Let $\eta $ be [-11pc] [-7pc]a closed real 1-form on a closed Riemannian n-manifold $(M,g)$. Let $d_z$, $\delta _z$ and $\Delta _z$ be the induced Witten’s type perturbations of the de Rham derivative and coderivative and the Laplacian, parametrized by $z=\mu +i\nu \in \mathbb C$ ($\mu ,\nu \in \mathbb {R}$, $i=\sqrt {-1}$). Let $\zeta (s,z)$ be the zeta function of $s\in \mathbb {C}$, defined as the meromorphic extension of the function $\zeta (s,z)=\operatorname {Str}({\eta \wedge }\,\delta _z\Delta _z^{-s})$ for $\Re s\gg 0$. We prove that $\zeta (s,z)$ is smooth at $s=1$ and establish a formula for $\zeta (1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $\zeta (1,z)$ converges to some $\mathbf {z}\in \mathbb {R}$ as $\mu \to +\infty $, uniformly on $\nu $. We describe $\mathbf {z}$ in terms of the instantons of an auxiliary Smale gradient-like vector field X and the Mathai–Quillen current on $TM$ defined by g. Any real 1-cohomology class has a representative $\eta $ satisfying the hypothesis. If n is even, we can prescribe any real value for $\mathbf {z}$ by perturbing g, $\eta $ and X and achieve the same limit as $\mu \to -\infty $. This is used to define and describe certain tempered distributions induced by g and $\eta $. These distributions appear in another publication as contributions from the preserved leaves in a trace formula for simple foliated flows, giving a solution to a problem stated by C. Deninger.

In Communications in Contemporary Mathematics 24 3, (2022),the authors have developed a method for constructing G-invariant partial differential equations (PDEs) imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie group G. In the present paper, the method is applied to the case when $G=\mathsf{PGL}(n+1)$ (respectively, $G=\mathsf{Aff}(n+1)$) and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ (respectively, affine $\mathbb{A}^{n+1}$) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in n variables with respect to the group $\mathsf{CO}(d,n-d)$ of conformal transformations of $\mathbb{R}^{d,n-d}$.

Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-preserving B-series method is the exact flow of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this work, we introduce a new algebraic tool, called the aromatic bicomplex, similar to the variational bicomplex in variational calculus. We prove the exactness of this bicomplex and use it to describe explicitly the key object in the study of volume-preserving integrators: the aromatic forms of vanishing divergence. The analysis provides us with a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator. In particular, we conclude that an aromatic Runge–Kutta method cannot preserve volume.

Let ${\mathcal {D}}$ and $T$ be, respectively, a $C^1$ distribution of $k$-planes and a normal $k$-current on ${\mathbb {R}}^n$. Then ${\mathcal {D}}$ has to be involutive at almost every superdensity point of the tangency set of $T$ with respect to ${\mathcal {D}}$.

Since the celebrated work by Cartan, distributions with small growth vector $(2,3,5)$ have been studied extensively. In the holomorphic setting, there is a natural correspondence between holomorphic $(2,3,5)$-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5. We generalize this correspondence to higher dimensions by studying nondegenerate lines on holomorphic contact manifolds and the corresponding class of distributions of small growth vector $(2m, 3m, 3m+2)$ for any positive integer m.

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable $C^{\infty }$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable $C^{\infty }$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.

Abstract prepared by Rodrigo Figueiredo.

E-mail: rodrigo@ime.usp.br

URL: https://doi.org/10.11606/T.45.2019.tde-28042019-181150

Let $\Delta $ denote a nondegenerate k-simplex in $\mathbb {R}^k$. The set $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices in $\mathbb {R}^k$ similar to $\Delta $ is diffeomorphic to $\operatorname {O}(k)\times [0,\infty )\times \mathbb {R}^k$, where the factor in $\operatorname {O}(k)$ is a matrix called the pose. Among $(k-1)$-spheres smoothly embedded in $\mathbb {R}^k$ and isotopic to the identity, there is a dense family of spheres, for which the subset of $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in \operatorname {O}(k)$. Further, the intersection of $\operatorname {\mathrm {Sim}}(\Delta )$ with the configuration space of $k+1$ distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in $\operatorname {O}(k)$ via the pose map. This gives a high-dimensional generalisation of classical results on inscribing families of triangles in plane curves. We use techniques established in our previous paper on the square-peg problem where we viewed inscribed simplices in spheres as transverse intersections of submanifolds of compactified configuration spaces.

Using a certain well-posed ODE problem introduced by Shilnikov in the sixties, Minervini proved the currential “fundamental Morse equation” of Harvey–Lawson but without the restrictive tameness condition for Morse gradient flows. Here, we construct local resolutions for the flow of a section of a fiber bundle endowed with a vertical vector field which is of Morse gradient type in every fiber in order to remove the tameness hypothesis from the currential homotopy formula proved by the first author. We apply this to produce currential deformations of odd degree closed forms naturally associated to any hermitian vector bundle endowed with a unitary endomorphism and metric compatible connection. A transgression formula involving smooth forms on a classifying space for odd K-theory is also given.

In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

Green–Griffiths–Kerr introduced Hodge representations to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford–Tate subdomains. We summarize how, given a fixed period domain $ \mathcal{D} $ , to enumerate the Hodge representations and corresponding Mumford–Tate subdomains $ D \subset \mathcal{D} $ . The procedure is illustrated in two examples: (i) weight two Hodge structures with $ {p}_g={h}^{2,0}=2 $ ; and (ii) weight three CY-type Hodge structures.

In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$, where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$, which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$. In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

As was shown by a part of the authors, for a given $(2,3,5)$-distribution $D$ on a five-dimensional manifold $Y$, there is, locally, a Lagrangian cone structure $C$ on another five-dimensional manifold $X$ which consists of abnormal or singular paths of $(Y,D)$. We give a characterization of the class of Lagrangian cone structures corresponding to $(2,3,5)$-distributions. Thus, we complete the duality between $(2,3,5)$-distributions and Lagrangian cone structures via pseudo-product structures of type $G_{2}$. A local example of nonflat perturbations of the global model of flat Lagrangian cone structure which corresponds to $(2,3,5)$-distributions is given.

This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$-principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

We show that derivations of the differential structure of a subcartesian space satisfy the chain rule and have maximal integral curves.

For a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

This paper deals with the analysis of the singularities arising from the solutions of the problem ${-}\,{\rm Curl\ } F=\mu $, where F is a 3 × 3 matrix-valued Lp-function ($1\les p<2$) and μ a 3 × 3 matrix-valued Radon measure concentrated in a closed loop in Ω ⊂ ℝ3, or in a network of such loops (as, for instance, dislocation clusters as observed in single crystals). In particular, we study the topological nature of such dislocation singularities. It is shown that $F=\nabla u$, the absolutely continuous part of the distributional gradient Du of a vector-valued function u of special bounded variation. Furthermore, u can also be seen as a multi-valued field, that is, can be redefined with values in the three-dimensional flat torus 𝕋3 and hence is Sobolev-regular away from the singular loops. We then analyse the graphs of such maps represented as currents in Ω × 𝕋3 and show that their boundaries can be written in term of the measure μ. Readapting some well-known results for Cartesian currents, we recover closure and compactness properties of the class of maps with bounded curl concentrated on dislocation networks. In the spirit of previous work, we finally give some examples of variational problems where such results provide existence of solutions.

We prove that, for conelike stratified diffeological spaces, a zero-perverse form is the restriction of a global differential form if and only if its index is equal to one for every stratum.

First, we extend the notion of stratified spaces to diffeology. Then we characterise the subspace of stratified differential forms, or zero-perverse forms in the sense of Goresky–MacPherson, which can be extended smoothly into differential forms on the whole space. For that we introduce an index which outlines the behaviour of the perverse forms on the neighbourhood of the singular strata.