Hyperbolicity for Log Smooth Families with Maximal Variation

We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.

3d gravity in Bondi-Weyl gauge: charges, corners, and integrability

We introduce a new gauge and solution space for three-dimensional gravity. As its name Bondi-Weyl suggests, it leads to non-trivial Weyl charges, and uses Bondi-like coordinates to allow for an arbitrary cosmological constant and therefore spacetimes which are asymptotically locally (A)dS or flat. We explain how integrability requires a choice of integrable slicing and also the introduction of a corner term. After discussing the holographic renormalization of the action and of the symplectic potential, we show that the charges are finite, symplectic and integrable, yet not conserved. We find four towers of charges forming an algebroid given by $$ \mathfrak{vir}\oplus \mathfrak{vir}\oplus $$ vir ⊕ vir ⊕ Heisenberg with three central extensions, where the base space is parametrized by the retarded time. These four charges generate diffeomorphisms of the boundary cylinder, Weyl rescalings of the boundary metric, and radial translations. We perform this study both in metric and triad variables, and use the triad to explain the covariant origin of the corner terms needed for renormalization and integrability.

Constructing new spectral systems from simplicial fibrations

In this work we present an ongoing project on the development and study of new spectral systems which combine filtrations associated to Serre and Eilenberg-Moore spectral sequences of different fibrations. Our new spectral systems are part of a new module for the Kenzo system and can be useful to deduce new relations on the initial spectral sequences and to obtain information about different filtrations of the homology groups of the fiber and the base space of the fibrations.

New types of almost contact metric submersions

We introduce the concept of conjugaison in contact geometry. This concept allows to define new structures which are used as base space of a Riemannian submersion. With these new structures, we study new three types of almost contact metric submersions.

MHD Equations in a Bounded Domain

We consider the MHD system in a bounded domain Ω ⊂ ℝ N , N = 2, 3, with Dirichlet boundary conditions. Using Dan Henry’s semigroup approach and Giga–Miyakawa estimates we construct global in time, unique solutions to fractional approximations of the MHD system in the base space (L 2(Ω)) N × (L 2(Ω)) N . Solutions to MHD system are obtained next as a limits of that fractional approximations.

Inverse monoids and immersions of Δ-complexes

An immersion [Formula: see text] between [Formula: see text]-complexes is a [Formula: see text]-map that induces injections from star sets of [Formula: see text] to star sets of [Formula: see text]. We study immersions between finite-dimensional connected [Formula: see text]-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szakács on immersions into 2-dimensional [Formula: see text]-complexes.

Right-angled Artin groups, polyhedral products and the -generating function

For a graph $\Gamma$ , let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$ . We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$ , we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$ . Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

Rigidity of the mod 2 families Seiberg–Witten invariants and topology of families of spin 4-manifolds

We show a rigidity theorem for the Seiberg–Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of $4$ -manifolds $M$ for which the inclusion maps $\operatorname {Diff}(M) \hookrightarrow \operatorname {Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin $4$ -manifolds.