We define a totally geodesic immersion of irreducible type from a symmetric space of compact type into a Grassmannian and classify such immersions. Any totally geodesic immersion is related to a homogeneous vector bundle with the canonical connection and the eigenspaces of the Laplace operator acting on the space of sections of the bundle.

We prove that for any element in the 𝛾-completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle, if its 𝛾-support is a smooth Lagrangian submanifold, then the element itself is a smooth Lagrangian. We also prove that if the 𝛾-support of an element in the completion is compact, then it is connected.

To each projective hypersurface which is not a cone, we associate an abelian linear algebraic group called the symmetrizer group of the corresponding symmetric form. This group describes the set of homogeneous polynomials with the same Jacobian ideal and gives a conceptual explanation of results by Ueda–Yoshinaga and Wang. In particular, the diagonalizable part of the symmetrizer group detects Sebastiani–Thom property of the hypersurface and its unipotent part is related to the singularity of the hypersurface.

Let (𝑢, 𝑣) be a solution to the Cauchy problem for a semilinear parabolic system

(P) \begin{cases} 𝜕_𝑡𝑢 = 𝐷₁ Δ𝑢 + 𝑣^𝑝 in ℝ^𝑁 × (0, 𝑇), 𝜕_𝑡𝑣 = 𝐷₂ Δ𝑣 + 𝑢^𝑞 in ℝ^𝑁 × (0, 𝑇), (𝑢(⋅,0), 𝑣(⋅,0)) = (𝜇, 𝜈) in ℝ^𝑁, \end{cases}

where 𝑁 ≥ 1, 𝑇 > 0, 𝐷₁ > 0, 𝐷₂ > 0, 0 < 𝑝 ≤ 𝑞 with 𝑝𝑞 > 1, and (𝜇, 𝜈) is a pair of nonnegative Radon measures or locally integrable nonnegative functions in ℝ^𝑁. In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem (P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces.

The quotient of a Hermitian symmetric space of non-compact type by a torsion-free cocompact arithmetic subgroup of the identity component of the group of isometries of the symmetric space is called an arithmetic fake compact Hermitian symmetric space if it has the same Betti numbers as the compact dual of the Hermitian symmetric space. Arithmetic fake compact Hermitian symmetric spaces of types other than 𝐴₁ and 𝐴₃ have been classified in our earlier work and the work of Cartwright–Steger. There are many known examples of type 𝐴₁ and there may not be a convenient way to describe them all, but there are already descriptions under some restrictions given in Shavel and Linowitz–Stover–Voight. In this article, we study the remaining type 𝐴₃. We show that there are possibly a handful of explicit candidates.

Recently, Maesaka, Seki and Watanabe discovered a surprising equality between multiple harmonic sums and certain Riemann sums which approximate the iterated integral expression of the multiple zeta values. In this paper, we describe the formula corresponding to the multiple zeta-star values and, more generally, to the Schur multiple zeta values of diagonally constant indices. We also discuss the relationship of these formulas with Hoffman's duality identity and an identity due to Kawashima.

We give a description of shape theory using finite topological 𝑇₀-spaces (finite partially ordered sets). This description serves as a first step towards developing computational methods in shape theory for future works. Additionally, we introduce the notion of core for inverse sequences of finite topological spaces and prove some properties.

We give a first example of a pair of 2-knots that share the same knot group but have different knot quandles. In fact, we give infinitely many triples of twist spins that share the same knot group but have mutually different knot quandles. To this end, we prove that the type of the knot quandle of an 𝑛-twist spin of a non-trivial knot is equal to 𝑛. By using the latter result, we also complete the classification of the twist spins with finite knot quandles by distinguishing the 2-knots in Inoue's list.

We prove that every knot type in ℝ³ can be parametrised by a smooth function 𝑓 : 𝑆¹ → ℝ³, 𝑓(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) such that all derivatives 𝑓^(𝑛)(𝑡) = ( 𝑥^(𝑛)(𝑡), 𝑦^(𝑛)(𝑡), 𝑧^(𝑛)(𝑡) ), 𝑛 ∈ ℕ, parametrise knots and every knot type appears in the corresponding sequence of knots. We also study knot types that arise as limits of such sequences.

We prove that a positive two-bridge knot other than the (2, 𝑘)-torus knot does not admit chirally cosmetic surgeries, a pair of Dehn surgeries along distinct slopes yielding orientation-reversingly homeomorphic 3-manifolds.

Spectral analysis is worked out on moment problems related to linear functionals of certain polynomial hypergroups.

In this paper, we make some progress about a boundary behavior of the almost-complete Kähler–Einstein metric of negative Ricci curvature on a quasi-projective manifold with semiample log-canonical bundle. First its volume growth near the boundary is investigated in terms of the Kodaira dimension of the boundary, and then we characterize the boundary to be of general type via the volume growth. Moreover the volume growth is determined in the case of a Calabi–Yau boundary. We also affirmatively solve a modified version of the conjecture suggested previously by the author about the residue of the Kähler–Einstein metric if the boundary is a smooth finite quotient of an abelian variety.

We investigate the combinatorial structure of the set of maximal antichains in a Boolean algebra ordered by almost refinement. We also consider the reaping relation and its associated cardinal invariants, focusing in particular on reduced powers of Boolean algebras. As an application, we obtain that, on the one hand, the ultrafilter number of the Cohen algebra is greater than or equal to the cofinality of the meagre ideal and, on the other hand, a suitable parametrized diamond principle implies that the ultrafilter number of the Cohen algebra is equal to ℵ₁.

We propose a combinatorial formula for the coproduct in a Hopf algebra of decorated multi-indices that recently appeared in the literature, which can be briefly described as the graded dual of the enveloping algebra of the free Novikov algebra generated by the set of decorations. Similarly to what happens for the Hopf algebra of rooted forests, the formula can be written in terms of admissible cuts. We also prove a combinatorial formula for the extraction-contraction coproduct for undecorated multi-indices, in terms of a suitable notion of covering subforest.

The notion of the Yau sequence was introduced by Tomaru, as an attempt to extend Yau's elliptic sequence for (weakly) elliptic singularities to normal surface singularities of higher fundamental genera. In this paper, we obtain the canonical cycle using the Yau cycle for certain surface singularities of degree two. Furthermore, we obtain a formula of arithmetic genera. We also give some properties about the classification of weighted dual graphs of certain surface singularities of degree two.