arXiv subjects
Publications and source records attributed to Alexander Plakhotnikov.
In this note we attempt to propose a categorical framework for the Ricci flow, treating it as a sequence of functors connecting the stack of Riemannian metrics to the category of geometric decompositions via singular flow spacetimes. To rigorize the domain of the flow, we adapt the definition of differentiable stacks to the site of Banach manifolds. We demonstrate that the Ricci flow defines a stratification of this stack.
This work investigates continuous embeddings for quantum Sobolev spaces $\mathfrak{H}_γ^{s,p}(G,H)$ into Schatten--von Neumann classes $S_r(H)$. We try to extend the results of Lakmon and Mensah to the case where the operators belong to Schatten classes $S_p(H)$ for $p \neq 2$. We establish that these quantum Sobolev spaces are Banach spaces and, by employing a duality argument, we define spaces for $p>2$.
The paper investigates the existence of a limit in the operator norm for a family of operators $T_z(H)= F(H-z)^{-1}F^*$ for $z$ tending to the real axis. The conditions for the $H$ operator and the rigging operator $F$ are established, under which the limit exists. Special attention is paid to the separation of cases when the limit point belongs and does not belong to the point spectrum $H$.
Let a finite set of points $\{ξ_1,...,ξ_k\}$ be chosen in a metric space $(X,d)$, and let the squared distance matrix $\mathfrak{D}=(\mathfrak{D}(ξ_i,ξ_j)^2)_{i,j=1}^{k}$ be constructed from them. We propose a geometric approach to studying the spectral properties of squared distance matrices of infinite size, constructed from a countable set of points $\{ξ_k\}_{k\in \mathbb{Z}}$ on Riemannian manifold $(M,g)$. We move from the discrete problem to a continuous one using walk matrices. We describe the structure of the spectrum and study the properties of spectral flows.