## Tuesday Seminar on Topology

Seminar information archive ～09/24｜Next seminar｜Future seminars 09/25～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

**Seminar information archive**

### 2020/01/28

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Existence problems for fibered links (JAPANESE)

**Nozomu Sekino**(The University of Tokyo)Existence problems for fibered links (JAPANESE)

[ 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.

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.

### 2020/01/28

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fibred cusp b-pseudodifferential operators and its applications (JAPANESE)

**Jun Watanabe**(The University of Tokyo)Fibred cusp b-pseudodifferential operators and its applications (JAPANESE)

[ 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 is a 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.

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 is a 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.

### 2020/01/14

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

SO(3)-invariant G

**Ryohei Chihara**(The University of Tokyo)SO(3)-invariant G

_{2}-geometry (JAPANESE)
[ Abstract ]

Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G

Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G

_{2}and Spin(7). Many authors have studied G_{2}- and Spin(7)-manifolds with torus symmetry. In this talk, we generalize the celebrated examples due to Bryant and Salamon and study G_{2}-manifolds with SO(3)-symmetry. Such torsion-free G_{2}-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.### 2020/01/14

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Algebraic entropy of sign-stable mutation loops (JAPANESE)

**Tsukasa Ishibashi**(The University of Tokyo)Algebraic entropy of sign-stable mutation loops (JAPANESE)

[ 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.

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.

### 2020/01/07

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Magnitude homology of crushable spaces (JAPANESE)

**Yasuhiko Asao**(The University of Tokyo)Magnitude homology of crushable spaces (JAPANESE)

[ 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.

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.

### 2020/01/07

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)

**Tomohiro Asano**(The University of Tokyo)Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)

[ Abstract ]

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.

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.

### 2019/12/17

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Symplectic homology of fiberwise convex sets and homology of loop spaces (JAPANESE)

**Kei Irie**(The University of Tokyo)Symplectic homology of fiberwise convex sets and homology of loop spaces (JAPANESE)

[ 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.

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.

### 2019/12/10

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

q-Deformation of a continued fraction and its applications (JAPANESE)

**Takeyoshi Kogiso**(Josai University)q-Deformation of a continued fraction and its applications (JAPANESE)

[ 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.

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.

### 2019/12/03

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Homotopy Gerstenhaber algebras, Courant algebroids, and Field Equations (ENGLISH)

**Anton Zeitlin**(Louisiana State University)Homotopy Gerstenhaber algebras, Courant algebroids, and Field Equations (ENGLISH)

[ 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.

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.

### 2019/11/26

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

$2+1$-TQFTs from non-semisimple modular categories (ENGLISH)

**Marco De Renzi**(Waseda University)$2+1$-TQFTs from non-semisimple modular categories (ENGLISH)

[ 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

Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.

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.

### 2019/11/19

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The smooth Gromov space and the realization problem (ENGLISH)

**Ramón Barral Lijó**(Ritsumeikan University)The smooth Gromov space and the realization problem (ENGLISH)

[ 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.

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.

### 2019/11/05

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Magnitude homology of geodesic space (JAPANESE)

**Kiyonori Gomi**(Tokyo Institute of Technology)Magnitude homology of geodesic space (JAPANESE)

[ 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.

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.

### 2019/10/29

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Strong stability of minimal submanifolds (ENGLISH)

**Chung-Jun Tsai**(National Taiwan University)Strong stability of minimal submanifolds (ENGLISH)

[ 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.

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.

### 2019/10/15

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Generalized Dehn twists on surfaces and surgeries in 3-manifolds (ENGLISH)

**Gwénaël Massuyeau**(Université de Bourgogne)Generalized Dehn twists on surfaces and surgeries in 3-manifolds (ENGLISH)

[ 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.

(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.

### 2019/10/08

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

How can we generalize hyperbolic dynamics to group actions? (JAPANESE)

**Masaki Tsukamoto**(Kyushu University)How can we generalize hyperbolic dynamics to group actions? (JAPANESE)

[ 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 $Z^2$ 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 $Z^2$-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 $Z^k$ 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 $Z^k$.

This talk is based on the joint works with Tom Meyerovitch and Mao Shinoda.

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 $Z^2$ 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 $Z^2$-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 $Z^k$ 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 $Z^k$.

This talk is based on the joint works with Tom Meyerovitch and Mao Shinoda.

### 2019/10/01

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Quantized SL(2) representations of knot groups (JAPANESE)

**Jun Murakami**(Waseda University)Quantized SL(2) representations of knot groups (JAPANESE)

[ 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.

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.

### 2019/07/16

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Seifert vs. slice genera of knots in twist families and a characterization of braid axes (JAPANESE)

**Kimihiko Motegi**(Nihon University)Seifert vs. slice genera of knots in twist families and a characterization of braid axes (JAPANESE)

[ Abstract ]

Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n)$ limits to $1$, then the winding number of $K$ about $c$ equals either zero or the wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family $\{\overline{K_n}\}$ contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that $c$ is a braid axis of $K$ if and only if both $\{K_n\}$ and $\{\overline{K_n}\}$ each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for $\{K_n\}$ to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore.

Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n)$ limits to $1$, then the winding number of $K$ about $c$ equals either zero or the wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family $\{\overline{K_n}\}$ contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that $c$ is a braid axis of $K$ if and only if both $\{K_n\}$ and $\{\overline{K_n}\}$ each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for $\{K_n\}$ to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore.

### 2019/07/09

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Mod 2 cohomology of moduli stacks of real vector bundles (ENGLISH)

**Florent Schaffhauser**(Université de Strasbourg)Mod 2 cohomology of moduli stacks of real vector bundles (ENGLISH)

[ Abstract ]

The rational cohomology ring of the moduli stack of holomorphic vector bundles of fixed rank and degree over a compact Riemann surface was studied by Atiyah and Bott using tools of differential geometry and algebraic topology: they found generators of that ring and computed its Poincaré series. In joint work with Chiu-Chu Melissa Liu, we study in a similar way the mod 2 cohomology ring of the moduli stack of real vector bundles of fixed topological type over a compact Riemann surface with real structure. The goal of the talk is to explain the principle of that computation, emphasizing the analogies and differences between the real and complex cases, and discuss applications of the method. In particular, we provide explicit generators of mod 2 cohomology rings of moduli stacks of vector bundles over a real algebraic curve.

The rational cohomology ring of the moduli stack of holomorphic vector bundles of fixed rank and degree over a compact Riemann surface was studied by Atiyah and Bott using tools of differential geometry and algebraic topology: they found generators of that ring and computed its Poincaré series. In joint work with Chiu-Chu Melissa Liu, we study in a similar way the mod 2 cohomology ring of the moduli stack of real vector bundles of fixed topological type over a compact Riemann surface with real structure. The goal of the talk is to explain the principle of that computation, emphasizing the analogies and differences between the real and complex cases, and discuss applications of the method. In particular, we provide explicit generators of mod 2 cohomology rings of moduli stacks of vector bundles over a real algebraic curve.

### 2019/07/02

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Brane coproducts and their applications (JAPANESE)

**Shun Wakatsuki**(The University of Tokyo)Brane coproducts and their applications (JAPANESE)

[ Abstract ]

The loop coproduct is a coproduct on the homology of the free loop space of a Poincaré duality space (or more generally a Gorenstein space). In this talk, I will introduce two kinds of brane coproducts which are generalizations of the loop coproduct to the homology of a sphere space (i.e. the mapping space from a sphere). Their constructions are based on the finiteness of the dimensions of mapping spaces in some sense. As an application, I will show the vanishing of some cup products on sphere spaces by comparing these two brane coproducts. This gives a generalization of a result of Menichi for the case of free loop spaces.

The loop coproduct is a coproduct on the homology of the free loop space of a Poincaré duality space (or more generally a Gorenstein space). In this talk, I will introduce two kinds of brane coproducts which are generalizations of the loop coproduct to the homology of a sphere space (i.e. the mapping space from a sphere). Their constructions are based on the finiteness of the dimensions of mapping spaces in some sense. As an application, I will show the vanishing of some cup products on sphere spaces by comparing these two brane coproducts. This gives a generalization of a result of Menichi for the case of free loop spaces.

### 2019/06/25

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Geometry of symplectic log Calabi-Yau surfaces (ENGLISH)

**Tian-Jun Li**(University of Minnesota)Geometry of symplectic log Calabi-Yau surfaces (ENGLISH)

[ Abstract ]

This is a survey on the geometry of symplectic log Calabi-Yau surfaces, which are the symplectic analogues of Looijenga pairs. We address the classification up to symplectic deformation, the relations between symplectic circular sequences and anti-canonical sequences, contact trichotomy, and symplectic fillings. This is a joint work with Cheuk Yu Mak.

This is a survey on the geometry of symplectic log Calabi-Yau surfaces, which are the symplectic analogues of Looijenga pairs. We address the classification up to symplectic deformation, the relations between symplectic circular sequences and anti-canonical sequences, contact trichotomy, and symplectic fillings. This is a joint work with Cheuk Yu Mak.

### 2019/06/18

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Filtered instanton homology and the homology cobordism group (JAPANESE)

**Masaki Taniguchi**(The University of Tokyo)Filtered instanton homology and the homology cobordism group (JAPANESE)

[ Abstract ]

We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.

We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.

### 2019/06/04

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Gluck twist on branched twist spins (JAPANESE)

**Mizuki Fukuda**(Tokyo Gakugei University)Gluck twist on branched twist spins (JAPANESE)

[ Abstract ]

A branched twist spin is an embedded two sphere in the four sphere and it is defined as the set of singular points of a circle action on the four sphere. Gluck showed that the set of isotopy classes of diffeomorphisms on $S^1 \times S^2$ is isomorphic to $Z_2$, and an operation of removing a neighborhood of 2-knot from the four sphere and regluing it by the generator of $Z_2$ is called a Gluck twist. It is known by Pao that the Gluck twist along a branched twist spin does not change the four sphere. In this talk, we give an another proof of Pao’s result by using a decomposition of $S^4$ associated with the circle action, and we show that the set of branched twist spins does not change by the Gluck twist.

A branched twist spin is an embedded two sphere in the four sphere and it is defined as the set of singular points of a circle action on the four sphere. Gluck showed that the set of isotopy classes of diffeomorphisms on $S^1 \times S^2$ is isomorphic to $Z_2$, and an operation of removing a neighborhood of 2-knot from the four sphere and regluing it by the generator of $Z_2$ is called a Gluck twist. It is known by Pao that the Gluck twist along a branched twist spin does not change the four sphere. In this talk, we give an another proof of Pao’s result by using a decomposition of $S^4$ associated with the circle action, and we show that the set of branched twist spins does not change by the Gluck twist.

### 2019/05/28

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Exotic four-manifolds via positive factorizations (ENGLISH)

**R. Inanc Baykur**(University of Massachusetts)Exotic four-manifolds via positive factorizations (ENGLISH)

[ Abstract ]

We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.

We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.

### 2019/05/21

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the dealternating number and the alternation number (ENGLISH)

**Maria de los Angeles Guevara**(Osaka City University)On the dealternating number and the alternation number (ENGLISH)

[ Abstract ]

Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.

Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.

### 2019/05/14

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Diagrammatic Algebra (ENGLISH)

**J. Scott Carter**(University of South Alabama, Osaka City University)Diagrammatic Algebra (ENGLISH)

[ Abstract ]

Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.

Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.