Tsuyoshi Kato

Covering monopole map and aspherical 10/8-inequality conjecture

Abstract: The aspherical 10/8-inequality conjecture is obtained by combining a covering 10/8-inequality conjecture and Singer conjecture.In this talk, I will present a survey of recent developments on related topics.

Tian-Jun Li

Existence of  Lagrangian surfaces in symplectic 4-manifolds

Abstract: We discuss the existence  of  Lagragian surfaces in symplectic 4-manifolds. We first briefly review the orientable case which has been extensively studied. In the lesser understood non-orientable case, two basic facts are, given a mod 2 degree 2 homology class, 

 (i) the complexity satisfies Audin's Mod 4 congruence, and 

 (ii) the complexity  increases by 4 via Givental's local surgery. 

In addition, we observe that every mod 2 degree 2 homology class is represented by a non-orientable Lagrangian surface. So a natural problem is to investigate the minimal complexity of such surfaces subject to Audin's congruence. This problem  will be discussed in Lecture 2. This is based on joint works with Bo Dai, Chung-I Ho, and Weiwei Wu.

Minimal complexity of non-orientable Lagrangian surfaces in symplectic rational surfaces

Abstract:  We first  introduce various cones in the degree 2 de Rham cohomology group  defined by the existence of non-orientable Lagrangian surfaces and discuss their properties. We then investigate  the minimal complexity  of non-orientable Lagragians in symplectic rational surfaces. For this problem we will focus on the existence of Lagrangian projective planes. In particular, we show that any rational surface,  except the complex projective plane, admits a symplectic structure with no Lagrangian real projective plane. This is based on joint works with  Chung-I Ho, Weiwei Wu, and Shuo Zhang.

Thomas Mark

Convexity and embedding problems in 4-dimensional symplectic topology I

Abstract: Given a closed 3-manifold,  it is natural to ask what is the ``simplest’’ 4-manifold into which it embeds. One version of this question asks which 3-manifolds embed smoothly in 4-dimensional Euclidean space. While a complete answer to this seems out of reach for the moment, joint work with Bulent Tosun (2021) showed that if one asks for an embedding whose image bounds a symplectically convex region of 4-space, then there is an obstruction coming from Floer theory. In particular, we used this to show that no Brieskorn homology sphere admits such a symplectically convex embedding. In this first lecture I will present these ideas and results, and describe how they fit in with other notions of convexity such as holomorphic convexity and rational convexity.

Convexity and embedding problems in 4-dimensional symplectic topology II

Abstract: Continuing with the topics from the first lecture, I will describe circumstances in which we can obtain an obstruction for a 3-manifold to bound a symplectically convex region inside a closed symplectic manifold. Combined with recent progress in complex geometry, this allows extensions some of the non-embedding results for C^2 to 4-manifolds other than Euclidean space, such as certain rational surfaces. This work, some of which is in progress, is joint with Bulent Tosun.

András Stipsicz

Exotic definite four-manifolds with fundamental group Z_2

Abstract: It is not known if any simply connected definite four-manifold admits exotic structures — this question contains the smooth 4-dimensional Poincare conjecture as special case. The situation is drastically different if we allow even the smallest non-trivial fundamental group. The constructions of the exotic examples use the rational blow-down method, knot surgery and torus surgery. This is joint work with Zoltán Szabó.

Surfaces in 4-manifolds

Abstract: Minimal genera of surfaces (satisfying some further constraint, eg from homology or on the boundary) in 4-manifolds seem to be intimately related to possible exotic phenomena. In the talk we review some simple constructions, and discuss the case when the compact four-manifold has nontrivial boundary, and the surface is supposed to intersect this boundary in a prescribed manner. We will also discuss classical sliceness obstructions from this new angle.

Bülent Tosun

Contact surgeries and symplectic fillability

Abstract: It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties (e.g. tightness, fillability, vanishing or non-vanishing of various Floer or symplectic homology classes) of contact structures are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, extending an earlier work of the speaker with Conway and Etnyre, I will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (positive) integer surgery along a Legendrian knot to yield a weakly fillable contact manifold. When specialized to knots in the three sphere with its standard tight structure, this result can be rather efficient to find many examples of fillable surgeries along with various obstructions and surprising topological applications. This will report on joint work with T. Mark.