2019年度年会(於:東京工業大学)

SUMMARY: Topological insulators are certain quantum states of matters which are experimentally observed as materials with insulating bulk and metallic edge. Topological K-theory is applied in Kitaev’s theoretical classification of topological insulators. This idea is generalized to the classification of topological crystalline insulators, and prompted calculations of twisted equivariant K-theory. An outcome of such calculations in my collaborating work with Ken Shiozaki and Masatoshi Sato suggests a torsion phase of topological insulators protected by glide symmetry. As an introduction to topological insulators for topologists, my talk will be an exposition of these K-theoretic classifications. If time permits, I would like to mention recent applications of equivariant generalized (co)homology theory to the classification of crystalline symmetry protected phases.

SUMMARY: Fiber structures such as open books and Lefschetz fibrations have played an important role in the study of the topology of contact and symplectic manifolds. In particular, since combinatorial techniques of mapping class groups of surfaces are well-developed, fiber structures enable us to construct various contact and symplectic manifolds. In contrast, little is known about mapping class groups of higher-dimensional manifolds, and hence this yields a big gap between low and higher dimensions. In the first part of this talk, we survey previous results on the topology of contact and symplectic manifolds from the point of view of fiber structures. In the second part, we discuss how to examine higher dimensions. We especially focus on symplectic manifolds which cannot admit Lefschetz fibrations but admit Lefschetz–Bott fibrations.

SUMMARY: In this talk, as a fundamental and important problem on the theory of Morse functions and their higher dimensional versions including fold maps, which are simplest generalizations of Morse functions, and application to differential topology of manifolds, we introduce history, known results, and new results on manifolds admitting explicit fold maps.

SUMMARY: In geometric theory of Morse functions, fold maps, which are higher dimensional versions of Morse functions, and more general good smooth maps and its application to geometry of manifolds, Reeb spaces between smooth maps are fundamental and important. They are defined as the spaces of all connected components of inverse images and in cosiderable cases, they are polyhedra of dimensions equal to those of the target spaces and have much information of homology groups etc.. Related to this branch, the top dimensional homology groups of Reeb spaces are shown to be non-trivial if there exists an inverse image of a regular value containing a manifold being not null-cobordant in 2012–14 by Hiratuka and Saeki. In this talk, an extension of this fact and explicit application will be presented.

SUMMARY: Let \((X^n, 0)\) be the germ at \(0 \in \mathbb {R}^n\) of the pair \((\mathbb {R}^n, \mathbb {R}^{n -1}\times \{0\})\). We shall use \((x_1, \ldots , x_{n -1}, y)\) coordinates on \(\mathbb {R}^n\). The boundary and interior of our manifold with boundary correspond to the part of \(y >0\) and \(y >0\) respectively. In this talk, we introduce the notions topologically \(\mathcal {B}\)-equivalent and topologically \(\mathcal {B}_+\)-equivalent among smooth map germs \((X^n, 0) \to (\mathbb {R}^2, 0)\). Then, we show that there are three invariants of topologically \(\mathcal {B}_+\)-equivalent (or three invariants of topologically \(\mathcal {B}\)-equivalent) among smooth map germs \((X^3, 0) \to (\mathbb {R}^2, 0)\).

SUMMARY: In this talk, we will give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney \(C^\infty \)-topology). We introduce the notion of end-triviality of smooth mappings, which concerns behavior of mappings around the ends of the source manifolds, and show that a Morse function is stable if it is end-trivial at any point in its discriminant. We further show that a Morse function \(f:N\to \mathbb {R}\) is strongly stable (i.e. there exists a continuous mapping \(g\mapsto (\Phi _g,\phi _g)\in \mathrm {Diff}(N)\times \mathrm {Diff}(\mathbb {R})\) such that \(\phi _g\circ g\circ \Phi _g =f\) for any \(g\) close to \(f\)) if (and only if) \(f\) is quasi-proper.

SUMMARY: The apparent contour of a smooth surface is considered as the set of the singularities of a projection mapping of the surface, and we can investigate it in terms of singularity theory. What information of a surface can we get from the apparent contour, especially when the apparent contour is singular? In this talk, we give some answers to the question considering the contact types of surfaces with smooth cylindrical surfaces.

SUMMARY: There are many studies about curves, surfaces and curves, surfaces with singular points by using height functions and distance squared functions. Height functions represents the contact with the plane, and distance squared functions do with the round sphere. These method works well for surfaces with singular points, for such cases, studying contact with singular objects also must work well. In this talk, we present an attempt studying contacts of singular surfaces with the standard cuspidal edge.

SUMMARY: Cuspidal edges are fundamental singularities of wave fronts. A surface with cuspidal edge singularities has a well-defined unit normal vector or the Gauss map even at cuspidal edges. Thus we can consider the extended distance squared function and the extended height function on cuspidal edges. In this talk, we give conditions that \(D_4\) singuarities appear on the extended distance squared and the extended height functions on cuspidal edges by geometric invariants of cuspidal edges. Moreover, we show relations among singularities of parallel surfaces, singularities of Gauss maps and conditions of \(D_4\) singularities appearing on such surfaces.

SUMMARY: We define the associated simplicial bundle of homotopy algebra models of fibers for a fiber bundle. The obstruction theory of simplicial bundles yields the obstruction class of the simplicial bundle. A characteristic class of a fiber bundle is obtained for each invariant anti-symmetric form on the homotopy group of a fiber of the simplicial bundle. For a surface bundle, this class is equivalent to twisted Morita–Mumford classes for the bundle. Thus, the characteristic classes include Morita–Mumford classes of a surface bundle. For another example, we can also obtain the Euler class of a specific sphere bundle by direct calculus.

SUMMARY: We generalize the Gysin formulas for flag bundles in the ordinary cohomology theory, which are due to Darondeau–Pragacz, to the complex cobordism theory.

SUMMARY: We will give two generalizations of the loop coproduct to the mapping spaces \(Map(S^k, M)\) from \(k\)-dimensional spheres. One generalization is defined when \(M\) is a \(k\)-connected space with finite dimensional rational homotopy group. This is based on the finiteness of the dimension of the \((k-1)\)-fold based loop space of \(M\) as a Gorenstein space. The other generalization is defined when \(M\) is a 1-connected Poincaré duality space. It is different from the previous generalization and is easier to compute. Moreover, it has an application to the TNCZ problem of fibrations, which is originally due to Menichi in the case of the loop coproduct.

SUMMARY: When we consider the square lattice \(\mathbb {Z}^2\) and half-planes of it distinguished by lines through the origin, \(y=\alpha x\) and \(y=\beta x\), a quarter plane appears as an intersection of two half-planes. Associated with this quarter-plane, we can consider quarter-plane Toeplitz operators. Its index theory is applied to the study of topologically protected corner states, which are also studied in condensed matter physics under the name of higher-order topological insulators. In this talk, we consider Toeplitz operators, associated with the sum of two half-planes and study its index theory. We further discuss its implications for the study of topologically protected corner states.

SUMMARY: The speaker constructed a pair of symplectic structures on the product of two copies of the space of univariate normal distributions and explained the relation between the relevant information geometry and Bayesian learning. We generalize this to the multivariate case by constructing a pair of Poisson structures on the similar product.

SUMMARY: It is known that for the natural algebraic torus actions on the Grassmannians, the closures of torus orbits are toric varieties, and that these toric varieties are smooth if and only if the corresponding matroid polytopes are simple. We shall report that simple matroid polytopes are products of simplices and smooth torus orbit closures in the Grassmannians are products of complex projective spaces. Moreover, it turns out that the smooth torus orbit closures are uniquely determined by the corresponding simple matroid polytopes.

SUMMARY: We carry out theoretical analysis of fluid dynamics on surfaces assuming the existence of a no-normal and nontrivial Killing vector field. To this end, we derive an exact solution of the hydrodynamic Green’s function using the homogeneous foliation generated by the Killing vector field. After that we discuss physical properties of the Killing vector field and potential vector field as a steady solution of the Euler equations.

SUMMARY: For a symmetrizable Kac–Moody Lie algebra \(\mathfrak {g}\), we construct a family of weighted quivers \(Q_m(\mathfrak {g})\) and realize the Weyl group as a subgroup of the corresponding cluster modular group. This is a generalization of the construction for type \(A_n\) and \(\tilde {A_n}\) given by Inoue–Lam–Pylyavskyy. Moreover if \(\mathfrak {g}\) is of finite type and \(m\) is the Coxeter number, then our quiver encodes the cluster structure of the moduli space of decorated \(G\)-local systems on the punctured disk with two marked points on its boundary. Using this we define an action of the Weyl group on the moduli space, and show that this action coincides with the one given by Goncharov–Shen.

SUMMARY: We discuss indices of power subgroups in the mapping class group of a punctured sphere and in the hyperelliptic mapping class group of an oriented closed surface. The main tool we use is a projective representation of the mapping class group obtained through the linear skein theory. We show that power subgroups of symmetric Dehn twists have infinite indices in hyperelliptic mapping class group in many cases. Our works are the study of “the remaining case” of Masbaum’s work and a generalization of Stylianakis’ work.

SUMMARY: As is well-known, by the Birman–Hilden theory, the mapping class group of a sphere with \(p\) marked points is virtually embedded in the mapping class group of a closed surface of genus \(g\) if \(p\) is not greater than \(2g+2\). Here, we say that a group \(H\) is virtually embedded in a group \(G\) if \(H\) has a finite index subgroup which can be embedded in \(G\). In this talk, using the curve graphs of surfaces and right-angled Artin groups in the mapping class groups, we prove that no finite index subgroup of the mapping class group of a sphere with at least \(2g+3\) marked points is embedded in the mapping class group of a closed surface of genus \(g\).

SUMMARY: A triangulation (resp., quadrangulation) on a surface \(\mathbb {F}\) is a map with each face bounded by a cycle of length 3 (resp., 4). It is known that every triangulation on \(\mathbb {F}\) has a quadrangulation as a spanning subgraph. For the spherical case, every quadrangulation is bipartite. On the other hand, there are both bipartite and nonbipartite ones on non-spherical surfaces. Kündgen and Thomassen asked whether the bipartiteness of a spanning quadrangulation of a given triangulation on \(\mathbb {F}\) can be controlled. We focus on the balancedness for spanning bipartite quadrangulations of a given triangulation when \(G\) has a spanning bipartite quadrangualation. In this talk we will give an evaluation for the balancedness. This is a joint work with Naoki Matsumoto (Seikei University).

SUMMARY: We study Voronoi tilings with general Archimedean spiral lattices related to phyllotactic patterns of typical plants such as sunflower. In this talk, we show that Voronoi tilings with the general Archimedean spiral lattices possesses annular patterns (the grain boundaries) which consist of heptagons, hexagons, and pentagons. Moreover, we show that the number of polygons in grain boundaries is represented by denominators of convergents of the divergence angle. Also, we consider the limit shapes of Voronoi polygons in grain boundaries. If the divergence angle is an irrational number, the limit shape is are rectangles. In particular, if the divergence angle is an irrational number which is equivalent to the golden section, the limit shape is the square.

SUMMARY: In this talk, we prove that if a rational homology 3-sphere X contains a homologically fibered knot of genus g, then X contains infinitely many such knots. The proof is based on the simplicial volume of 3-manifolds whose boundaries consist of tori. Combining this result with our previous work, we conclude that every lens space contains infinitely many homologically fibered knots of genus one.

SUMMARY: In this talk, we report on our results on separating incompressible surfaces and incompressible subsurfaces of Heegaard surfaces in a 3-manifold.

SUMMARY: We study the problem of whether or not a symmetry of a compact orientable 3-manifold given by an orientation-reversing periodic diffeomorphism can be reflected on a corresponding framed link, called a surgery description.

SUMMARY: Using a completed HOMFLY-PT skein algebra and Heegaard splittings, we construct the universal \(sl(n)\) quantum invariant for integral homology 3-spheres.

SUMMARY: Shimizu introduced a region crossing change unknotting operation for knot diagrams. As extensions, two integral region choice problems are proposed and the existences of solutions of the problems are shown for all non-trivial knot diagrams by Ahara and Suzuki, and Harada. We relate both integral region choice problems with an Alexander index for regions of a link diagram, and discuss the problems on link diagrams.

SUMMARY: We give a computation of finite type invariants of spherical curves by the aid of computers, and compare it with the dimension of Vassiliev invariants of knots.

SUMMARY: We determine the set of alternating knots with the crosscap number two.

SUMMARY: Goussarov, Polyak, and Viro conjectured that every finite type invariant of classical knots could be extended to a finite type invariant of long virtual knots (Goussarov–Polyak–Viro Conjecture). For the degree-three case of the conjecture, we give an answer with a new viewpoint by introducing a reduced Polyak algebra for classical knots.

SUMMARY: In this talk, in order to define an augmented Alexander matrix for a handlebody-link, we define an extension of a multiple conjugation quandle and introduce an MCQ Alexander pair which gives a linear extension of a multiple conjugation quandle.

SUMMARY: Given a shadow biquandle \((B,X)\) composed of a biquandle \(B\) and a strongly connected \(B\)-set \(X\), we have a local biquandle structure on \(X\). The (co)homology groups of such shadow biquandles are isomorphic to those of the corresponding local biquandles. Moreover, cocycle invariants, of oriented links and oriented surface-links, using such shadow biquandles coincide with those using the corresponding local biquandles. The results imply that for some cases, the Niebrzydowski’s theory related to knot-theoretic ternary quasigroup is the same as shadow biquandle theory.

SUMMARY: We define a Fox derivative for quandles and construct an invariant of quandles. In particular, by taking knot quandles, we obtain an invariant of pairs of knots and quandle representations.

SUMMARY: A \(G\)-cork (reps. a weakly equivariant \(G\)-cork) is a pair \((C,G)\) of a compact contractible 4-manifold \(C\) and a subgroup \(G\) of the diffeomorphism group (resp. the mapping class group) of \(\partial C\) such that any nontrivial element of \(G\) does not extend over \(C\). In this talk, we will explain how to construct a \(G\)-cork for any extension \(G\) of \(\mathbb Z^m\) by a finite subgroup of \(\mathrm {SO}(4)\) and a weakly equivariant \(G\)-cork for any extension \(G\) of \(\mathbb Z^m\) by a finite solvable group.

SUMMARY: In this talk, we compute the twisted Alexander polynomials of tunnel number one Montesinos knots associated to their \(SL_2(\mathbb {C})\) representations. We give presentations of the knot groups which has two generators and one relation. We will also give the holonomy representations.

SUMMARY: We study some properties on twisted Alexander polynomials of twist knots for non-abelian \({\rm SL}_2(\mathbb {C})\)-representaions and discuss some relations with certain Dehn surgeries.

SUMMARY: The twisted Alexander polynomial is defined as a rational function, not necessarily a polynomial. It is shown that for a ribbon 2-knot, the twisted Alexander polynomial associated to a irreducible representation of the knot group to SL(2, F) is always a polynomial. Also, if a ribbon 2-knot of 1-fusion is fibered then its twisted Alexander is monic.

SUMMARY: A \(2\)-knot is a surface in \(\mathbf {R^4}\) that is homeomorphic to \(\mathbf {S^2}\), the standard sphere in \(3\)-space. A ribbon \(2\)-knot is a \(2\)-knot obtained from \(m\) \(2\)-spheres in \(\mathbf {R^4}\) by connecting them with \(m-1\) annuli. Let \(\mathbf {K^2}\) be a ribbon \(2\)-knot. The ribbon crossing number, denoted by \(r-cr(K^2)\) is a numerical invariant of the ribbon 2-knot \(\mathbf {K^2}\). It is known that the degree of the Alexander polynomial of \(\mathbf {K^2}\) is less than or equal to \(r-cr(K^2)\). In this lecture, we show that \(r-cr(K^2)\) is evaluated by coefficients in the Alexander polynomial of \(\mathbf {K^2}\). Furthermore, applying this fact, for a classical knot \(\mathbf {k^1}\), we also evaluate the crossing number, denoted by \(cr(k^1)\).