Tuesday Seminar on Topology (April -- July, 2024)
[Japanese] [Past Programs]
17:00 -- 18:30 Graduate School of Mathematical Sciences,
The University of Tokyo (with live streaming on Zoom)
or
17:00 -- 18:00 Online seminar on Zoom.
Last updated September 24, 2024
Information :
Kazuo Habiro
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai
April 9, 17:00-18:30 -- Room 056 with live streaming on Zoom
Shouhei Honda (The University of Tokyo)
Topological stability theorem and Gromov-Hausdorff convergence
Abstract: Gromov-Hausdorff distance defines a distance on the set of
all isometry classes of compact metric spaces.
It is natural to ask about topological relationships between two compact metric spaces whose Gromov-Hausdorff distance is small.
Cheeger-Colding provided a striking result about this question,
under a (lower) curvature bound on Ricci curvature.
In this talk we will improve this result sharply.
This is a joint work with Yuanlin Peng (Tohoku University).
If time permits, along this direction,
we will also discuss a recent work about a topological stability result to flat tori via harmonic maps,
where this is a joint work with
Christian Ketterer (University of Freiburg),
Ilaria Mondello (Université de Paris Est Créteil),
Chiara Rigoni (University of Vienna) and Raquel Perales (CIMAT).
April 16, 17:00-18:30 -- Online on Zoom
Hiroaki Karuo (Gakushuin University)
Skein algebras and quantum tori in view of pants decompositions
Abstract: To understand the algebraic structures of skein algebras and their
generalizations,
we usually try to embed these algebras into quantum tori using ideal triangulations of a surface and the splitting map.
However, such a construction does not work for the skein algebras of closed surfaces and the Roger-Yang skein algebras of punctured surfaces.
In the talk, we define filtrations on these algebras using pants decompositions and embed the associated graded algebras into quantum tori.
As a consequence, Roger-Yang skein algebras are quantizations of decorated Teichmuller spaces.
This talk is based on a joint work with Wade Bloomquist (Morningside University)
and Thang Le (Georgia Institute of Technology).
April 23, 17:00-18:30 -- Room 056 with live streaming on Zoom
Tatsumasa Suzuki (Meiji University)
Pochette surgery on 4-manifolds and the Ozsváth-Szabó d-invariants of Brieskorn homology 3-spheres
Abstract: This talk consists of the following two research contents:
I. The boundary sum of S1×D3 and D2×S2 is called a pochette.
The pochette surgery, which is a generalization of Gluck surgery and a special case of torus surgery,
was discovered by Zjuñici Iwase and Yukio Matsumoto in 2004.
For a pochette P embedded in a 4-manifold X,
a pochette surgery on X is the operation of removing the interior of
P and gluing P by a diffeomorphism of the boundary of P.
In this talk, we focus on the fact that pochette surgery is a surgery with a cord and the 2-sphere S2,
and attempt to classify the diffeomorphism type of pochette surgery on the 4-sphere S4.
II. In 2003, Peter Ozsváth and Zoltán Szabó introduced a homology cobordism invariant for homology 3-spheres called a d-invariant.
In this talk,
we present new computable examples by refining the Karakurt--Şavk
formula for any Brieskorn homology 3-sphere ∑(p,q,r) with p is odd and pq+pr-qr=1.
Furthermore, by refining the Can-Karakurt formula for the d-invariant of any ∑(p,q,r),
we also introduce the relationship with the d-invariant of ∑(p,q,r) and those of lens spaces.
This talk includes contents of joint work with Motoo Tange (University of Tsukuba).
May 7, 17:00-18:00 -- Online on Zoom
Ingrid Irmer (Southern University of Science and Technology)
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.
May 14, 17:00-18:00 -- Online on Zoom
Noriyuki Hamada (Institute of Mathematics for Industry, Kyushu University)
Exotic 4-manifolds with signature zero
Abstract: We will talk about our novel examples of symplectic 4-manifolds,
which are homeomorphic but not diffeomorphic to the standard simply-connected closed 4-manifolds with signature zero.
In particular, they provide such examples with the smallest Euler characteristics known to date. Our method employs the time-honored approach of reverse-engineering,
while the key new ingredients are the model manifolds that we build from scratch as Lefschetz fibrations.
Notably, our method greatly simplifies pi_1 calculations, typically the most intricate aspect in existing literature.
This is joint work with Inanc Baykur (University of Massachusetts Amherst).
May 21, 17:30-18:30 -- Room 056 with live streaming on Zoom
Yuichi Ike (Institute of Mathematics for Industry, Kyushu University)
γ-supports and sheaves
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).
May 28, 17:00-18:00 -- Online on Zoom
Andreani Petrou (Okinawa Institute of Science and Technology)
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.
June 4, 17:00-18:00 -- Online on Zoom
Katsumi Ishikawa (RIMS, Kyoto University)
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.
June 11, 17:00-18:30 -- Room 056 with live streaming on Zoom
Nariya Kawazumi (The University of Tokyo)
A topological proof of Wolpert's formula of the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates
Abstract: Wolpert explicitly described the Weil-Petersson symplectic form on the Teichmüller space in terms of the Fenchel-Nielsen coordinate system,
which comes from a pants decomposition of a surface.
By introducing a natural cell-decomposition associated with the decomposition,
we give a topological proof of Wolpert's formula,
where the symplectic form localizes near the simple closed curves defining the decomposition.
June 20 (Thu) [Joint with RIKEN iTHEMS], 17:00-18:30 -- Room 002 with live streaming on Zoom
Dominik Inauen (University of Leipzig)
Rigidity and Flexibility of Iosmetric Embeddings
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.
June 25 [Joint with RIKEN iTHEMS],
17:00-18:30 -- Room 056 with live streaming on Zoom
Emmy Murphy (University of Toronto)
Liouville symmetry groups and pseudo-isotopies
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.
July 2, 17:00-18:30 -- Room 117 with live streaming on Zoom
Kokoro Tanaka (Tokyo Gakugei University)
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.
July 9, 17:00-18:00 -- Online on Zoom
Inasa Nakamura (Saga University)
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).
July 23, 17:00-18:30 -- Room 056 with live streaming on Zoom
Keiko Kawamuro (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.