The Thom-Boardman symbol was first introduced by Thom in 1956 to classify singularities of differentiable maps. It was later generalized by Boardman to a more general setting. Although the Thom-Boardman symbol is realized by a sequence of non-increasing, nonnegative integers, to compute those numbers is, in general, extremely difficult. In the case of polynomial multiplication maps, Robert Varley conjectured that computing the Thom-Boardman symbol for polynomial multiplication reduces to computing the successive quotients and remainders for the Euclidean algorithm applied to the degrees of the two polynomials. In this paper, we confirm Varley’s conjecture.

Motivated by the classical Theorems of Picard and Siegel and their generalizations, we define the notion of an essentially large effective divisor and derive some of its arithmetic and function-theoretic consequences. We then investigate necessary and sufficient criteria for divisors to be essentially large. In essence, we prove that on a nonsingular irreducible projective variety $X$ with $\mathrm{Pic}(X) = \mathbb{Z}$, every effective divisor with $\operatorname{dim}X + 2$ or more components in general position is essentially large.

The link between Frobenius manifolds and singularity theory is well known, with the simplest examples coming from the simple hypersurface singularities. Associated with any such manifold is a function known as the $G$-function. This plays a role in the construction of higher-genus terms in various theories. For the simple singularities the $G$-function is known explicitly: $G = 0$ . The next class of singularities, the unimodal hypersurface or elliptic hypersurface singularities consists of three examples, $\tilde{E}_6 , \tilde{E}_7 , \tilde{E}_8$ (or equivalently $P_8 ,X_9 , J_10$). Using a result of Noumi and Yamada on the flat structure on the space of versal deformations of these singularities the $G$-function is explicitly constructed for these three examples. The main property is that the function depends on only one variable, the marginal (dimensionless) deformation variable. Other examples are given based on the foldings of known Frobenius manifolds. Properties of the $G$-function under the action of the modular group is studied, and applications within the theory of integrable systems are discussed.

New thoughts about the first and second integral bounds of Hans F. Weinberger for Green’s functions of uniformly elliptic equations are presented by extending the bounds to two optimal monotone principles, but also further explored via: (i) discovering two new sharp Green-function- involved isoperimetric inequalities; (ii) verifying the lower dimensional Pólya conjecture for the lowest eigenvalue of the Laplacian; (iii) sharpening an eccentricity-based lower bound for the Mahler volumes of the origin-symmetric convex bodies.

Let $M^H_X (u)$ be the moduli space of semi-stable pure sheaves of class $u$ on a smooth complex projective surface $X$. We specify $u = (0, L, \xi(u) = 0)$, i.e. sheaves in $u$ are of dimension 1. There is a natural morphism $\pi$ from the moduli space $M^H_X (u)$ to the linear system $|L|$. We study a series of determinant line bundles $\lambda_{c_n^r}$ on $M^H_X (u)$ via $\pi$. Denote $g_L$ the arithmetic genus of curves in $|L|$. For any $X$ and $g_L \le 0$, we compute the generating function $Z^r(t) = \sum_n h^0(M^H_X (u), \lambda_{c_n^r})t^n$. For $X$ being $\mathbb{P}^2$ or $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} (−e))$ with $e = 0, 1$, we compute $Z^1(t)$ for $g_L \gt 0$ and $Z^r(t)$ for all $r$ and $g_L = 1, 2$. Our results provide a numerical check to Strange Duality in these specified situations, together with Göttsche’s computation. And in addition, we get an interesting corollary (Corollary 4.2.13) in the theory of compactified Jacobian of integral curves.

In this paper we study the homogeneous Kähler manifolds (h.K.m.) which can be Kähler immersed into finite or infinite dimensional complex space forms. On the one hand we completely classify the h.K.m. which can be Kähler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. On the other hand, we extend known results about Kähler immersions into the finite dimensional complex projective space to the infinite dimensional setting.

We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type. Generalized Harish-Chandra modules are $(\mathfrak{g}, \mathfrak{k})$-modules of finite type where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k} \subset \mathfrak{g}$ is a reductive in $\mathfrak{g}$ subalgebra. A first result of the present paper is that a fundamental series module is a $\mathfrak{g}$-module of finite length. We then define the notions of strongly and weakly reconstructible simple $(\mathfrak{g}, \mathfrak{k})$-modules $M$ which reflect to what extent $M$ can be determined via its appearance in the socle of a fundamental series module. In the second part of the paper we concentrate on the case $\mathfrak{k} \simeq sl(2)$ and prove a sufficient condition for strong reconstructibility. This strengthens our main result from Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type for the case $\mathfrak{k} = sl(2)$. We also compute the $sl(2)$-characters of all simple strongly reconstructible (and some weakly reconstructible) $(\mathfrak{g}, sl(2))$-modules. We conclude the paper by discussing a functor between a generalization of the category $\mathcal{O}$ and a category of $(\mathfrak{g}, sl(2))$-modules, and we conjecture that this functor is an equivalence of categories.

Ideas and methods of Clemens C. H., Griffiths Ph. The intermediate Jacobian of a cubic threefold are applied to a Fano threefold $X$ of genus 6 — intersection of $G(2, 5) \subset P^9$ with $P^7$ and a quadric. Main results:

1. The Fano surface $F(X)$ of $X$ is smooth and irreducible. Hodge numbers and some other invariants of $F(X)$ are calculated.

2. Tangent bundle theorem for $X$ is proved, and its geometric interpretation is given. It is shown that $F(X)$ defines $X$ uniquely.

3. The Abel-Jacobi map $\Phi : \operatorname{Alb} F(X) \to J^3(X)$ is an isogeny.

4. As a necessary step of calculation of $h^{1,0}(F(X))$ we describe a special intersection of 3 quadrics in $P^6$ (having 1 double point) whose Hesse curve is a smooth plane curve of degree 6.

5. $\operatorname{im} \Phi(F(X)) \subset J^3(X)$ is algebraically equivalent to $\frac{2\Theta^8}{8!}$ where $\Theta \subset J^3(X)$ is a Poincaré divisor (a sketch of the proof).