Related papers: On the spectral type of some class of rank one flo…

It is shown that a certain class of Riesz product type measures on $\mathbb{R}$ is realized a spectral type of rank one flows. As a consequence, we will establish that some class of rank one flows has a singular spectrum. Some of the…

We study a class of generalized Riesz products connected to the spectral type of some class of rank one flows on $\R$. Applying a Central Limit Theorem of Kac, we exhibit a large class of singular generalized Riesz products on the Bohr…

We introduce a new tool to study the spectral type of rank one transformations using the method of central limit theorem for trigonometric sums. We get some new applications.

We extend partially the Kakutani-Zygmund dichotomy theorem to a class of generalized Riesz-product type measures by proving that the generalized Riesz-product is singular if and only if its Mahler measure is zero. As a consequence, we…

Let $T$ be a locally finite tree equipped with a flow measure $m$. Let $\mathcal L$ be the flow Laplacian on $(T,m)$. We prove that the first order Riesz transform $\nabla \mathcal L^{-1/2}$ is bounded on $L^p(m)$ for $p\in (1,\infty)$.…

Let $M$ be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on $M$ to a small tubular neighborhood of each connected component of its singular stratum is foliated-diffeomorphic to an…

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that almost every such locally Hamiltonian flow with only simple saddles has singular…

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of…

Via continuous deformations based on natural flow evolutions, we prove several novel monotonicity results for Riesz-type nonlocal energies on triangles and quadrilaterals. Some of these results imply new and simpler proofs for known…

We study the convergence of a modified Kaeher-Ricci flow defined by Zhou Zhang. We show that the flow converges to a singular metric when the limit class is degenerate. This proves a conjecture of Zhang.

We study a class of dynamical systems given by measure preserving actions of the group $Z^d$ or $R^d$ and generating a set of spectral measures with an extremal rate of the Fourier coefficient decay: $\Hat\sigma(n) = O(|n|^{-1/2+\epsilon})$…

We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible…

We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further…

Recently, with Mussardo we defined a quantum mechanical problem of a single particle scattering with impurities wherein the quantized energy levels $E_n (\sigma)$ are exactly equal to the zeros of the Riemann $\zeta (s)$ where $\sigma = \Re…

For the K\"ahler-Ricci flow on a compact K\"ahler manifold with semi-ample canonical line bundle, we prove the singularity type at infinity does not depend on the choice of the initial metric. We also provide new simple proofs for some…

We prove analogs of Peyri\`ere's mutual singularity theorem for standard and generalized Riesz products on the unit sphere of $\mathbb{C}^n$, $n\ge 2$. As a corollary, we obtain an analog of Zygmund's dichotomy for the Riesz products under…

In this paper we study the $s$-dimensional Riesz transform of a finite measure $\mu$ in $\mathbf{R}^d$, with $s\in (d-1,d)$. We show that the boundedness of the Riesz transform of $\mu$ implies that a nonlinear potential of exponential type…

The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both $L^2$ spaces and weighted-$L^2$ spaces. As a consequence, an example of a flow admitting a…

We study the generalizations of Jonathan King's rank-one theorems (Weak-Closure Theorem and rigidity of factors) to the case of rank-one R-actions (flows) and rank-one Z^n-actions. We prove that these results remain valid in the case of…

We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which,…