Schedule for: 23w5123 - Interactions between Symplectic and Holomorphic Convexity in 4 Dimensions

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

This introductory lecture will explore the basic smooth topology of manifolds admitting Stein structures, with a focus on Stein surfaces (i.e., those of real dimension 4). Guided by the natural Morse functions carried by Stein manifolds, we will unpack Eliashberg's topological characterization of Stein manifolds, Gompf's handlebody construction of Stein surfaces, and the adjunction inequality. We will close with an application to the existence of exotic smooth structures on 4-space.

We will discuss several general tools for finding strictly pseudoconvex subsets of complex surfaces. An open subset U is smoothly isotopic to a Stein open subset if and only if its inherited complex structure is homotopic (through almost-complex structures) to a Stein structure on U. If we allow topological isotopy (homotopy through homeomorphic embeddings with no differentiability assumed), the condition on the complex structure can be dropped, and it is only necessary for U to admit a topological Morse function whose critical points have index at most 2. A deeper version of this shows that every finite 2-complex in a complex surface is topologically isotopic to a Stein compact, in fact, to a nested intersection of uncountably many homeomorphic Stein open subsets. This leads to a notion of pseudoconvexity for unsmoothably embedded 3-manifolds. We discuss examples and applications of such phenomena, with the hope of encouraging further exploration with these tools.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Although exotic blow-ups of the complex projective plane at n points have been constructed for every n>1, the only examples known by means of rational blowdowns satisfy n>4. It has been an intriguing problem whether it is possible to decrease n. In this talk, I will show how to construct it for n=4 from a configuration of 8 lines and 2 conics in a special position. This is part of a bigger picture to construct exotic $p \C \pP^2 \# q \bar{\C \pP^2}$ via the construction of particular Kollár--Shepherd-Barron--Alexeev (KSBA) singular surfaces. This is done by explicitly analyzing obstructions coming from configurations of rational curves, and the use of computer searchers. This connection between the geography of configuration of rational curves and exotic 4-manifolds from KSBA surfaces leads to, I believe, a new view on this problem. There is a lot of data out of these searches, showing an intricate picture for KSBA surfaces. I hope to show that in this talk too. This is joint work with Javier Reyes.

A classic question in algebraic geometry asks what are the possible singularities for a plane curve of a given degree and genus. This is closely related to existence questions for singular Lagrangian surfaces in the affine or projective complex plane, which in turn connect with questions about the topology of rationally convex domains. In this talk I will describe a construction of various new families of rational plane curves with prescribed singularities, and I will wax poetic about how this ties in with the themes of this workshop.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A Lagrangian cobordism between Legendrian knots is an important notion in symplectic geometry. Many questions, including basic structural questions about these surfaces are yet unanswered. For instance, while it is known that these cobordisms form a preorder, and that they are not symmetric, it is not known if they form a partial order on Legendrian knots. The idea of a Lagrangian quasi-cobordism was first defined by Sabloff, Vela-Vick, and Wong. Roughly, for two Legendrians of the same rotation number, it is the smooth composition of a sequence of alternatingly ascending and descending Lagrangian cobordisms which start at one knot and ends at the other. This forms a metric monoid on Legendrian knots, with distance given by the minimal genus between any two Legendrian knots. In this talk, I will discuss some new results about Lagrangian quasi-cobordisms, based on some work in progress with Sabloff, Vela-Vick, and Wong.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In 1906, F. Hartogs discovered the existence of domains in $\mathbb{C}^n$ for which every holomorphic function can be extended to a larger domain. Domains that do not admit this extension phenomenon satisfy a complex type of convexity, known as pseudoconvexity. This type of convexity can be viewed as convexity "with respect to holomorphic functions", as opposed to geometric convexity, which is convexity "with respect to linear functions". In this talk, we will motivate and define pseudoconvexity. We will also compare and contrast its many equivalent formulations with that of classical convexity. We will also introduce a class of "pseudoconvex" manifolds and discuss their many properties. Notions of convexity with respect to other classes of functions will also be discussed.

In the first half of the talk I will give an overview of polynomial and rational convexity: I will give basic definitions, examples and outline some fundamental properties of polynomial and rationally convex compacts. In the second half of the talk I will discuss characterization of rational convexity of real submanifolds in complex Euclidean spaces and related problems.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Kodaira classified all singular fibers that can arise in algebraic elliptic fibrations. Later, Ogg, Iitaka and then Namikawa and Ueno gave a classification for genus two fibrations. In this work, we split these algebraic genus two fibrations into Lefschetz fibrations and determine the monodromies. More specifically, we look at four families of hypersurface singularities in \C^3. Each hypersurface comes equipped with a fibration by genus 2 algebraic curves which degenerate into a single singular fiber. We determine the resolution of each of the singularities in the family and find a flat deformation of the resolution into simpler pieces, resulting in a fibration of Lefschetz type. We then record the description of the Lefschetz fibration as a positive factorization in Dehn twists. This gives us a dictionary between configurations of curves and monodromy factorizations for some singularities of genus 2 fibrations. This is joint work with J. Van Horn-Morris.

A compact subset of $\mathbb{C^n}$ is said to be polynomially convex if it is cut out by a family of polynomial inequalities. Polynomial convexity grants certain approximation-theoretic properties to the underlying set. When the set is a real submanifold of $\mathbb{C^n}$, its convexity properties are partly influenced by its topology, and the local and global structure of its CR (complex-real) singularities. The minimum complex dimension into which all compact real manifolds of a fixed dimension admit smooth polynomially convex embeddings is not known (although some bounds can be deduced from the literature). In this talk, we will discuss some recent improvements on the previously known bounds. We will especially focus on the case where the h-principle has proved useful for producing the desired embeddings. This is joint work with R. Shafikov.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this introductory lecture, we focus on a special case of symplectic reduction, in which the reduction is compatible with a toric group action. We recall the basic notions, discuss an example that will come up in Jonny's lecture and, if time permits, give further applications.

Let K be a Lagrangian submanifold and L_t be a family of Lagrangian submanifolds. Suppose you can displace K from each L_t. Can you displace K from all L_t simultaneously? If not, from how many L_t can you simultaneously displace K? We will discuss some specific problems which have this flavour and give some small results in this direction.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Thom conjecture, proven by Kronheimer and Mrowka in 1994, states that complex curves in CP2 are genus minimizers in their homology class. We will show that an analogous statement does not hold for complex hypersurfaces in CP3. This is joint work with Ruberman and Strle.

The embedded contact homology (ECH) chain complex has several natural filtrations, and applications of ECH to symplectic and contact geometry often rely on computing the associated spectral invariants. When the three-manifold is spherical (or more generally, toric), this means there is a precise function from ECH index to the minimal filtration value among cycles representing that index's homology class. ECH practitioners attempt to compute or estimate these functions. We will explain why those attempts are much more successful in the case of convex toric contact forms, including applications of the ECH knot filtration to surface dynamics and the ECH action filtration (aka the ECH spectrum) to symplectic embedding problems. The latter project is based on joint work with several coauthors, in arXiv:2010.08567, arXiv:2203.06453, and arXiv:2210.15069.

I will motivate several notions of convexity that play important roles in Hamiltonian dynamics and in the theory of symplectic embeddings. In particular, I will focus on the mysterious role convexity plays in Viterbo’s conjecture on the systolic ratio and the symplectic capacities of convex domains in Euclidean space. I will end my talk by reviewing some recent progress towards this conjecture.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Twenty years ago, Giroux gave an influential result on the equivalence of contact structures in dimension 3 and open book decompositions up to stabilization. At the time, Giroux and Mohsen also partially extended the correspondence to all dimensions, albeit with different technology. From one point of view, the existence of open book decompositions can be viewed as a convexity statement for contact manifolds, and there are natural connections to symplectic convexity. In this talk, I will describe forthcoming joint work with Ko Honda and Yang Huang on establishing the Giroux correspondence in all dimensions using convex hypersurface theory.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The contact connected sum is a well-understood operation for contact manifolds. I will focus on the 3-dimensional case and the Weinstein 1-handle model for the contact connected sum. I will discuss how pseudo-holomorphic curves in the symplectization behave under this operation. After reviewing embedded contact homology, we will see how this results in a chain-level description of the embedded contact homology of a connected sum.

Peculiar features of low-dimensional differential, symplectic, and contact topology affect the theory of holomorphic functions of two complex variables. The purpose of the talk will be to illustrate this principle with a few token examples and discuss open problems and possible research directions in this area.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.