今野 北斗 From families Seiberg-Witten theory to corks
コルクとそれに関連する事柄への族のSeiberg-Witten理論によるアプローチを紹介する．
族のSeiberg-Witten理論を使って（強い意味での）コルクを検出できること，また「コルクの1-パラメータ版」の初めての例を検出できることを紹介する．
時間が許せば，コルクと関係するエキゾチックな曲面の検出も述べる．Abhishek Mallick氏, 谷口正樹氏との共同研究である．
作間 誠 Three and four dimensional equivariant genera of strongly invertible knots,
and the equivariant cobordism group
A knot $K$ in the 3-sphere is strongly invertible if there is an orientation-preserving smooth involution h of the 3-sphere
such that $h(K)=K$ and $h$ reverses the orientation of $K$.
In my paper “On strongly invertible knots” published in 1986,
I introduced the notion of a direction on a strongy invertoble knot
and defined the equivariant cobordism group of directed strongly invertible knots.
I then observed that the natural homomorphism from the equivariant cobordism group to the cobordism group has a nontrivial kernel,
by using a certain polynomial invariant based on the Kojima-Yamasaki eta-invarinat.
To my pleasant surprise, the strongly invertible knots recently attract attention of various researchers,
in particular, significant progresses have been made on the 4-dimensional equivariant genera
of strongly invertible knots by using modern technology.
On the other hand, in my recent joint works with Mikami Hirasawa and Ryota Hiura,
we studied 3-dimensional equivariant genera of directed strongly invertible knots, by using classical techniques.
In this talk, I quickly recall basic facts on strongly invertible knots,
and then give a survey of recent developments.
(Please understand that I cannot mention to the technical part using modern technology
by the lack of my knowledge and ability. )
谷口 正樹 Corks, Instantons, and Chern-Simons functional
(joint work with Abhishek Mallick (Rutgers University), Irving Dai (University of Texas at Austin) and Antonio Alfieri (CRM-ISM))
We present a variant of involutive instanton Floer theory that obstructs the existence of diffeomorphisms on certain 4-manifolds with boundaries.
Our research yields strong corks which survive after stabilization by $n{\mathbb C}P^2$ or $-n{\mathbb C}P^2$ and new strong corks obtained as a family of surgeries along slice knots, and establishes topological constraints concerning non-orientable surfaces with boundaries. Moreover, we prove that any smooth self-homology cobordism of the boundary of the Akbulut cork is not simply connected if there is a diffeomorphism extending involutions on the boundaries.
This is joint work with Abhishek Mallick, Irving Dai, and Antonio Alfieri.
高橋 夏野 Relative trisections for corks and exotic 4-manifolds
An important conjecture in trisection theory is that “if two 4-manifolds
are exotic, then they have the same trisection genus”. In this talk, we
present an exotic pair of 4-manifolds with boundary whose trisection
genera are both 4. Among the known supporting evidence for the
conjecture, our exotic pair has the smallest trisection genus. Our
trisections of the exotic pair are derived from two minimal genus
relative trisections of the Akbulut cork.
若槙 洋平 Stabilizations of some small simply-connected closed 4-manifolds
An oriented 4-manifold is said to be almost completely
decomposable (ACD) if it becomes diffeomorphic to $\#a{\mathbb C}P^2 \#
(-b){\mathbb C}P^2$ after a single ${\mathbb C}P^2$-stabilization
(i.e., taking a connected sum with ${\mathbb C}P^2$ for some integer
$a,b \geq 0$. It is a conjecture of Mandelbaum that any simply-connected
closed 4-manifold is ACD for at least one orientation. While many
examples are known to be ACD, those with small second Betti numbers
b2 are quite rare. In this talk, we give new examples of ACD
4-manifolds with $b_2 = 7,8,9$, and $10$. We also discuss a similar
result for $S^2\times S^2$-stabilization.
安井弘一 Genus functions of 4-manifolds and their applications
Typical applications of corks are constructions of exotic 4-manifolds, and their smooth structures are often distinguished by their genus functions. Here, the genus function of a smooth oriented 4-manifold is the function that maps a second homology class to the minimal genus of an embedded closed oriented surface representing the class. In this talk, I will briefly review genus functions, and give applications to exotic smooth structures and properties of smooth 4-manifolds such as the geometric simple connectivity and the mod 2 simple type conjecture. I will also discuss a limitation of genus functions and its applications.
Roberto Ladu On the universal cork conjecture
I will discuss some recent results on the universal cork conjecture including a sufficient condition for a cork to be not universal and the non-existence of a universal cork in the case of manifolds with boundary.
Sungkyung Kang
(Talk 1) Bordered Floer homology and the invariant splitting principle
We introduce bordered Floer theory and its involutive version, as well as their applications to knot complements.
We will sketch the proof that invariant splittings of $CFK$ and those of $CFD$ correspond to each other under the Lipshitz-Ozsvath-Thurston correspondence, via invariant splitting principle, which is an ongoing work with Gary Guth.
Sungkyung Kang
(Talk 2) One is not enough, with a view towards two
Using the invariant splitting principle, we construct an infinite family of exotic pairs of contractible 4-manifolds which survive one stabilization.
We argue that some of them are potential candidates for surviving two stabilizations.
