Representations of the Lie algebra of vector fields on a sphere

For an affine algebraic variety X we study a category of modules that admit compatible actions of both the algebra A of functions on X and the Lie algebra of vector fields on X. In particular, for the case when X is the sphere ${\mathbb{S}}^2$, we construct a set of simple modules that are finitely generated over A. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational ${\mathrm{GL}}_2$-modules.     

A generalization of lifting non-proper tropical intersections

Let X and $X^{\prime }$ be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles in the sense of [11, §2] between the tropicalization of the intersection product $X?X^{\prime }$ and the stable intersection $\mathrm{trop}(X)?\mathrm{trop}(X^{\prime })$, when restricted to (the inverse image under the tropicalization map of) a connected component C of $\mathrm{trop}(X)\cap \mathrm{trop}(X^{\prime })$. This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in [11, §2]. As a result we get a numerical equivalence between $\stackrel{￣}{X}?{\stackrel{￣}{X}}^{\prime }{|}_{\stackrel{￣}{C}}$ and $\stackrel{￣}{\mathrm{trop}(X)?\mathrm{trop}(X^{\prime }){|}_C}$ via the compactified stable intersection, where the closures are taken inside the compactifications of T and ${\mathbb{R}}^n$. In particular, when X and $X^{\prime }$ have complementary codimensions, this equivalence generalizes [15, Theorem 6.4], in the sense that $X\cap X^{\prime }$ is allowed to be of positive dimension. Moreover, if $\stackrel{￣}{X}\cap {\stackrel{￣}{X}}^{\prime }$ has finitely many points which tropicalize to $\stackrel{￣}{C}$, we prove a similar equation as in [15, Theorem 6.4] when the ambient space is a reduced subscheme of T (instead of T itself).     

Reeb posets and tree approximations

A well known result in the analysis of finite metric spaces due to Gromov says that given any metric space $(X,d_X)$ there exists a tree metric $t_X$ on $X$ such that ${|d_X-t_X|}_{\infty }$ is bounded above by twice $\mathrm{hyp}\left(X\right)\cdot log\left(2\left|X\right|\right)$. Here $\mathrm{hyp}\left(X\right)$ is the hyperbolicity of $X$, a quantity that measures the treeness of 4-tuples of points in $X$. This bound is known to be asymptotically tight.We improve this bound by restricting ourselves to metric spaces arising from filtered posets. By doing so we are able to replace the cardinality appearing in Gromov’s bound by a certain poset theoretic invariant which can be much smaller thus significantly improving the approximation bound.The setting of metric spaces arising from posets is rich: For example, every finite metric graph can be induced from a filtered poset. Since every finite metric space can be isometrically embedded into a finite metric graph, our ideas are applicable to finite metric spaces as well.At the core of our results lies the adaptation of the Reeb graph and Reeb tree constructions and the concept of hyperbolicity to the setting of posets, which we use to formulate and prove a tree approximation result for any filtered poset.     

Nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality

A Banach space X is said to be isomorphic to another Y with respect to the structure of Birkhoff-James orthogonality, denoted by $X{～}_{BJ}Y$, if there exists a (possibly nonlinear) bijection between X and Y that preserves Birkhoff-James orthogonality in both directions. It is shown that $X?Y$ if either X or Y is finite dimensional and $X{～}_{BJ}Y$, and that ${?}_p{?}_{BJ}{?}_q$ if $1<p<q<\infty$. Moreover, if H is a Hilbert space with $\dim ?H\ge 3$ and $H{～}_{BJ}X$, then $H=X$. In the two-dimensional case, it turns out that ${?}_{p,q}^2{～}_{BJ}{?}_2^2$, which indicates that nonlinear Birkhoff-James orthogonality preservers between Banach spaces are not necessarily scalar multiples of isometric isomorphisms.     

Malliavin smoothness on the Lévy space with Hölder continuous or BV functionals

We consider Malliavin smoothness of random variables $f\left(X_1\right)$, where $X$ is a pure jump Lévy process and the function $f$ is either bounded and Hölder continuous or of bounded variation. We show that Malliavin differentiability and fractional differentiability of $f\left(X_1\right)$ depend both on the regularity of $f$ and the Blumenthal–Getoor index of the Lévy measure.     

On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the n-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length n to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.     

L-orthogonality in Daugavet centers and narrow operators

We study the presence of L-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that if $\mathrm{dens}(Y)?{\omega }_1$ and $G:X?Y$ is a Daugavet center with separable range then, for every non-empty $w^{?}$-open subset W of $B_{X^{??}}$, it follows that $G^{??}(W)$ contains some L-orthogonal to Y. In the context of narrow operators, we show that if X is separable and $T:X?Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^{?}$-open subset W of $B_{X^{??}}$ then W contains some L-orthogonal u so that $T^{??}(u)=T(y)$. In the particular case that $T^{?}(Y^{?})$ is separable, we extend the previous result to $\mathrm{dens}(X)={\omega }_1$. Finally, we prove that none of the previous results holds in larger density characters (in particular, a counterexample is shown for ${\omega }_2$ under the assumption $2^c={\omega }_2$).     

On Yau's problem of evolving one curve to another: Convex case

A revised Yau's Curvature Difference Flow is considered to deform one convex curve $X_0$ to another one $\stackrel{?}{X}$. It is proved that this flow exists globally on time interval $[0,+\infty )$ and the evolving curve, preserving its convexity and bounded area A, converges to a fixed limiting curve $X_{\infty }$ (congruent to $\sqrt{A/\stackrel{?}{A}}\stackrel{?}{X}$) as time tends to infinity, where $\stackrel{?}{A}$ is the area bounded by the target curve $\stackrel{?}{X}$.     

Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }{X}_{+}$, where $X_{+}=X??$, for a smooth scheme $X\in {\mathrm{Sm}}_k$ over an infinite perfect field k, is computed.The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $({\mathbb{A}}^l\times X)/(({\mathbb{A}}^l?0)\times X)$, $X\in {\mathrm{Sm}}_k$, is one of ingredients in the theory. In the article we extend this result to the case of a pair $(X,U)$ given by a smooth affine variety X over k and an open subscheme $U?X$.The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }(X_{+}/U_{+})$ of the quotient-sheaf $X_{+}/U_{+}$.     

Higher Laplacians on pseudo-Hermitian symmetric spaces

One of the most fundamental operators studied in geometric analysis is the classical Laplace–Beltrami operator. On pseudo-Hermitian manifolds, higher Laplacians $L_m$ are defined for each positive integer m, where $L_1$ coincides with the Laplace–Beltrami operator. Despite their natural definition, these higher Laplacians have not yet been studied in detail. In this paper, we consider the setting of simple pseudo-Hermitian symmetric spaces, i.e., let $X=G/H$ be a symmetric space for a real simple Lie group G, equipped with a G-invariant complex structure. We show that the higher Laplacians $L_1,L_3,\dots ,L_{2r?1}$ form a set of algebraically independent generators for the algebra ${\mathbb{D}}_G(X)$ of G-invariant differential operators on X, where r denotes the rank of X. For higher rank, this is the first instance of a set of generators for ${\mathbb{D}}_G(X)$ defined explicitly in purely geometric terms, and confirms a conjecture of Engli? and Peetre, originally stated in 1996 for the class of Hermitian symmetric spaces.     

Lengths of involutions in finite Coxeter groups

Let W be a finite Coxeter group and X a subset of W. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t)={\sum }_{x\in X}{t}^{?(x)}$, where ? is the length function on W. If $X=\{x\in W:x^2=1\}$ then we call $L_{W,X}(t)$ the involution length polynomial of W. In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W. In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type $B_n$ and $D_n$. Moreover, we give a counterexample to a unimodality conjecture stated in [11].