
On the stabilized symplectic embedding problem for ellipsoids Open
This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has `eccentricity' of the form 3n-1 for some integer n.
View article: Strict orbifold atlases and weighted branched manifolds
Strict orbifold atlases and weighted branched manifolds Open
This note revisits the ideas in an earlier (2007) paper on orbifolds and branched manifolds, showing how the constructions can be simplified by using a version of the Kuranishi atlases recently developed by McDuff--Wehrheim. We first show …
View article: The fundamental class of smooth Kuranishi atlases with trivial isotropy
The fundamental class of smooth Kuranishi atlases with trivial isotropy Open
Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build suc…