In this paper, we obtain local energy decay estimates and $L_p$-$L_q$ estimates of the solutions to the Stokes equations with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equations. Comparing with the non-slip boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null force at the boundary.

Let $(X,o)$ be a normal complex surface singularity. We define an invariant $p_e(X,o)$ for $(X,o)$ in terms of pencils of compact complex curves. Similarly, for a pair of $(X,o)$ and $h\in {𝔪}_{X,o}$ (the maximal ideal of ${𝒪}_{X,o}$), we define an invariant $p_e(X,o,h)$. We call $p_e(X,o)$ (resp. $p_e(X,o,h)$) the pencil genus of $(X,o)$ (resp. a pair of $(X,o)$ and $h$). In this paper, we give a method to construct pencils of compact complex curves by gluing a resolution space of $(X,o)$ and resolution spaces of some cyclic quotient singularities. Using this, we prove some formulae on $p_e(X,o,h)$ and estimate $p_e(X,o)$. We also characterize Kodaira singularities in terms of $p_e(X,o,h)$.

We introduce an equivalence of plane curve germs which is weaker than Zariski's equisingularity and prove that the set of all Newton diagrams of a germ is an invariant of this equivalence. Then we show how to construct all Newton diagrams of a plane many-branched singularity starting with some invariants of branches and their orders of contact.

We study a stable suspension order of a universal phantom map out of a space. We prove that it is infinite if $X$ is a non-trivial finite Postnikov space, a classifying space of connected Lie group or a loop space on a connected Lie group with torsion. We also show that the loop spaces on the exceptional Lie groups $E_6$ and $E_7$ are stably indecomposable.

Two-by-two matrix functions, which are the lifts of the local solutions of the matrix hypergeometric differential equation of $SL$ type at $0,1,\mathrm{\infty }$ to the upper half plane by the lambda function, are introduced. Each component of these matrix functions is represented by a definite integral with a power product of theta functions as integrand, which we call in this paper Wirtinger integral. Transformations of the matrix functions under some modular transformations are established by exploiting classical formulas of theta functions. These are regarded as formulas of monodromy or connection of the hypergeometric function of Gauss.

We investigate the nonstationary Navier-Stokes equations for an exterior domain $\mathrm{\Omega }\subset {\mathbf{R}}^3$ in a solution class $L^s(0,T;L^q(\mathrm{\Omega }\left)\right)$ of very low regularity in space and time, satisfying Serrin's condition $\frac{2}{s}+\frac{3}{q}=1$ but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by $u{|}_{\partial \mathrm{\Omega }}=g\in L^s(0,T;W^{-1/q,q}(\partial \mathrm{\Omega }\left)\right)$, and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition $k=÷u\in L^s(0,T;L^r(\mathrm{\Omega }\left)\right)$, where $\frac{1}{3}+\frac{1}{q}=\frac{1}{r}$.

We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras $𝔤$ to the fixed point subalgebras ${𝔤}^{\sigma }$ of outer automorphisms $\sigma$. The representations have highest weights which are scalar multiples of fundamental weights or linear combinations of two scalar ones. Our list of pairs of Lie algebras $(𝔤,{𝔤}^{\sigma })$ includes an exceptional symmetric pair $(E_6,F_4)$ and also a non-symmetric pair $(D_4,G_2)$ as well as a number of classical symmetric pairs. Some of the branching rules were known and others are new, but all the rules in this paper are proved by a unified method. Our key lemma is a characterization of the ``middle'' cosets of the Weyl group of $𝔤$ in terms of the subalgebras ${𝔤}^{\sigma }$ on one hand, and the length function on the other hand.

We prove that the sets of homotopy minimal periods for expanding maps on $n$-dimensional infra-nilmanifolds are uniformly cofinite,i.e., there exists a positive integer $m_0$, which depends only on $n$, such that for any integer $m\ge m_0$, for any $n$-dimensional infra-nilmanifold $M$, and for any expanding map $f$ on $M$, any self-map on $M$ homotopic to $f$ has a periodic point of least period $m$, namely, $[m_0,\mathrm{\infty })\subset \mathrm{H}\mathrm{P}\mathrm{e}\mathrm{r}\left(f\right)$. This extends the main result, Theorem 4.6, of [13] from periods to homotopy periods.

Simple homomorphisms to elliptic modular forms are defined on the ring of Siegel modular forms and linear relations on the Fourier coefficients of Siegel modular forms are implied by the codomains of these homomorphisms. We use the linear relations provided by these homomorphisms to compute the Siegel cusp forms of degree $n$ and weight $k$ in some new cases: $(n,k)=(4,14)$, $(4,16)$, $(5,8)$, $(5,10)$, $(6,8)$. We also compute enough Fourier coefficients using this method to determine the Hecke eigenforms in the nontrivial cases. We also put the open question of whether our technique always succeeds in a precise form. As a partial converse we prove that the Fourier series of Siegel modular forms are characterized among all formal series by the codomain spaces of these homomorphisms and a certain boundedness condition.

We define new $L^2$-invariants which we call secondary Novikov-Shubin invariants.We calculate the first secondary Novikov-Shubin invariants of finitely generated groups by using random walk on Cayley graphs and see in particular that these are invariant under quasi-isometry.

We find new generalizations of Dirichlet's theorem to real quadratic fields and quartic fields with two complex places, which are closely related to geometry of products of hyperbolic spaces.

After Gálvez, Martínez and Milán discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic $3$-space $H^3$, the first, third and fourth authors here gave a framework for complete flat fronts with singularities in $H^3$. In the present work we broaden the notion of completeness to weak completeness, and of front to p-front. As a front is a p-front and completeness implies weak completeness, the new framework and results here apply to a more general class of flat surfaces.

This more general class contains the caustics of flat fronts --- shown also to be flat by Roitman (who gave a holomorphic representation formula for them) --- which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only p-fronts. Using the new framework, we obtain characterizations for caustics.