11:00-12:00, Masaki Taniguchi (Kyoto University)
Title: An Adjunction Inequality and the Topology of Symplectic Caps
Abstract: While the topology of symplectic fillings of contact 3-manifolds has been extensively studied, much less is known about the topology of symplectic caps. For the standard tight contact structure on $S^3$, Hom–Lidman and Yasui independently established notable constraints on the topology of such caps, which, in fact, yield constraints on closed symplectic 4-manifolds.
In this talk, we present analogous constraints for symplectic caps of lens spaces, using Seiberg–Witten theory. In addition, we establish a Thurston–Bennequin type inequality for symplectic caps. This is joint work with Anubhav Mukherjee and Nobuo Iida, and builds on our earlier paper: arXiv:2102.02076.
14:00-15:00, Jongil Park (Seoul National University)
Title: Algebraic Montgomery–Yang problems and smooth obstructions
Abstract: A normal projective surface with the same Betti numbers of the projective plane $\mathbb{CP}^2$ is called a rational homology projective plane or a $\mathbb{Q}$-homology $\mathbb{CP}^2$. People working in algebraic geometry and topology have long studied a $\mathbb{Q}$-homology $\mathbb{CP}^2$ with possibly
quotient singularities.
It is now known that it has at most five such singular points, but it is so mysterious that there are still many unsolved problems left.
Among them, the (algebraic) Montgomery–Yang problem is one of the most famous conjectures in this field.
In this talk, I’ll review some known results and recent developments on the algebraic Montgomery–Yang problem obtained by utilizing gauge theoretic results such as Donaldson diagonalization theorem, Heegaard Floer correction terms and so on.
This is a joint work with Woohyeok Jo and Kyungbae Park.
15:30-16:30, Tomohiro Asano (Kyoto University)
Title: Microlocal Sheaf Theory and the Square Peg Problem
Abstract: The square peg problem asks whether every simple closed curve in the plane contains four points that form the vertices of a square. This problem was proposed by Toeplitz in 1911 and it remains open at present.
In recent years, Greene and Lobb discovered that symplectic geometry can be effectively applied to this problem and obtained new results.
In this talk, I will explain how microlocal sheaf theory allows us to further extend this approach and affirmatively solve the problem for a large class of curves, including all curves of finite length. This talk is based on joint work with Yuichi Ike.
10:00-11:00, Jin Miyazawa (RIMS)
Title: Recent development of Real Seiberg–Witten theory
Abstract: Real Seiberg–Witten theory is a variant of Seiberg–Witten theory and it has recently developed rapidly . If there is an involution on a 3- or 4-manifold and if the involution satisfies some topological conditions, we can consider a twisted lift of the involution to Seiberg–Witten equations. By looking at the solutions of Seiberg–Witten equations that are fixed by the twisted lift of the involution, we can construct some invariants of 3- or 4-manifold and the involution on it. As an application, we can prove the existence of exotic $P^2$-knots and non-sliceness of certain knots. In this talk, we explain the outline of the Real Seiberg–Witten theory and its applications to low-dimensional topology including recent work with JungWang Park and Masaki Taniguchi about satellite formula of Floer homotopy type.