arXiv research

Defines formal exponentials for graded manifolds and linearizes QP-manifolds.

We introduce, for every $\mathbb{Z}$-graded manifold, a formal exponential map defined in a purely algebraic way and study its properties. As an application, we give a simple new construction of a Fedosov type resolution of the algebra of smooth functions of $\mathbb{Z}$-graded manifolds and we extend the Emmrich--Wein…

The paper explores connections between dg manifolds and homotopy Lie algebras.

We recall the construction of non-formal deformation quantization of the Poincare Group ISO(1,1) on its coadjoint orbit and exhibit the associated non-formal star-exponentials.

A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle $TX$ of a complex manifold $X$ makes the shifted tangent bundle $TX[-1]$ into a Lie algebra object in the derived category $D(X)$. Moreover, he showed that there is an $L_\infty$-algebra structure on the Dolbeault resolution of $TX[-1]$…

Research covers geometry, analysis, and integration on infinite-dimensional spaces.

Deform quantization recovers scalar curvature in complex structures.

Study the exponential map on surfaces using fluid dynamics.

We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the lo…

Study on stability of harmonic maps with sub-Riemannian geometry.

Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group provides a precise mathematical framework for lifted inference in the general expone…

We prove that for a fibration of simply-connected spaces of finite type $F\hookrightarrow E\to B$ with $F$ being positively elliptic and $H^*(F,\qq)$ not possessing non-trivial derivations of negative degree, the base $B$ is formal if and only if the total space $E$ is formal. Moreover, in this case the fibration map i…

Proves properties of sub-Riemannian exponential map, showing it's not injective.

Deform moment map on symplectic connections using star product algebras.

Develops a new exponential map for time-varying vector fields.

Analogous exponential map defined for Hopf algebras.

Exponential rate of convergence for harmonic heat flow maps.

B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flatte…

In this paper we introduce a new type of exponential map in semi-simple compact Lie groups, which is related to the sub-Riemannian geometry generated by the orthogonal complement of a Cartan subalgebra in a similar way to how the group exponential map is related to the Riemannian geometry.

Exponential proportion of pseudo-Anosovs in mapping class groups.

There are solved standard problems related to Formal (Holomorphic) Segre preserving Mappings of non-trivial Real-Formal Hypersurfaces in $\mathbb{C}^{2}$.

The exponential map fails to be injective near critical points in sub-Riemannian geometry.

Paper develops a new algorithm to find shortest paths on surfaces.

We show that the mapping class group of an orientable finite type surface has uniformly exponential growth, as well as various closely related groups. This provides further evidence that mapping class groups may be linear.

Develops a lifting theory for exponential maps in semi-Riemannian geometry.

Researchers improve NCE by addressing its flat loss landscape issues.

This paper summarizes closed-form relations for SE(3) maps and their derivatives.

We prove that the Riemannian exponential map of the right-invariant $L^2$ metric on the group of volume-preserving diffeomorphisms of a two-dimensional manifold with a nonempty boundary is a nonlinear Fredholm map of index zero.

Extended logarithm for solvable elements in mapping class groups.

New non-Kähler examples of generalized Kähler manifolds constructed via mapping tori.

We introduce Markov substitute processes, a new model at the crossroad of statistics and formal grammars, and prove its main property : Markov substitute processes with a given support form an exponential family.

We show that the probability that a finitely supported random walk on a non-elementary subgroup of the the mapping class group gives a non-pseudo-Anosov element decays exponentially in the length of the random walk. More generally, we show that if R is a set of mapping class group elements with an upper bound on their …

We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension $m$ and with first Betti n…

The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.

The MAP estimate's log-likelihood sub-optimality is hard to bound in general.

We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup $H$. We further assume that $H$ is not consisting only of lifts with respect to any one covering. Then w…

We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details

Formality of Dolbeault DGAs on complex nilmanifolds restricted to tori.

Study shows how to section map between holonomic and formal solutions.

Geodesic flow on submanifolds is shown to be $C^{k-1}$.

We develop a new method to price SOFR futures contracts considering convexity, skew, and smile.

Herbert Gr{ö}tzsch is the main founder of the theory of quasicon-formal mappings. We review five of his papers, written between 1928 and 1932, that show the progress of his work from conformal to quasiconformal geometry. This will give an idea of his motivation for introducing quasicon-formal mappings, of the problems …

Shows Euler-like vector fields come from specific embeddings.

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 …

In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of the equivariant multidifferential operators and the complex of equivaria…

Given a pseudo-Riemannian metric of regularity $C^{1,1}$ on a smooth manifold, we prove that the corresponding exponential map is a bi-Lipschitz homeomorphism locally around any point. We also establish the existence of totally normal neighborhoods in an appropriate sense. The proofs are based on regularization, combin…

In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of ba…