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 $\textrm{rblo}(\mu)$.