Tuesday Seminar on Topology (October, 2019 -- January, 2020)
[Japanese] [Past Programs]
17:00 -- 18:30 Graduate School of Mathematical Sciences,
The University of Tokyo
Tea: 16:30 -- 17:00 Common Room
Last updated March 13, 2020
Information :
Toshitake Kohno
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai
October 1 -- Room 056, 17:00 -- 18:30
Jun Murakami (Waseda University)
Quantized SL(2) representations of knot groups
Abstract: Let K be a knot and G be a group.
The representation space of K for the group G means the space of homomorphisms from the knot group to G
and is defined by using the group ring C[G],
where C[G] is the ring of functions on G and has a commutative Hopf algebra structure.
This construction can be generalized to any commutative Hopf algebras.
In this talk, we extend this construction to any braided Hopf algebras with braided commutativity.
A typical example is BSL(2), which is the braided SL(2) introduced by S. Majid.
Applying the above construction to BSL(2), we get the space of BSL(2) representations,
which provides a quantization of SL(2) representations of a knot.
This is joint work with Roloand van der Veen.
October 8 -- Room 056, 17:30 -- 18:30
Masaki Tsukamoto (Kyushu University)
How can we generalize hyperbolic dynamics to group actions?
Abstract: Hyperbolicity is one of the most fundamental concepts in the study of dynamical systems.
It provides rich (expansive and positive entropy) and yet controllable (stable and having some nice measures) dynamical systems.
Then, can we generalize this to group actions?
A naive approach seems difficult.
For example, suppose Z2 smoothly acts on a finite dimensional compact manifold.
Then it is easy to see that its entropy is zero.
In other words, there is no rich Z2-actions in the ordinary finite dimensional world.
So we must go to infinite dimension.
But what kind structure can we expect in the infinite dimensional world?
The purpose of this talk is to explain that mean dimension seems to play an important role in such a research direction.
In particular, we explain the following principle:
If Zk acts on a space X with some hyperbolicity,
then we can control the mean dimension of the sub-action of any rank (k-1) subgroup G of Zk.
This talk is based on the joint works with Tom Meyerovitch and Mao Shinoda.
October 15 -- Room 056, 17:00 -- 18:30
Gwénaël Massuyeau (Université de Bourgogne)
Generalized Dehn twists on surfaces and surgeries in 3-manifolds
Abstract: (Joint work with Yusuke Kuno.)
Given an oriented surface S and a simple closed curve C in S,
the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist.
If the curve C is not simple, this transformation of S does not make sense anymore,
but one can consider two possible generalizations:
one possibility is to use the homotopy intersection form of S to
"simulate" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S;
another possibility is to view C as a curve on the top boundary of the cylinder S×[0,1],
to push it arbitrarily into the interior so as to obtain,
by surgery along the resulting knot, a new 3-manifold.
In this talk, we will relate two those possible generalizations of a Dehn twist
and we will give explicit formulas using a "symplectic expansion" of the fundamental group of S.
October 29 -- Room 056, 17:00 -- 18:30
Chung-Jun Tsai (National Taiwan University)
Strong stability of minimal submanifolds
Abstract: It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function.
One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario.
Besides a strong local uniqueness property,
a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow.
We will also discuss some famous local (complete, non-compact) models.
This is based on a joint work with Mu-Tao Wang.
November 5 -- Room 056, 17:00 -- 18:30
Kiyonori Gomi (Tokyo Institute of Technology)
Magnitude homology of geodesic space
Abstract: Magnitude is an invariant which counts `effective number of points' on a metric space.
Its categorification is magnitude homology.
This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton,
and then for any metric spaces by Leinster and Shulman.
The definition of the magnitude homology is easy, but its calculation is rather difficult.
For example, the magnitude homology of the circle with geodesic metric was known partially.
In my talk,
I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption.
In this result, the magnitude homology is described in terms of geodesics.
Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.
November 19 -- Room 056, 17:00 -- 18:30
Ramón Barral Lijó (Ritsumeikan University)
The smooth Gromov space and the realization problem
Abstract: The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds.
In this talk we will present the definition and basic properties of this space as well as two different applications:
The first addresses the following classical problem in foliation theory.
Realization problem: what kind of manifolds can be leaves of compact foliations?
Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.
Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor.
Moreover, we can assume that X has trivial holonomy and is transversely Cantor.
The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.
November 26 -- Room 056, 17:00 -- 18:30
Marco De Renzi (Waseda University)
2+1-TQFTs from non-semisimple modular categories
Abstract: Non-semisimple constructions have substantially generalized the standard approach of Witten,
Reshetikhin, and Turaev to quantum topology, producing powerful invariants and TQFTs with unprecedented properties.
We will explain how to use the theory of modified traces
to renormalize Lyubashenko’s closed 3-manifold invariants coming from
finite twist non-degenerate unimodular ribbon categories.
Under the additional assumption of factorizability, our renormalized invariants extend to 2+1-TQFTs,
unlike Lyubashenko’s original ones.
This general framework encompasses important examples of non-semisimple modular categories
which were left out of previous non-semisimple TQFT constructions.
Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.
December 3 -- Room 056, 17:00 -- 18:30
Anton Zeitlin (Louisiana State University)
Homotopy Gerstenhaber algebras, Courant algebroids, and Field Equations
Abstract: I will talk about the underlying homotopical structures within field equations,
which emerge in string theory as conformal invariance conditions for sigma models.
I will show how these, often hidden, structures emerge from the homotopy Gerstenhaber algebra associated to vertex and Courant algebroids,
thus making all such equations the natural objects within vertex algebra theory.
December 10 -- Room 056, 17:00 -- 18:30
Takeyoshi Kogiso (Josai University)
q-Deformation of a continued fraction and its applications
Abstract: A kind of q-Deformation of continued fractions was introduced by Morier-Genoud and Ovsienko.
The most important application of this q-deformation of regular (or negative)
continued fraction expansion of rational number r/s is to calculate the Jones polynomial of the rational link of r/s.
Moreover we can apply the q-deformation of this continued fraction to quadratic irrational number theory and combinatorics.
On the other hand, there exist another recipes for determing the Jones polynomials by using Lee-Schiffler's snake graph and by using Kogiso-Wakui's Conway-Coxeter frieze method.
Therefore, another approach of the result due to Morier-Genoud and Ovsienko can be considered from the viewpoint of a Conway-Coxeter frieze and a snake graph.
Furthermore, we can consider a cluster-variable transformation of continued fractions as a further generalization by using ancestoral triangles used in the Kogiso-Wakui.
December 17 -- Room 056, 17:00 -- 18:30
Kei Irie (The University of Tokyo)
Symplectic homology of fiberwise convex sets and homology of loop spaces
Abstract: For any (compact) subset in the symplectic vector space,
one can define its symplectic capacity by using symplectic homology,
which is a version of Floer homology.
In general, it is very difficult to compute or estimate this capacity directly from its definition,
since the definition of Floer homology involves counting solutions of nonlinear PDEs (so called Floer equations).
In this talk, we consider the symplectic vector space as the cotangent bundle of the Euclidean space,
and show a formula which computes symplectic homology and capacity of fiberwise convex sets from homology of loop spaces.
We also explain two applications of this formula.
January 7 -- Room 056
17:00 -- 18:00
Yasuhiko Asao (The University of Tokyo)
Magnitude homology of crushable spaces
Abstract: The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al.
They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of
"persistence of points clouds".
However, little property of them has been revealed.
In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy.
We use techniques from singular homology theory.
18:00 -- 19:00
Tomohiro Asano (The University of Tokyo)
Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory
Abstract: The microlocal sheaf theory due to Kashiwara-Schapira has been applied to symplectic geometry for these ten years.
Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds.
In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory.
This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory.
This is joint work with Yuichi Ike.
January 14 -- Room 056
17:00 -- 18:00
Ryohei Chihara (The University of Tokyo)
SO(3)-invariant G2-geometry
Abstract: Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G2 and Spin(7).
Many authors have studied G2- and Spin(7)-manifolds with torus symmetry.
In this talk, we generalize the celebrated examples due to Bryant and Salamon and study G2-manifolds with SO(3)-symmetry.
Such torsion-free G2-structures are described as a dynamical system of SU(3)-structures on an SO(3)-fibration over a 3-manifold.
As a main result, we reduce this system into a constrained Hamiltonian dynamical system on the cotangent bundle over the space of all Riemannian metrics on the 3-manifold.
The Hamiltonian function is very similar to that of the Hamiltonian formulation of general relativity.
18:00 -- 19:00
Tsukasa Ishibashi (The University of Tokyo)
Algebraic entropy of sign-stable mutation loops
Abstract: Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics,
unifying several combinatorial operations as well as their positivity notions.
A mutation loop induces several dynamical systems via cluster transformations,
and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.
We introduce a new property of mutation loops called the sign stability,
with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation.
A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor",
in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface.
We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop,
and conclude that these two coincide with the logarithm of the cluster stretch factor.
This talk is based on a joint work with Shunsuke Kano.
January 28 -- Room 056
17:00 -- 18:00
Nozomu Sekino (The University of Tokyo)
Existence problems for fibered links
Abstract: It is known that every connected orientable closed 3-manifold has a fibered knot. However,
finding (and classifying) fibered links whose fiber surfaces are fixed homeomorphism type in a given 3-manifold is difficult in general.
We give a criterion of a simple closed curve on a genus 2g Heegaard surface being a genus g fibered knot in terms of its Heegaard diagram.
As an application, we can prove the non-existence of genus one fibered knots in some Seifert manifolds.
There is one generalization of fibered links, homologically fibered links.
This requests that the complement of the "fiber surface" is a homologically product of a surface and an interval.
We give a necessary and sufficient condition for a connected sums of lens spaces of having a homologically fibered link whose fiber surfaces are some fixed types as some algebraic equations.
18:00 -- 19:00
Jun Watanabe (The University of Tokyo)
Fibred cusp b-pseudodifferential operators and its applications
Abstract: Melrose's b-calculus and its variants are important tools to study index problems on manifolds with singularities.
In this talk, we introduce a new variant "fibred cusp b-calculus",
which isa generalization of fibred cusp calculus of Mazzeo-Melrose and b-calculus of Melrose.
We discuss the basic property of this calculus and give a relative index formula.
As its application, we prove the index theorem for a Z/k manifold with boundary,
which is a generalization of the mod k index theorem of Freed-Melrose.