We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal upper bound for the total Gauss curvature of the surface which depends only on the orders of the ODEs and we show that the total Gauss curvature of the surface vanishes if the ODEs are second order. We examine when the surfaces are asymptotically minimal.

Let $F$ be a non-Archimedean local field, and let $G$ be an inner form of $\mathrm{GL}_N(F)$ with $N \ge 1$. Let $\boldsymbol{\mathrm{JL}}$ be the Jacquet--Langlands correspondence between $\mathrm{GL}_N(F)$ and $G$. In this paper, we compute the invariant $s$ associated with the essentially square-integrable representation $\boldsymbol{\mathrm{JL}}^{-1}(\rho)$ for a cuspidal representation $\rho$ of $G$ by using the recent results of Bushnell and Henniart, and we restate the second part of a theorem given by Deligne, Kazhdan, and Vignéras in terms of the invariant $s$. Moreover, by using the parametric degree, we present a proof of the first part of the theorem.

We prove that a projective surface of globally $F$-regular type defined over a field of characteristic zero is of Fano type.

The generalized Wintgen inequality was conjectured by De Smet, Dillen, Verstraelen and Vrancken in 1999 for submanifolds in real space forms. It is also known as the DDVV conjecture. It was proven recently by Lu (2011) and by Ge and Tang (2008), independently. The present author established a generalized Wintgen inequality for Lagrangian submanifolds in complex space forms in 2014. In the present paper we obtain the DDVV inequality, also known as generalized Wintgen inequality, for Legendrian submanifolds in Sasakian space forms. Some geometric applications are derived. Also we state such an inequality for contact slant submanifolds in Sasakian space forms.

The guiding thread of the present work is the following result, in the vain of Grothendieck's conjecture for differential equations : if the reduction modulo almost all prime $p$ of a given linear Mahler equation with coefficients in $\mathbb{Q}(z)$ has a full set of algebraic solutions, then this equation has a full set of rational solutions. The proof of this result, given at the very end of the paper, relies on intermediate results of independent interest about Mahler equations in characteristic zero as well as in positive characteristic.

Based on the analogies between knot theory and number theory, we study a deformation theory for ${\rm SL}_2$-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-${\rm SL}_2$-representations, we prove the existence of the universal deformation of a given ${\rm SL}_2$-representation of a finitely generated group $\Pi$ over a perfect field $k$ whose characteristic is not 2. We then show its connection with the character scheme for ${\rm SL}_2$-representations of $\Pi$ when $k$ is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.

In this paper, we study deformations of Brieskorn polynomials of two variables obtained by adding linear terms consisting of the conjugates of complex variables and prove that the deformed polynomial maps have only indefinite fold and cusp singularities in general. We then estimate the number of cusps appearing in such a deformation. As a corollary, we show that a deformation of a complex Morse singularity with real linear terms has only indefinite folds and cusps in general and the number of cusps is 3.

It is an interesting question whether a given equation of motion has a periodic solution or not, and in the positive case to describe it. We investigate periodic magnetic curves in elliptic Sasakian space forms and we obtain a quantization principle for periodic magnetic flowlines on Berger spheres. We give a criterion for periodicity of magnetic curves on the unit sphere ${\mathbb{S}}^3$.

The derivatives of the Bott class and those of the Godbillon-Vey class with respect to infinitesimal deformations of foliations, called infinitesimal derivatives, are known to be represented by a formula in the projective Schwarzian derivatives of holonomies [3], [1]. It is recently shown that these infinitesimal derivatives are represented by means of coefficients of transverse Thomas-Whitehead projective connections [2]. We will show that the formula can be also deduced from the latter representation.

The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\mathbb{R}^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\mathbb{R}^{n+2}$. On the level of their conformal Gauss maps into $Gr_{1,3}(\mathbb{R}^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of $\mathbb{R}^{1,n+3}$, contains a fixed lightlike vector or where it does not contain such a ``constant lightlike vector''. Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into $S^{n+2}$ which are not conformal to a minimal surface in $\mathbb{R}^{n+2}$. It turns out that our proof also works analogously for minimal immersions into the other space forms.