Related papers: Measures on differentiable stacks
This article discusses two recent works by the author, one with Brown and Hurtado on Zimmer's conjecture and one with Bader, Miller and Stover on totally geodesic submanifolds of real and complex hyperbolic manifolds. The main purpose of…
Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit…
We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…
On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with…
Mode separation, namely how sharply a distribution fragments into barrier-separated clusters, is a fundamental geometric property of densities, difficult to quantify in high dimensions. It is structurally distinct from dispersion, yet…
In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a perfect and separable metric space (thus,…
In this manuscript, we show how conformal invariance can be incorporated in a classical theory of gravitation, in the context of metric measure space. Metric measure space involves a geometrical scalar $f$, dubbed as density function, which…
In this paper, we consider the large deviations of invariant measure for the 3D stochastic hyperdissipative Navier-Stokes equations driven by additive noise. The unique ergodicity of invariant measure as a preliminary result is proved using…
In this paper we develop an intrinsic formalism to study the topology, smooth structure, and Riemannian geometry of the Wasserstein space of a closed Riemannian manifold. Our formalism allows for a new characterisation of the Weak topology…
We study $p$-adic manifolds associated with twisted points of quotient stacks $\mathcal{X} = [U/G]$ and their quotient spaces $\pi:\mathcal{X} \to X$. We prove several structural results about the fibres of $\pi$ and derive in particular a…
We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of geometric invariant theory. We apply this to two…
It is perhaps not widely recognized that certain common notions of distance between probability measures have an alternative dual interpretation which compares corresponding functionals against suitable families of test functions. This dual…
Building upon the well-posedness results in \cite{snse1}, in this note we prove the existence of invariant measures for the stochastic Navier-Stokes equations with stable L\'evy noise. The crux of our proof relies on the assumption of…
We study the density of the invariant measure of the Hurwitz complex continued fraction from a computational perspective. It is known that this density is piece-wise real-analytic and so we provide a method for calculating the Taylor…
Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $\theta$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_\theta(G) \subseteq V$ of semistable points. We give necessary,…
We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc…
In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a…
In this paper, we establish the sheafified version of the cohomological integrality conjecture for stacks obtained as a quotient of a smooth affine symmetric algebraic variety by a reductive algebraic group equipped with an invariant…
In this paper we analyze the derivative nonlinear Schr\"odinger equation on $\mathbb{T}$ with randomized initial data in $\cap_{s < \frac{1}{2}} H^{s}(\mathbb{T})$ according to a Wiener measure. We construct an invariant measure at each…
The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising from nonlinear regression problems is considered. Extending investigations from arXiv:2009.05298, local and global…