One of the fundamental results in convex geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach–Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over $\mathbb{C}$ using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on $\mathbb{N}$-graded Lie algebras of maximal class. As shown by $\text{A}$. Fialowski there are only three isomorphism types of $\mathbb{N}$-graded Lie algebras $L\,=\,\oplus _{i=1}^{\infty }\,{{L}_{i}}$ of maximal class generated by ${{L}_{i}}$ and ${{L}_{2}}$, $L\,=\,\left\langle {{L}_{1}},\,{{L}_{2}} \right\rangle$. Vergne described the structure of these algebras with the property $L\,=\,\left\langle {{L}_{1}} \right\rangle$. In this paper we study those generated by the first and $q$-th components where $q\,>\,2$, $L\,=\,\left\langle {{L}_{1}},\,{{L}_{q}} \right\rangle$. Under some technical condition, there can only be one isomorphism type of such algebras. For $q=\,3$ we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

Let $\mathbf{x}\,=\,\left( {{x}_{0}},\,{{x}_{1}},.\,.\,. \right)$ be a $N$-periodic sequence of integers $\left( N\,\ge \,1 \right)$, and $\mathbf{s}$ a sturmian sequence with the same barycenter (and also $N$-periodic, consequently). It is shown that, for affine functions $\alpha :\,\mathbb{R}_{(N)}^{\mathbb{N}}\,\to \,\mathbb{R}$ which are increasing relatively to some order ${{\le }_{2}}$ on $\mathbb{R}_{(N)}^{\mathbb{R}}$ (the space of all $N$-periodic sequences), the average of $\left| \alpha \right|$ on the orbit of $\mathbf{x}$ is greater than its average on the orbit of $\mathbf{s}$.

In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

We obtain results on the unitary equivalence of weak contractions of class ${{C}_{0}}$ to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool in this paper is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.

Combings of compact, oriented, 3-dimensional manifolds $M$ are homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. A combing is called torsion if this Euler class is a torsion element of ${{H}^{2}}(M;\,\mathbb{Z})$. Gompf introduced a $\mathbb{Q}$-valued invariant ${{\theta }_{G}}$ of torsion combings on closed 3-manifolds, and he showed that ${{\theta }_{G}}$ distinguishes all torsion combings with the same $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. We give an alternative definition for ${{\theta }_{G}}$ and we express its variation as a linking number. We define a similar invariant ${{p}_{1}}$ of combings for manifolds bounded by ${{S}^{2}}$. We relate ${{p}_{1}}$ to the $\Theta$-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula $\Theta \,=\,\frac{1}{4}{{p}_{1}}\,+\,6\text{ }\!\!\lambda\!\!\text{ }\left( {\hat{M}} \right)$, where $\text{ }\!\!\lambda\!\!\text{ }$ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.

The work of Reid, Chinburg–Hamilton–Long–Reid, Prasad–Rapinchuk, and the author with Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic manifold. The main results of this article show that generalizations of these results to other arithmetic manifolds will require a wide range of data. Specifically, we prove that certain incommensurable arithmetic manifolds arising from the semisimple Lie groups of the form ${{\left( \text{SL(d,}\,\mathbf{R}) \right)}^{r}}\,\times \,{{\left( \text{SL(d,}\,\mathbf{C}) \right)}^{s}}$ have the same commensurability classes of totally geodesic submanifolds coming from a fixed field. This construction is algebraic and shows the failure of determining, in general, a central simple algebra from subalgebras over a fixed field. This, in turn, can be viewed in terms of forms of $\text{S}{{\text{L}}_{d}}$ and the failure of determining the form via certain classes of algebraic subgroups.

We consider Tate cycles on an Abelian variety $A$ defined over a sufficiently large number field $K$ and having complexmultiplication. We show that there is an effective bound $C\,=\,C(A,\,K)$ so that to check whether a given cohomology class is a Tate class on $A$, it suffices to check the action of Frobenius elements at primes $v$ of norm $\le \,C$. We also show that for a set of primes $v$ of $K$ of density 1, the space of Tate cycles on the special fibre ${{A}_{v}}$ of the Néron model of $A$ is isomorphic to the space of Tate cycles on $A$ itself.

Let $F$ be a p-adic field of odd residual characteristic. Let $\overline{GS{{p}_{2n}}(F)}$ and $\overline{S{{p}_{2n}}(F)}$ be the metaplectic double covers of the general symplectic group and the symplectic group attached to the $2n$ dimensional symplectic space over $F$, respectively. Let $\sigma$ be a genuine, possibly reducible, unramified principal series representation of $\overline{GS{{p}_{2n}}(F)}$. In these notes we give an explicit formula for a spanning set for the space of Spherical Whittaker functions attached to $\sigma$. For odd $n$, and generically for even $n$, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of $\overline{S{{p}_{2n}}(F)}$. If $n$ is odd, then each element in the set has an equivariant property that generalizes a uniqueness result proved by Gelbart, Howe, and Piatetski-Shapiro.Using this symmetric set, we construct a family of reducible genuine unramified principal series representations that have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for $n$ even.