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The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis…

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold $(M, g)$ with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar and…

Given a compact and connected four dimensional smooth Riemannian manifold $(M,g_0)$ with $k_P := \int_M Q_{g_0} dV_{g_0} <0$ and a smooth non-constant function $f_0$ with $\max_{p\in M}f_0(p)=0$, all of whose maximum points are…

It is well known from the Perron-Frobenius theory that the spectral gap of a positive square matrix is positive. In this paper, we give a more quantitative characterization of the spectral gap. More specifically, using a complex extension…

Quantum field theories (QFT's) on noncommutative spacetimes are currently under intensive study. Usually such theories have world sheet noncommutativity. In the present work, instead, we study QFT's with commutative world sheet and…

In this paper, we focus on the following question. Assume $\phi$ is a discrete Gaussian free field (GFF) on $\Lambda \subset \frac 1 n \mathbb{Z}^2$ and that we are given $e^{iT \phi}$, or equivalently $\phi \pmod{\frac {2\pi} T}$. Can we…

We introduce the $R$ cut-off covering spectrum and the cut-off covering spectrum of a complete length space or Riemannian manifold. The spectra measure the sizes of localized holes in the space and are defined using covering spaces called…

We prove the Focal Index Lemma and the Rauch and Berger comparison theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop spaces of a compact manifold with the L^2…

We introduce the notion of a positive spectral measure on a $\sigma$-algebra, taking values in the positive projections on a Banach lattice. Such a measure generates a bounded positive representation of the bounded measurable functions. If…

Following Parisi \& Wu's paradigm of stochastic quantization, we constructed in \cite{BDFT} a $\Phi^4$ measure on an arbitrary closed, compact Riemannian manifold of dimension $3$ as an invariant measure of a singular stochastic partial…

We give a short proof of the following fact. Let $\Sigma$ be a connected, finitely connected, noncompact manifold without boundary. If $g$ is a complete Riemannian metric on $\Sigma$ whose Gaussian curvature $K$ is nonnegative at infinity,…

We seek for an alternative to the metric tensor $g_{\mu\nu}$ as a fundamental geometrical object in four-dimensional Riemannian manifolds. We suggest that the metric tensor $g_{\mu\nu}(P)$ at a given point $P$ of a manifold may be replaced…

We explore the possibility that, in a quantum field theory with Planck scale cutoff Lambda=Mp, observable quantities for low-energy processes respect the Lorentz symmetry. In particular, we compute the one-loop radiative correction Pi to…

Let $I=(a,b)\times(c,d)\subset {\mathbb R}_{+}^2$ be an index set and let $\{G_{\alpha}(x) \}_{\alpha \in I}$ be a collection of Gaussian functions, i.e. $G_{\alpha}(x) = \exp(-\alpha_1 x_1^2 - \alpha_2 x_2^2)$, where $\alpha = (\alpha_1,…

The main object under consideration is a class $\Phi_n\backslash\Phi_{n+1}$ of radial positive definite functions on $\R^n$ which do not admit \emph{radial positive definite continuation} on $\R^{n+1}$. We find certain necessary and…

We conjecture an intrinsic UV cutoff for the validity of the effective field theory with a large number of species coupled to gravity. In four dimensions such a UV cutoff takes the form $\Lambda=\sqrt{\lambda/ N}M_p$ for $N$ scalar fields…

We present a critical reappraisal of the available results on the broken phase of $\lambda(\Phi^4)_4$ theory, as obtained from rigorous formal analyses and from lattice calculations. All the existing evidence is compatible with Spontaneous…

In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $\lambda$-Lipschitz…

Let $\mu$ be a self-similar measure generated by iterated function system of four maps of equal contraction ratio $0<\rho<1$. We study when $\mu$ is a spectral measure which means that it admits an exponential orthonormal basis $\{e^{2\pi i…

In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval…