To investigate a concentration of measure phenomena on metric measure spaces in terms of Gromov-Ledoux's expansion coefficients on this space as well as Ledoux's per se, we studied a concentration function in concert with their expansion coefficients. Further investigation into an exponential concentration in terms of Ledoux's expansion coefficient on a bounded and volume doubling metric measure space enables us to derive an upper bound for its diameter in terms of both the Ledoux's expansion coefficient and doubling constant, provided that Ledoux's expansion coefficient $> 1$. In this study, we let Ledoux's expansion coefficient $> 1$ on a metric measure space, which is ensured by adopting Poincaré inequality. We demonstrated that on a metric measure space, Gromov-Ledoux's expansion coefficients with Ledoux's expansion coefficient $> 1$ give rise to an exponential concentration in terms of themselves. We further showed that on a bounded and volume doubling metric measure space, a Ledoux's expansion coefficient of order bounded from above in terms of both the doubling constant $> 1$ and its diameter is bounded from above in terms of the doubling constant per se. We applied this upper diameter bound to a closed smooth Riemannian manifold with non-negative Ricci curvature. This upper bound is described in terms of both the spectral gap and dimension.

We consider some parameterized planar sets with unbounded digits. We investigate these sets by using the method of “transversality", which is the main tool in investigating self-similar sets with overlaps. We calculate the Hausdorff dimension of these sets for typical parameters in some region with respect to the 2-dimensional Lebesgue measure. In addition, we estimate the local dimension of the exceptional set of parameters.

The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter, Jelsovsky, Kamada, Langford and Saito in Transactions of the American Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant. In this article we show that the ribbon cocycle invariant is a quantum invariant. We do so by constructing a ribbon category from a TSD set whose twisting and braiding morphisms entail a given TSD $2$-cocycle. Then we show that the quantum invariant naturally associated to this braided category coincides with the cocycle invariant. We generalize this construction to symmetric monoidal categories and provide classes of examples obtained from Hopf monoids and Lie algebras. We further introduce examples from Hopf-Frobenius algebras, objects studied in quantum computing.

We will give a sequence of irreducible mutations converting Brauer star algebra to any two-restricted star-to-tree complex.

The purpose of this article is to show a simple method of finding $\mathbb Z_2$-Betti numbers of oriented Grassmannians. The actual calculation is performed by computing $\mathbb Z_2$-ranks of matrices defined in a combinatorial way. For the reduction of problem from topology to matrix rank, we will analyse Morse homology of certain functions on the oriented Grassmannians.

We obtain a combinatorial expression for the boundary map coefficients of real isotropic and odd orthogonal Grassmannians. It provides a natural generalization of the known formulas for Lagrangian and maximal isotropic Grassmannians. The results derive from the classification of Schubert cells into four types of covering pairs when identified with signed $k$-Grassmannian permutations. Our formulas show that the coefficients depend on the changed positions for each permutation pair type. We apply this to obtain an orientability criterion and compute the first and second homology groups for these Grassmannians. Furthermore, we exhibit an apparent symmetry of the boundary map coefficients.

Extending theorems of J. E. Greene [Invent. Math. \textbf{192} (2013), 717-750] and A. S. Lipson [Enseign. Math. (2) \textbf{36} (1990), 93-114], we prove that the equivalence class of a classical link $L$ under mutation is determined by Goeritz matrices associated to diagrams of $L$.

In this paper, we consider the mean curvature flow starting from closed submanifolds in rank one symmetric spaces satisfying some pinching condition for the norm of the second fundamental form. We prove that, under some additional condition, the closed submanifold satisfying the pinching condition collapses to a round point in finite time or converges to a totally geodesic submanifold in infinite time along the mean curvature flow.