Titles and abstracts:
Isabelle Charton (University of Haifa):
The equivariant cohomology of complexity one spaces with isolated fixed points
Oliver Goertsches (University of Marburg), Leopold Zoller (LMU Munich) and Panagiotis Konstantis (University of Marburg):
Low-dimensional GKM theory in three acts
In GKM theory, one associates a labelled graph to certain torus actions on smooth manifolds. This graph encodes many topological properties of the action, such as the equivariant cohomology and characteristic classes, and in dimension 6 even the diffeomorphism type of the manifold. On the other hand, motivated by the Delzant correspondence, one can ask the question which GKM graphs are realized by GKM actions.
In this series of talks we will, after reviewing previous results on GKM realization of GKM fiber bundles in dimension 6 and non-equivariant
rigidity in dimensions 6 and 8, focus on the general GKM realization and equivariant rigidity problem in dimension 6. Prominent roles will be
played by the surface one obtains by gluing discs to the 3-valent GKM graph along the connection paths, equivariant Dehn twists, nonstandard
embeddings of spheres in $S^4$, and the extension problem for elements of the mapping class group of connected sums of $S^2 \times S^1$.
Tara Holm (Cornell University)
Equivariant cohomology and the (symplectic) diffeotype of complexity-one four-manifolds
In this talk, we will explore the relationship between the geometry and topology of a complexity-one four-manifold and the combinatorial data that encode it. We will use a generators and relations description for the even part of the equivariant cohomology of the manifold to see what geometric aspects the equivariant cohomology determines. Namely, it allows us to reconstruct the diffeotype but not the complex structure. The talk will be driven by specific examples. It is based on joint work with Liat Kessler; and with Liat Kessler and Susan Tolman.
Liat Kessler (University of Haifa)
An algorithm for a soft proof of finiteness of Hamiltonian torus actions
We recall McDuff-Borisov's proof of the finiteness of toric actions on compact symplectic manifolds. We extract the algorithm behind it and the data required for the algorithm to apply to torus actions of greater complexity: an integrable almost complex structure that is compatible with both the action and the symplectic form, and well behaved generators of the equivariant cohomology over the integers. We deduce a soft proof of the finiteness in case the torus is a circle and the manifold is of dimension four, in joint work with Tara Holm. We will discuss how to extend this proof to a more general case.
Nicholas Lindsay (University of Cologne)
On the topology of manifolds admitting a Hamiltonian circle action
I will discuss some results and questions regarding the topology of closed symplectic manifolds admitting a Hamiltonian circle action. In particular, I will highlight some applications of a result of Jones and Rawnsley about the intersection form of such manifolds. In a previous work I was able to weaken the assumption on the fixed point set in Jones and Rawnsley's result, under certain assumptions on the Betti numbers. I will discuss a new result in this direction.
Nicole Magill (Cornell University)
Symplectic embeddings and almost toric fibrations
The four dimensional ellipsoid embedding function of a toric symplectic manifold M measures when a symplectic ellipsoid embeds into M. It generalizes the Gromov width and ball packing numbers. In 2012, McDuff and Schlenk computed this function for a ball. The function has a delicate structure known as an infinite staircase. Based on work with McDuff, Pires, and Weiler, we will discuss the classification of which Hirzebruch surfaces have infinite staircases. We will focus on the part of the argument where symplectic embeddings are constructed via almost toric fibrations. We will see how in many cases, we expect almost toric fibrations to give all symplectic embeddings for ellipsoids with small eccentricity (where there are obstructions other than the volume).
Grace Mwakyoma-Oliveira (MSRI)
Hamiltonian circle actions on 4-orbifolds
In this talk I intend to explain the classification of 4-dimensional symplectic closed orbifolds with isolated cyclic singularities admitting a Hamiltonian circle action - which are called Hamiltonian S1-orbi-spaces. This was my thesis work and part of a collaboration with Leonor Godinho and Daniele Sepe. If time permits I may also briefly comment on extensions of this work which we will be pursuing.
Susan Tolman (University of Illinois at Urbana-Champaign)
Non-Hamiltonian circle actions with fewer isolated fixed points
Let the circle act on a closed manifold $M$, preserving a symplectic form $\omega$.
We say that the action is Hamiltonian if there exists a moment map, that is, a map $\Psi \colon M \to R$ such that $\iota_\xi \omega = - d \Psi$, where $\xi$ is the vector field that generates the action. In this case, a great deal of information about the manifold is determined by the fixed set. Therefore, it is very important to determine when symplectic actions are Hamiltonian. There has been a great deal of research on this question, but it left the following question, usually called the ``McDuff conjecture": Does there exists a non-Hamiltonian symplectic circle action with isolated fixed points on a closed, connected symplectic manifold? If so, how many fixed points? I answered the first question by constructing such an example with 32 fixed points. (This construction relies in part on joint work with J. Watts.) I will discuss my work with D. Jang to answer the second by reducing the number of fixed points. We have already constructed an example with as few as 10 fixed points, and are now working on constructing an example with only two fixed points, which is the smallest possible number.
Nikolas Wardenski (University of Marburg)
Multiplicity free U(2)-actions and triangles
We explicitly construct all compact, connected and locally free multiplicity free Hamiltonian U(2)-manifolds whose momentum polytope is a triangle.
Morgan Weiler (Cornell University)
ECH cobordism maps and 4D toric symplectic embeddings
In 1985 Gromov proved that volume is not the only obstruction to symplectic embedding, and that J-holomorphic curves, which are symplectically immersed surfaces if they are not multiply covered, can provide much stronger obstructions. In 2011, McDuff introduced a technique using symplectically embedded spheres to obstruct 4D symplectic embeddings, while Hutchings defined "ECH capacities," using J-holomoprhic curve cobordism maps to also obstruct 4D symplectic embeddings. The correspondence between these methods is not perfect, even in the case of toric domains (we will provide examples), but is occasionally fruitful: ECH capacities are more quickly computable, while symplectically embedded spheres often carry more information. We will explain how using both methods simultaneously allowed us to identify new infinite staircases of symplectic embeddings of toric domains (with Bertozzi, Holm, Maw, McDuff, Mwakyoma, and Pires and with Magill and McDuff). We will then discuss what hope there might be for a geometric proof of the correspondence between the methods in the case of toric domains, in light of previous work by Cristofaro-Gardiner--Hind and Cristofaro-Gardiner--Hind--McDuff.
Catalin Zara (UMass Boston)
The concept of assignments was introduced in Ginzburg et al. (1999) as a method for extracting geometric information about group actions on manifolds from combinatorial data encoded in the infinitesimal orbit-type stratification. For a torus acting in a Hamiltonian fashion on a compact symplectic manifold, the assignment ring is an extension of the equivariant cohomology ring and it is modeled on the GKM description of the equivariant cohomology of a GKM space. We will discuss results and questions relating the two structures, including works with Guillemin and Sabatini (2014), and Guillemin and Tolman (2020).