Abstract:
We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case of integrable orthogonal distribution the flow velocity is proportional to the mixed scalar curvature. We observe that the mean curvature vector $H$ of $D$ satisfies along the leaves the forced Burgers equation, this reduces to the linear Schr\"{o}dinger equation, whose potential function is a certain "non-umbilicity" measure of $D$. On order to show convergence of the solution metrics $g_t$ as $t\to\infty$, we normalize the flow, and instead of a foliation consider a fiber bundle $\pi: M\to B$ of a Riemannian manifold $(M, g_0)$. In this case, if the "non-umbilicity" of $D$ is smaller in a sense then the "non-integrability", then the limit mixed scalar curvature function is positive. For integrable $D$, we give examples with foliated surfaces and twisted products.

Abstract:
We introduce and study the flow of metrics on a foliated Riemannian manifold $(M,g)$, whose velocity along the orthogonal distribution is proportional to the mixed scalar curvature, $\Sc_{\,\rm mix}$. The flow is used to examine the question: When a foliation admits a metric with a given property of $\Sc_{\,\rm mix}$ (e.g., positive or negative)\/? We observe that the flow preserves harmonicity of foliations and yields the Burgers type equation along the leaves for the mean curvature vector $H$ of orthogonal distribution. If $H$ is leaf-wise conservative, then its potential obeys the non-linear heat equation $\dt u=n\Delta_\calf\,u +(n\beta_{\mathcal D}+\Phi)\,u+\Psi^\calf_1 u^{-1}-\Psi^\calf_2 u^{-3}$ with a leaf-wise constant $\Phi$ and known functions $\beta_{\mathcal D}\ge0$ and $\Psi^\calf_i\ge0$. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of leaf-wise Schr\"{o}dinger operator $\mathcal{H}_\calf=-\Delta_\calf -\beta_{\mathcal D}\id$) there exists a unique global solution $g_t$, whose $\Sc_{\rm mix}$ converges exponentially as $t\to\infty$ to a leaf-wise constant. The metrics are smooth on $M$ when all leaves are compact and have finite holonomy group. Hence, in certain cases, there exist ${\mathcal D}$-conformal to $g$ metrics, whose $\Sc_{\rm mix}$ is negative or positive.

Abstract:
The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we determine its scalar curvature and Ricci tensor and prove a lower bound for the curvature on the submanifold of trace one matrices. This bound is achieved for the maximally mixed state, a further hint for the quantum statistical meaning of the scalar curvature.

Abstract:
When undergraduates ask me what geometric group theorists study, I describe a theorem due to Gromov which relates the groups with an intrinsic geometry like that of the hyperbolic plane to those in which certain computations can be efficiently carried out. In short, I describe the close but surprising connection between negative curvature and efficient computation. This theorem was one of the clearest early indications that applying a metric perspective to traditional group theory problems might lead to new and important insights.

Abstract:
On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove that if the manifold with the critical point metric has harmonic curvature, then it is isometric to a standard sphere.

Abstract:
We need much better understanding of information processing and computation as its primary form. Future progress of new computational devices capable of dealing with problems of big data, internet of things, semantic web, cognitive robotics and neuroinformatics depends on the adequate models of computation. In this article we first present the current state of the art through systematization of existing models and mechanisms, and outline basic structural framework of computation. We argue that defining computation as information processing, and given that there is no information without (physical) representation, the dynamics of information on the fundamental level is physical/ intrinsic/ natural computation. As a special case, intrinsic computation is used for designed computation in computing machinery. Intrinsic natural computation occurs on variety of levels of physical processes, containing the levels of computation of living organisms (including highly intelligent animals) as well as designed computational devices. The present article offers a typology of current models of computation and indicates future paths for the advancement of the field; both by the development of new computational models and by learning from nature how to better compute using different mechanisms of intrinsic computation.

Abstract:
Let (M,J) be a minimal compact complex surface of Kaehler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a KAEHLER metric of positive scalar curvature. This extends previous results of Witten and Kronheimer.

Abstract:
We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than 2, as we construct the counter-examples for all k greater than 2. Our proof relies on a new geometric inequality which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete non-compact hypersurfaces of finitely many regular ends with nonnegative scalar curvature are weakly mean convex, and prove a positive mass theorem for such hypersurfaces.

Abstract:
We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterised by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of alignment theory and the bivector form of the Weyl operator in higher dimensions, and introduce the important notions of diagonalisability and (complex) analytic metric extension. We show that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterised by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.

Abstract:
Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $M^n$ satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature.