Abstract: 本講演内容は以下の2つの研究内容から構成される:
I. S1×D3とD2×S2との境界連結和をポシェットと呼ぶ。
Gluck手術の一般化でありトーラス手術の特別な場合に相当するポシェット手術が2004年に岩瀬順一氏と松本幸夫氏により導入された。
4次元多様体 X に埋め込まれたポシェット P に対して、
X 上のポシェット手術とは P の内部を取り除き P の境界の微分同相写像で
P を再接着する操作のことである。本講演では、
ポシェット手術がコードと2次元球面 S2 を用いた手術であることに着目し、
4次元球面 S4 上のポシェット手術の微分構造の分類を試みる。
II. 2003年に Peter Ozsváth氏とZoltán Szabó氏は
d 不変量と呼ばれる3次元ホモロジー球面に対するホモロジー同境不変量を導入した。
本講演では、任意の p が奇数かつ pq+pr-qr=1 を満たす3次元Brieskornホモロジー球面
∑(p,q,r) に対するKarakurt-Şavkの公式を精密化することで新たに計算可能になった例について紹介する。
更に、任意の ∑(p,q,r) の
d 不変量に対するCan-Karakurtの公式を精密化することで現れた、
∑(p,q,r) とレンズ空間の d 不変量との関係についても紹介する。
本講演は丹下基生氏（筑波大学）との共同研究の内容を含む。
5月7日 -- オンライン開催, 17:00 -- 18:00
Ingrid Irmer (南方科技大学)
The Thurston spine and the Systole function of Teichmüller space
Abstract: The systole function fsys on Teichmüller space Tg of a closed genus g
surface is a piecewise-smooth map Tg → R
whose value at any point is the length of the shortest geodesic on the corresponding hyperbolic surface.
It is known that fsys gives a mapping class group-equivariant handle decomposition of Tg
via an analogue of Morse Theory.
This talk explains the relationship between this handle decomposition and the Thurston spine of Tg.
Abstract: The space of smooth compact exact Lagrangians of a cotangent bundle carries the spectral metric γ,
and we consider its completion. With an element of the completion,
Viterbo associated a closed subset called γ-support.
In this talk, I will explain how we can use sheaf-theoretic methods to explore the completion and γ-supports.
I will show that we can associate a sheaf with an element of the completion,
and its (reduced) microsupport is equal to the γ-support through the correspondence.
With this equality, I will also show several properties of γ-supports.
This is joint work with Tomohiro Asano (RIMS),
Stéphane Guillermou (Nantes Université),
Vincent Humilière (Sorbonne Université),
and Claude Viterbo (Université Paris-Saclay).
5月28日 -- オンライン開催, 17:00 -- 18:00
Andreani Petrou (沖縄科学技術大学院大学)
Knot invariants and their Harer-Zagier transform
Abstract: The Harer-Zagier (HZ) transform is a discrete Laplace transform that can be applied to knot polynomials,
mapping them into a rational function of two variables λ and q.
The HZ transform of the HOMFLY-PT polynomial has a simple form, as it can be written as a sum of factorised terms.
For some special families of knots, it can be fully factorised and it is completely determined by a set of exponents.
There is an interesting relation between such exponents and Khovanov homology.
Moreover, we conjecture that there is an 1-1 correspondence with such factorisability and a relation between the HOMFLY-PT and Kauffman polynomials.
Furthermore, we suggest that by fixing the variable λ = qn for some "magical" exponent n,
the HZ transform of any knot can obtain a factorised form in terms of cyclotomic polynomials.
Finally, the zeros of the HZ transform show an interesting behaviour, which shall be discussed.
6月4日 -- オンライン開催, 17:00 -- 18:00
石川 勝巳 (京都大学数理解析研究所)
The trapezoidal conjecture for the links of braid index 3
Abstract: The trapezoidal conjecture is a classical famous conjecture posed by Fox,
which states that the coefficient sequence of the Alexander polynomial of
any alternating link is trapezoidal.
In this talk, we show this conjecture for any alternating links of braid index 3.
Although the result holds for any choice of the orientation,
we shall mainly discuss the case of the closures of alternating 3-braids with parallel orientations.
Abstract: The problem of embedding abstract Riemannian manifolds isometrically
(i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of
whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same.
As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. C1)
than at high regularity (i.e. C2). For example,
by the famous Nash--Kuiper theorem it is possible to find C1
isometric embeddings of the standard 2-sphere into arbitrarily small balls in R3,
and yet, in the C2 category there is (up to translation and rotation) just one isometric embedding,
namely the standard inclusion.
Analoguous to the Onsager conjecture in fluid dynamics,
one might ask if there is a sharp regularity threshold in the Hölder scale which distinguishes these flexible and rigid
behaviours.
In my talk I will review some known results and argue why the Hülder exponent 1/2 can be seen as a critical exponent in the problem.
Abstract: Even though Cn is the most basic symplectic manifold,
when n>2 its compactly supported symplectomorphism group remains mysterious.
For instance, we do not know if it is connected.
To understand it better,
one can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them.
This leads us to the Liouville pseudo-isotopy group of a contact manifold,
important for relating (for instance) compactly supported symplectomorphisms of Cn,
and contactomorphisms of the sphere at infinity.
After explaining this background, the talk will focus on a new result:
that the pseudo-isotopy group is connected, under a Liouville-vs-Weinstein hypothesis.
The second quandle homology group of the knot n-quandle
Abstract: We compute the second quandle homology group of the knot n-quandle for each integer n>1,
where the knot n-quandle is a certain quotient of the knot quandle
(of an oriented classical knot in the 3-sphere).
Although the second quandle homology group of the knot quandle can only detect the unknot,
it turns out that that of its 3-quandle can detect the unknot,
the trefoil and the cinqfoil. This is a joint work with Yuta Taniguchi.
7月9日 -- オンライン開催, 17:00 -- 18:00
中村 伊南沙 (佐賀大学)
Knitted surfaces in the 4-ball and their chart description
Abstract: Knits (or BMW tangles) are tangles in a cylinder generated by generators of the BMW (Birman-Murakami-Wenzl) algebras,
consisting of standard generators of the braid group and their inverses,
and splices of crossings called pairs of hooks.
We give a new construction of surfaces in D2 × B2, called knitted surfaces (or BMW surfaces),
that are described as the trace of deformations of knits,
and we give the notion of charts for knitted surfaces, that are finite graphs in B2.
We show that a knitted surface has a chart description.
Knitted surfaces and their chart description include 2-dimensional braids and their chart description.
This is joint work with Jumpei Yasuda (Osaka University).
7月23日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
川室 圭子 (University of Iowa)
Shortest word problem in braid theory
Abstract: Given a braid element in B_n,
searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem.
Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997.
In this talk I will discuss recent development of this problem for braid index 5 or higher.
I will also show diagrammatic computational technique of the Left Canonical Form of a given braid,
that is a key to the three fundamental problems in braid theory;
the Word Problem, the Conjugacy Problem and the Shortest Word Problem.
This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.
