arXiv research

Computes deformations of parabolic structures on Riemann surfaces.

Real slices of parabolic opers on Riemann surfaces are studied.

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.

Introduces new spectral triples for parabolic geometry.

We prove monotonicity of a parabolic frequency on manifolds. This is a parabolic analog of Almgren's frequency function. Remarkably we get monotonicity on all manifolds and no curvature assumption is needed. When the manifold is Euclidean space and the drift operator is the Ornstein-Uhlenbeck operator this can been see…

In this paper, we introduce a generalization of G-opers for arbitrary parabolic subgroups P<G. For parabolic subgroups associated to even nilpotents, we parameterize (G,P)-opers by an object generalizing the base of the Hitchin fibration. In particular, we describe families of opers associated to higher Teichmuller spa…

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

This article studies a discrete geometric structure on triangulated manifolds and an associated curvature flow (combinatorial Yamabe flow). The associated evolution of curvature appears to be like a heat equation on graphs, but it can be shown to not satisfy the maximum principle. The notion of a parabolic-like operato…

Study of BGG sequences on foliated manifolds with transverse parabolic geometry.

The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the…

This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying struc…

The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.

In this expository article, we discuss various monotonicity formulas for parabolic and elliptic operators and explain how the analysis of the function spaces and the geometry of the underlining spaces are intertwined. After briefly discussing some of the well-known analytical applications of monotonicity for parabolic …

We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expresse…

The paper solves a complex financial optimization problem using a novel mathematical technique.

New calculus for invariant differential operators in parabolic geometries.

We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types…

We introduce the notion of Ricci-corrected differentiation in parabolic geometry, which is a modification of covariant differentiation with better transformation properties. This enables us to simplify the explicit formulae for standard invariant operators given in work of Cap, Slovak and Soucek, and at the same time e…

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

We present an analytic approach to solve a degenerate parabolic problem associated to the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second…

In this thesis, we show the existence of a sequence of differential operators starting with with the Dirac operator in k Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,§)\to C^\infty((\R^n)^k,§)$ ($§$ is the spinor module). This operator is the Cauchy-Riemann operato…

This is the second part in a series of two papers. The $k$-Dirac complex is a complex of differential operators which are natural to a particular $|2|$-graded parabolic geometry. In this paper we will consider the $k$-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove …

The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi-Yau manifolds. This…

Study first BGG operators on homogeneous geometries.

The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential ope…

The paper proves gap results for self-shrinkers in $r$-mean curvature flow.

We apply the Cartan-Kahler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for k = 2 the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-D…

Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current ar…

The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,α}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.

The paper proves Schauder estimates for Laplace-Beltrami on manifolds with fibered boundaries.

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, http://dx.doi.org/10.1007/s00031-01…

New conditions for parabolicity and hyperbolicity of conductive Riemannian manifolds established.

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula co…

Using an algebraic Fourier transform of operators, we develop a method (F-method) to obtain explicit highest weight vectors in the branching laws by differential equations. This article gives a brief explanation of the F-method and its applications to a concrete construction of some natural equivariant operators that a…

We consider the problem of recovering material parameters in a transversely isotropic medium from the qP and qSV waves' travel times, given the axis of isotropy and the material parameters associated to the qSH wave speed. The operators obtained from the pseudolinearization argument are of parabolic type, and so we dis…

Study asymptotically almost periodic solutions on real hyperbolic manifolds.

The article constructs differential operators for parabolic geometries.

We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of "elliptic operators" as defined by Y. Colin de Verdiere. For such operators, we discuss analogs of inequalities of C…

For a real or complex semisimple Lie group $G$ and two nested parabolic subgroups $Q\subset P\subset G$, we study parabolic geometries of type $(G,Q)$. Associated to the group $P$, we introduce a class of relative natural bundles and relative tractor bundles and construct some basic invariant differential operators on …

We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(Δ+q)$ on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of such functions is 1-dimensional. We obtain consequences for orientable, complete stable…

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities for the positive solutions to the Schödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequali…

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handl…

Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.

We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extend…

We construct new families of conformally invariant differential operators acting on densities. We introduce a simple, direct approach which shows that all such operators arise via this construction when the degree is bounded by the dimension. The method relies on a study of well-known transformation laws and on Weyl's …

For the geometry of oriented $(2, 3, 5)$ distributions $(M, {\mathbf D})$, which correspond to regular, normal parabolic geometries of type $(\mathrm{G}_2, P)$ for a particular parabolic subgroup $P < \mathrm{G}_2$, we develop the corresponding tractor calculus and use it to analyze the first BGG operator $Θ_0$ associa…

In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the composition series and branching problems for objects in the Bernstein-Gelfand-Gelfand pa…

We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\ma…