Let $A$ be a semi-abelian variety with an exponential map $\exp : \mathop{\mathrm{Lie}}(A) \to A$. The purpose of this paper is to explore Nevanlinna theory of the entire curve $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f := (\exp f, f) : \mathbf{C} \to A \times \mathop{\mathrm{Lie}}(A)$ associated with an entire curve $f : \mathbf{C} \to \mathop{\mathrm{Lie}}(A)$. Firstly we give a Nevanlinna theoretic proof to the analytic Ax–Schanuel Theorem for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax–Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that the elements of the vector-valued function $f(z) - f(0) \in \mathop{\mathrm{Lie}}(A) \cong \mathbf{C}^n$ are $\mathbf{Q}$-linearly independent in the case of $A = (\mathbf{C}^*)^n$. Our proof is based on the Log Bloch–Ochiai Theorem and a key estimate which we show.

Our next aim is to establish a Second Main Theorem for $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f$ and its $k$-jet lifts with truncated counting functions at level one. We give some applications to a problem of a type raised by S. Lang and the unicity. The results clarify a relationship between the problems of Ax–Schanuel type and Nevanlinna theory.

We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne–Mostow theory and periods of $K3$ surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of complex hyperbolic balls. We show in Theorem 7.4 that the two ball-quotient constructions can be unified in a geometric way.

This paper studies the obstructions to deforming a map from a complex variety to another variety which is an immersion of codimension one. We extend the classical notion of semiregularity of subvarieties to maps between varieties, and prove the unobstructedness of the deformation of such a map, even when the image is non-reduced. As an application, we give a simple but effective criterion for the vanishing of the obstructions to equisingular deformations of nodal curves on surfaces.

We give the inversion formula and the Plancherel formula for the hypergeometric Fourier transform associated with a root system of type $BC$, when the multiplicity parameters are not necessarily non-negative.

For any $1 < q < p < \infty$, we identify the best constant $K_{p,q}$ with the following property. If $\mathcal{H}$ is the Hilbert transform on the unit circle $\mathbb{T}$ and $A \subset \mathbb{T}$ is an arbitrary measurable set, then $\int_{A} | \mathcal{H}f | \mathrm{d}m \leq K_{p,q} \| f \|_{L^{p}(\mathbb{T},m)} m(A)^{1-1/q}$. The proof rests on the construction of certain special superharmonic functions on the plane, which are of independent interest.

We show the strong cohomological rigidity of Hirzebruch surface bundles over Bott manifolds. As a corollary, we have that the strong cohomological rigidity conjecture is true for Bott manifolds of dimension 8.

An element of a Weyl group of classical type is skew if it is the left factor in a reduced factorization of a Grassmannian element. The skew Grothendieck polynomials are those which are indexed by skew elements of the Weyl group. We define set-valued tableaux which are fillings of the associated skew Young diagrams and use them to prove tableau formulas for the skew double Grothendieck polynomials in all four classical Lie types. We deduce tableau formulas for the Grassmannian Grothendieck polynomials and the $K$-theoretic analogues of the (double mixed) skew Stanley functions in the respective Lie types.

In this paper we give an explicit formula of the Fourier coefficients of the Siegel Eisenstein series of degree 3, level $p$ with quadratic character. For the calculation we use the result of the generalized Gauss sum computed by Hiroshi Saito. After the tedious calculation we can get a rather simple result.

A Hurwitz group is a conformal automorphism group of a compact Riemann surface with precisely $84(g - 1)$ automorphisms, where $g$ is the genus of the surface. Our starting point is a result on the smallest Hurwitz group $\mathrm{PSL}(2,\mathbb{F}_{7})$ which is the automorphism group of the Klein surface. In this paper, we generalize it to various classes of simple Hurwitz groups and discuss a relationship between the surface symmetry and spectral asymmetry for compact Riemann surfaces. To be more precise, we show that the reducibility of an element of a simple Hurwitz group is equivalent to the vanishing of the $\eta$-invariant of the corresponding mapping torus. Several wide classes of simple Hurwitz groups which include the alternating group, the Chevalley group and the Monster, which is the largest sporadic simple group, satisfy our main theorem.

Using Hahn series, one can attach to any linear Mahler equation a basis of solutions at 0 reminiscent of the solutions of linear differential equations at a regular singularity. We show that such a basis of solutions can be produced by using a variant of Frobenius method.

We show that a Busemann space $X$ covered by parallel bi-infinite geodesics is homeomorphic to a product of another Busemann space $Y$ and the real line. We also show that a semi-simple isometry on $X$ preserving the foliation by parallel geodesics canonically induces a semi-simple isometry on $Y$.

Infinite-dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes a Lévy process, and the interaction between particles is determined by the long-range interaction potential. The potential is of Ruelle's class or logarithmic. We discuss the existence and uniqueness of strong solutions of the ISDEs.