arXiv research

Introduces Relational Privacy (RP) to control relation memorization in question answering models.

We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invar…

Paper discusses the Fisher metric and differentiability in statistical models.

We study two notions of relative differential cohomology, using the model of differential characters. The two notions arise from the two options to construct relative homology, either by cycles of a quotient complex or of a mapping cone complex. We discuss the relation of the two notions of relative differential cohomo…

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential in…

We show that the 2-jet bundle of local Riemannian metrics on an arbitrary differentiable manifold admits a section which pointwise fulfills the curvature relation sec(g)=a for any real number a. It follows by Gromov's h-principle for open, invariant differential relations that every noncompact differentiable manifold c…

The paper studies tautological rings of strata of differentials, proving cohomological stability and no relations in specific degrees.

Study uses graph techniques to understand meromorphic quadratic differential strata.

This paper solves a complex differential relation using a novel 'avoidance trick'.

In view of the Segal construction each category with a coherent operation gives rise to a cohomology theory. Similarly each open stable differential relation $R$ imposed on smooth maps of manifolds determines cohomology theories $k^*$ and $h^*$; the cohomology theory $k^*$ describes invariants of solutions of $R$, whil…

The relation between differential geometry of surfaces and some Heisenberg ferromagnet models is considered.

We discuss relations between the para-CR structures and differential equations (both ODEs and PDEs of finite type).

The Euclidean cone metrics coming from q-differentials on a closed surface of genus g > 1 define an equivalence relation on homotopy classes of closed curves declaring two to be equivalent if they have the equal length in every such metric. We prove an analog of the result of Randol for hyperbolic metrics (building on …

Paper proposes a novel RRL framework that learns from images and incorporates expert knowledge.

This paper bounds min-entropy leakage for Blowfish privacy using graph symmetries.

Survey talk on certain aspects of the subject, stressing the neighbor relation as a basic notion in differential geometry.

We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a gen…

Three types of equations of mathematical physics, namely, the equations, which describe any physical processes, the equations of mechanics and physics of continuous media, and field-theory equations are studied in this paper. In the first and second case the investigation is reduced to the analysis of the nonidentical …

This work establishes properties on diffeological structures for set-valued maps and measures.

Analyzes geometric structures on profinite diffeological spaces.

We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids.

Discusses geometry problems for fun and learning.

Researchers found differential invariants for Kundt spacetimes.

We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in se…

Introduces tractors for basic examples and modern differential calculus.

Review and generalize Haefliger's differentiable cohomology for diffeomorphisms and flat Cartan groupoids.

Study local commutation relation on almost complex manifolds.

This paper formalizes the h-principle and sphere eversion in differential topology.

Develops TCD maps to relate discrete differential geometry and cluster algebras.

This expository survey describes how holomorphic quadratic differentials arise in several aspects of Teichmüller theory, highlighting their relation with various geometric structures on surfaces. The final section summarizes results for non-compact surfaces of finite type, when the quadratic differential has poles of f…

We consider commensurability of quadratic differentials on surfaces. Each commensurability class has a natural order by the covering relation. We show that each commensurability class contains a unique (orbifold) element. We also discuss the relationship between commensurability of quadratic differentials and fibered c…

Recalls and refines the concept of algebraically rectifiable curves.

We discuss a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov--Zabolotskaya equation and the Boyer--Finley equation.

Proves formal self-adjointness of certain differential operators.

We construct a versal family of deformations of CR structures in five dimensions, using a differential complex closely related to the differential form complex introduced by Rumin for contact manifolds.

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold $(M,g)$ and the $L_{q,p}$-cohomology of that manifold. The $L_{q,p}$-cohomology of $(M,g)$ is defined to be the quotient of the space of closed differential forms in $L^p(M)$ modulo the exact forms which are exterior diff…

These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting f…

In this paper we study the Taylor series of an operator-valued function related to the differential of the exponential map. For a smooth manifold $\mathcal{M}$ with a torsion-free affine connection the operator $\mathcal{E}_p(v)$ acting on the space $T_p\mathcal{M}$ is defined to be the composition of the differential …

Lectures on topological field theories and differential cohomology.

The paper calculates large genus limits for quadratic differential volumes and constants.

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. …

The paper solves a 25-year-old problem about maximal growth distributions on manifolds.

The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a nove…

We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition of the corresponding coboundary rel…

Segre embedding was introduced by C. Segre (1863--1924) in his famous 1891 article \cite{segre}. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geo…

New insights into X-ray transform on hyperbolic disk, with functional relations and range characterizations.

For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite …

This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with di…