Motivated by index formulas for Dirac-type operators over negatively curved Riemannian manifolds of finite volume, we study $\eta$-functions of certain differential operators on nilmanifolds.

A tautological system, introduced in Period Integrals and Tautological Systems and Period Integrals of CY and General Type Complete Intersections, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. In this article, we introduce two formulas—one purely algebraic, the other geometric—to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic version gives the local solution space as a Lie algebra homology group, while the geometric one as the middle de Rham cohomology of the complement of a hyperplane section in $X$. We use both formulas to find certain degenerate points for which the rank of the solution sheaf becomes $1$. These rank $1$ points appear to be good candidates for the so-called large complex structure limits in mirror symmetry. The formulas are also used to prove a conjecture of Hosono, Lian, and Yau, on the completeness of the extended GKZ system when $X$ is $\mathbb{P}^n$, GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces.

We study the geometry of the twistor space of the universal hyperkähler implosion $Q$ for $SU(n)$. Using the description of $Q$ as a hyperkähler quiver variety, we construct a holomorphic map from the twistor space $\mathcal{Z}_Q$ of $Q$ to a complex vector bundle over $\mathbb{P}^1$, and an associated map of $Q$ to the affine space $\mathcal{R}$ of the bundle’s holomorphic sections. The map from $Q$ to $\mathcal{R}$ is shown to be injective and equivariant for the action of $SU(n) \times T^{n-1} \times SU(2)$. Both maps, from $Q$ and from $\mathcal{Z}_Q$, are described in detail for $n = 2$ and $n = 3$. We explain how the maps are built from the fundamental irreducible representations of $SU(n)$ and the hypertoric variety associated to the hyperplane arrangement given by the root planes in the Lie algebra of the maximal torus. This indicates that the constructions might extend to universal hyperkähler implosions for other compact groups.

We consider the integrable system on the moduli space of Higgs bundles restricted to the subvariety corresponding to representations of a surface group into certain noncompact real forms, and in doing so encounter as the fiber moduli spaces of rank $2$ bundles on a spectral curve. This contrasts with the general case where the fiber is an abelian variety.

The set of all error-correcting block codes over a fixed alphabet with $q$ letters determines a recursively enumerable set of rational points in the unit square with coordinates $(R, \delta):=$ (relative transmission rate, relative minimal distance). Limit points of this set form a closed subset, defined by $R \leq \alpha_q(\delta)$, where $\alpha_q(\delta)$ is a continuous decreasing function called the asymptotic bound. Its existence was proved by the first-named author in 1981 (Renormalization and computation I: Motivation and background), but no approaches to the computation of this function are known, and in A computability challenge: Asymptotic bounds and isolated errorcorrecting codes, it was even suggested that this function might be uncomputable in the sense of constructive analysis.

In this note we show that the asymptotic bound becomes computable with the assistance of an oracle producing codes in the order of their growing Kolmogorov complexity. Moreover, a natural partition function involving complexity allows us to interpret the asymptotic bound as a curve dividing two different thermodynamic phases of codes.

In this paper we study a constrained minimization problem for the Willmore functional. For prescribed surface area, we consider smooth embeddings of the sphere into the unit ball. We evaluate the dependence of the the minimal Willmore energy of such surfaces on the prescribed surface area and prove corresponding upper and lower bounds. Interesting features arise when the prescribed surface area just exceeds the surface area of the unit sphere. We show that (almost) minimizing surfaces cannot be a $C^2$-small perturbation of the sphere. Indeed, they have to be nonconvex and there is a sharp increase in Willmore energy with a square root rate with respect to the increase in surface area.

Catanese surfaces are regular surfaces of general type with $p_g = 0$. They specialize to double covers of Barlow surfaces. We prove that the $CH_0$ group of a Catanese surface is equal to $\mathbb{Z}$, which implies the same result for the Barlow surfaces.

We give a reformulation of the recent results of Candelas et al., Lines on the Dwork pencil of quintic threefolds, describing pencils of lines on the quintic threefold $$ \{ (x_1 : \dots : x_5) \in \mathbb{P}^4 (\mathbb{C}) \ {} \vert \ {} x^5_1 + \dots + x^5_5 = 5{\psi}x_1 \dots x_5 \} $$ in terms of the moduli space $M_{0,5}$ of curves of genus $0$ with $5$ marked points, and a generalization to pencils of lines on the degree $n$ hypersurfaces $$ \{ (x_1 : \dots : x_n) \in \mathbb{P}^{n-1} (\mathbb{C}) \ {} \vert \ {} x^n_1 + \dots + x^n_n = n{\psi}x_1 \dots x_n \} $$ in $\mathbb{P}^{n−1}(\mathbb{C})$ in terms of the moduli space $M_{0,n}$ for any odd integer $n \geq 5$.