arXiv research

Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.

Study shows stability of Schrödinger operator spectral data on a manifold.

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric c…

Stable solution found for manifold topology from boundary data.

Proves a Gel'fand-Kolmogoroff type result for vector bundles and polynomial functions.

In this paper, we investigate the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4. In the case of formal Hamiltonian vector fields on R^2, we computed the relative Gel'fand-Kalinin-Fuks cohomology groups of weight <20 in the paper by Mikami-Nakae-Kodama. The main strategy…

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $Δ_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We t…

In "The Gel'fand-Kalinin-Fuks class and characteristic classes of transversely symplectic foliations", arXiv:0910.3414, (October 2009) by D.Kotschick and S.Morita, the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields without constant vector fields on 2n-plane were characterized b…

Study shows stability of travel time data reconstruction from closed subsets.

We study the moduli spaces of polygons in R^2 and R^3, identifying them with subquotients of 2-Grassmannians using a symplectic version of the Gel'fand-MacPherson correspondence. We show that the bending flows defined by Kapovich-Millson arise as a reduction of the Gel'fand-Cetlin system on the Grassmannian, and with t…

By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm, we propose a new state-integral model for the Teichmüller TQFT, where the circle valued state variables live on the edges of oriented leveled shaped triangulations.

During the last decades algebraization of space turned out to be a promising tool at the interface between Mathematics and Theoretical Physics. Starting with works by Gel'fand-Kolmogoroff and Gel'fand-Naimark, this branch developed as from the fortieth in two directions: algebraic characterization of usual geometric sp…

We consider the moduli space M_r of polygons with fixed side lengths in five-dimensional eucledian space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between M_r and a weighted quotient of the n-fold product of the quaternionic projective line HP^1 by the diagonal PSL(2,H…

In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $Σ$ into two components and we know the eigenvalues $λ_j$ of the Laplace ope…

Study tackles inverse problems on low-dimensional manifolds, proving stability and proposing a reconstruction algorithm.

In "The {G}el'fand-{K}alinin-{F}uks class and characteristic classes of transversely symplectic foliations" arXiv:0910.3414, Kotschick and Morita showed that the Gel'fand-Kalinin-Fuks class in $\ds\HGF{7}{2}{}{8}$ is decomposed as a product $η\wedge ω$ of some leaf cohomology class $η$ and a transverse symplectic class…

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two …

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first resu…

In \cite{KOT:MORITA}, Kotschick and Morita showed that the Gel'fand-Kalinin-Fuks class in $\ds \HGF{7}{2}{}{8}$ is decomposed as a product $η\wedge ω$ of some leaf cohomology class $η$ and a transverse symplectic class $ω$. In other words, the Kontsevich homomorphism $\dsω\wedge :\HGF{5}{2}{0}{10} \rightarrow\HGF{7}{2}…

We show that discrete $W_m$ lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler-Gel'fand-Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structure defined on the space of geometric invar…

A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are con…

The geometric approach [1312.1262] to iterated variations of local functionals -- e.g., of the (master-)action functional -- resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory [IHES/M/15/13]. It also allowed of a rigorous proof ([1312.1262],[1210.0726]) fo…

For a one-parameter family of simple metrics of constant curvature ($4κ$ for $κ\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also t…

The main aim of this paper is to study soliton surfaces immersed in Lie algebras associated with ordinary differential equations (ODE's) for elliptic functions. That is, given a linear spectral problem for such an ODE in matrix Lax representation, we search for the most general solution of the wave function which satis…

Develops Chern-Weil theory for singular foliations.

The paper establishes correspondences between quaternionic spinors, Minkowski flags, and hyperbolic horospheres.

We provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichmüller TQFT, and we prove it in detail in the case of the knot complements of $4_{1}$ and $5_2$. The conjecture states that the level-$N$ Andersen-Kashaev invariant, $J^{(\mathrm{b},N)}_{M,K}$, is annihilated by the non-homogeneous $\hat{…

New method improves estimation of complex models from conditional moment restrictions.

We construct a local action of the group of rational maps from $S^2$ to $GL(n,C)$ on local solutions of flows of the ZS-AKNS $sl(n,C)$-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Bäcklund transformations, and we derive a permutability formula from different fact…

The biological processes involved in a drug's mechanisms of action are oftentimes dynamic, complex and difficult to discern. Time-course gene expression data is a rich source of information that can be used to unravel these complex processes, identify biomarkers of drug sensitivity and predict the response to a drug. H…

We introduce DeepMoD, a Deep learning based Model Discovery algorithm. DeepMoD discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives. A neural network approximates the data and constructs the function library, b…

New method for estimating parameters in inverse problems using double robustness.

The paper explains how microlocal analysis solves geometric inverse problems.

MCGDiff uses SGM to guide SMC for solving ill-posed linear inverse problems.

Study uses machine learning to solve photoacoustic tomography's inverse problem.

Machine learning models solve inverse eigenvalue problems for symmetric potentials and refractive indices.

EnKG solves inverse problems without derivatives, using diffusion models.

Study inverse problems for twisted geodesic flows on manifolds.

We solve image inverse problems using a flow-based noise model.

Proof of convergence for multi-objective optimization using inverse reinforcement learning.

Study solves inverse problems for real principal type operators using unique data sets and ray transforms.

Variational Gaussian Processes solve linear inverse problems efficiently.

Study solves inverse problems for equations with fractional nonlinearities.

Paper explores stability, regularization, and gradient flows for stochastic inverse problems.

New method tackles video inverse problems using image diffusion models.

Machine learning improves solving inverse problems and integrating data.

Diffusion models tackle noisy inverse problems with posterior sampling.

Deep neural networks solve noisy, complex problems accurately.