arXiv research

Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.

Paper develops a weighted linearization approach for vector fields.

The paper proves symplectic neighbourhood theorems for stratified subspaces.

The paper explores similarities in even and odd-dimensional geometry.

We provide sufficient conditions for the existence of Darboux charts on weakly symplectic bounded Fréchet manifolds by using the Moser's trick.

Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.

We give a construction to obtain canonically an ``isotropic average'' of given $C^1$-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem (obtained in collaboration with H. Karcher) and apply ``Moser's trick''. We also present an application …

Paper proves constants for Moser-Trudinger inequality on surfaces.

Proves a theorem similar to Moser's using a normalization method.

Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Moser's theorem on simplices. The proof is based on Banyag…

Sharp inequalities on curved spaces with bounded curvature.

Extends symplectic reduction and theorem to Lie algebroids.

We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that t…

There are proven few analogues of the Theorem of Moser using The Approximation Theorem of Artin.

The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…

Kuranishi's proof of complex deformation theory revisited

This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties of PMCTs, which already show th…

We prove a stability result for volume forms on fiber bundles with compact base and noncompact fibers. This generalizes the classical results of Moser and Greene--Shiohama, and recent work by the authors.

New theorems prove uniqueness of solutions to geometric PDEs.

The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.

We propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-C…

Theorems adapted for prequantum systems, showing symplectomorphism and gauge transformation.

We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kahler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group.

New inequality criterion for a mean field equation on spheres.

In [16], we established Trudinger-Moser inequalities for complete noncompact Riemannian manifold on which the Ricci curvature has lower bound and the injectivity radius is strictly positive. In this note, we improve those inequalties when the manifold is the hyperbolic space. The method we used here is still gluing loc…

We reconsider the (rational) Calogero-Moser system from the point of view of bi-Hamiltonian geometry. By using geometrical tools of the latter, we explicitly construct set(s) of spectral canonical coordinates, that is, complete sets of Darboux coordinates defined by the eigenvalues and the eigenvectors of the Lax matri…

This thesis is divided into two parts. In the first part we study completely integrable systems, and their underlying structures, in detail. We study their deformation theory and the different equivalence relations surrounding it. We motivate the definition of weak equivalence (found in the literature) by studying diff…

We show how the classical Moser Lemma from symplectic geometry extends to generalized complex structures (GCS) on arbitrary Courant algebroids. For this, we extend the notion of Lie derivative to sections of the tensor bundle $(\otimes^i E)\otimes(\otimes^j E^*)$ with respect to sections of the Courant algebroid $E$ us…

Researchers found counterexamples to a 2-jet determination theorem in higher codimension.

Researchers solve a Riemannian geometry problem using warped products.

Study of skateboard flips as continuous curves in $SO(3)$ group.

The paper proves a Moser-Trudinger inequality for zero-mean functions in 2D.

Paper analyzes solutions to quasilinear elliptic equations on manifolds using Nash-Moser iteration.

Nash's theorem proved with Günther's trick

Explains Conway's tangle trick and its mathematical origins.

The paper uses Nash-Moser iteration to prove gradient estimates for nonlinear equations on Riemannian manifolds.

We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary …

Unified framework for gradient estimation in combinatorial spaces.

Note on advancements in nonlinear elliptic equations' regularity theory.

We prove all knots can be transformed into a trefoil using special diagrams.

Geometric trick simplifies link homotopy and concordance.

Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.

We verify a conjecture of Gillet-Soulé. We prove that the determinant of the Laplacian on a line bundle over $\mathbb{CP}^{1}$ is always bounded from above. This can also be viewed as a multi-particle generalization of the Moser-Trudinger Inequality. Furthermore, we conjecture that this functional achieves its maximum …

Outlier based Robust Principal Component Analysis (RPCA) requires centering of the non-outliers. We show a "bias trick" that automatically centers these non-outliers. Using this bias trick we obtain the first RPCA algorithm that is optimal with respect to centering.

The Gumbel-max trick and its extensions simplify sampling from categorical distributions in machine learning.

A new gradient estimator for categorical distributions reduces bias and variance.

Gradient estimates for special harmonic functions on manifolds.