Very ampleness criteria for rank $2$ vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q=1$, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa--Intriligator type formulas for the Gromov--Witten invariants.

Let $L$ be an oriented Lagrangian submanifold in an $n$-dimensional Kähler manifold~$M$. Let $u \colon D \to M$ be a minimal immersion from a disk $D$ with $u(\partial D) \subset L$ such that $u(D)$ meets $L$ orthogonally along $u(\partial D)$. Then the real dimension of the space of admissible holomorphic variations is at least $n+\mu(E,F)$, where $\mu(E,F)$ is a boundary Maslov index; the minimal disk is holomorphic if there exist $n$ admissible holomorphic variations that are linearly independent over $\mathbb{R}$ at some point $p \in \partial D$; if $M = \mathbb{C}P^n$ and $u$ intersects $L$ positively, then $u$ is holomorphic if it is stable, and its Morse index is at least $n+\mu(E,F)$ if $u$ is unstable.

Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let $\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov boundary $S$ of $\mathcal{D}$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform $P_{l,\lambda}f$ of an element $f$ in the space $\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$ Namely, if $F$ is the Poisson transform of some $f\in \mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any $l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that $\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and only if $f$ satisfies the growth condition $$ \|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty. $$

An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are $674,\!688$ such varieties.

In this paper we construct an analogue of Iwahori--Hecke algebras of $\operatorname{SL}_2$ over $2$-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori--Matsumoto type relations.

We give an explicit construction of polynomial (of arbitrary degree) $(\alpha,\beta)$-metrics with scalar flag curvature and determine their scalar flag curvature. These Finsler metrics contain all non-trivial projectively flat $(\alpha,\beta)$-metrics of constant flag curvature.

In this paper we prove holomorphy for certain intertwining operators arising from the theory of Eisenstein series. To do that we need to normalize using the Langlands--Shahidi's normalization arising from the twisted endoscopy and the associated representations of the general linear group.

We calculate the group of homotopy classes of homotopy self-equivalences of $4$-manifolds with free fundamental group and obtain a classification of such $4$-manifolds up to $s$-cobordism.

We investigate equality cases in inequalities for Sylvester-type functionals. Namely, it was proven by Campi, Colesanti, and Gronchi that the quantity $$ \int_{x_0\in K}\cdots\int_{x_n\in K}[V(\textrm{conv}\{x_0,\dots,x_n\})]^pdx_0\cdots dx_n , n\geq d, p\geq 1 $$ is maximized by triangles among all planar convex bodies $K$ (parallelograms in the symmetric case). We show that these are the only maximizers, a fact proven by Giannopoulos for $p=1$. Moreover, if $h$: $\mathbb{R}_+\rightarrow \mathbb{R}_+$ is a strictly increasing function and $W_j$ is the $j$-th quermassintegral in $\mathbb{R}^d$, we prove that the functional $$ \int_{x_0\in K_0}\cdots\int_{x_n\in K_n}h(W_j(\textrm{conv}\{x_0,\dots,x_n\}))dx_0\cdots dx_n , n \geq d $$ is minimized among the $(n+1)$-tuples of convex bodies of fixed volumes if and only if $K_0,\dots,K_n$ are homothetic ellipsoids when $j=0$ (extending a result of Groemer) and Euclidean balls with the same center when $j>0$ (extending a result of Hartzoulaki and Paouris).

Let $\mu$ be a nonnegative Radon measure on $\mathbb{R}^d$ that satisfies the growth condition that there exist constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and $r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\textrm {BMO}$-type space $\textrm{RBMO}(\mu)$ of Tolsa, then the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an initial cube) and the inhomogeneous maximal function $\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube) associated with a given approximation of the identity $S$ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from $\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$-type space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous maximal operator $\mathcal{M}_S$ is bounded from the local $\textrm {BMO}$-type space $\textrm{rbmo}(\mu)$ to the local $\textrm {BLO}$-type space $\textrm{rblo}(\mu)$.