arXiv research

Simplicial sets deformation retract onto transverse simplices.

We introduce a combinatorial energy for maps of triangulated surfaces with simplicial metrics and analyze the existence and uniqueness properties of the corresponding harmonic maps. We show that some important applications of smooth harmonic maps can be obtained in this setting.

Smooth structures on diffeological spaces and sheaves, resolving conjectures.

Constructs a universal Chern-Weil map for infinite dimensional Lie groups.

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed conn…

Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each …

We present two approaches to constructing an integration map for smooth Deligne cohomology. The first is defined in the simplicial model, where a class in Deligne cohomology is represented by a simplicial form, and the second in a related but more combinatorial model.

We show that the integral foliated simplicial volume of a connected compact oriented smooth manifold with a regular foliation by circles vanishes.

We describe an $A_\infty$-quasi-equivalence of dg-categories between the first authors' $\mathcal{P}_{\mathcal{A}}$ ---the category of category of prefect $A^0$-modules with flat $\Z$-connection, corresponding to the de Rham dga $\mathcal{A}$ of a compact manifold $M$--- and the dg-category of \emph{infinity-local syst…

Study shows simplicial volume of certain fiber bundles is zero.

Parallel transport in a fibre bundle with respect to smooth paths in the base space B have recently been extended to representations of the smooth singular simplicial set Sing_{smooth}(B). Inspired by these extensions,I revisit the development of a notion of `parallel' transport in the topological setting of fibrations…

We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebr…

Localizes smooth spaces to study their homotopy properties.

We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification initiated by Kontsevich. Our constructions are elementary a…

Finite simplicial complexes dominate certain manifolds with a bounded number of simplices.

Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.

Unified probabilistic foundation for fuzzy simplicial sets in dimensionality reduction.

A class of 3d $\mathcal{N}=2$ supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class…

New techniques prove bounded acyclicity results for semi-simplicial sets.

Develops combinatorial theory of vector bundles on simplicial complexes.

Discrete exterior calculus shows natural properties of wedge product and averaging.

Derived differential manifolds are constructed using the usual homotopy theory of simplicial rings of smooth functions. They are proved to be equivalent to derived differential manifolds of finite type, constructed using homotopy sheaves of homotopy rings (D.Spivak), thus preserving the classical cobordism ring. This r…

We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the mul…

Local Kan conditions enable differentiation of simplicial manifolds.

This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to students familiar with just the fundamentals of algebraic topology.

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular,…

Study shows range of simplicial volumes for open manifolds.

Alternative proof of simplicial volume bound using area-minimizing sets.

We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of extensions treat the equivariant and the nonequivariant case in the same setting. …

A C-infinity ring is a set equipped with n-ary operations corresponding to smooth n-ary functions on the real line (satisfying natural axioms). We prove that the cosimplicial abelian group associated to the de Rham complex of Euclidean space has the structure of a cosimplicial C-infinity ring. We also analyse the notio…

We introduce a smooth quadratic conformal functional and its weighted version $W_2=\sum_e β^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)β^2(e),$ where $β(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of the mesh sharing the edge $e=(ij)$ and $n_i$ is the valence of vertex $i$. Besides minimizing…

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying m…

We consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal entropy, for certain classes of 3-manifolds. Among other resulsts, we show that if M is a closed, orientable, geometrizable 3-manifold with zero simplicial volume, then the minimal entropy can be solved for M…

We compute the rational cohomology of the universal family of smooth cubic surfaces using Vassiliev's method of simplicial resolution. Modulo embedding, the universal family has cohomology isomorphic to that of $\mathbb{P}^2$. A consequence of our theorem is that over the finite field $\mathbb{F}_q$, away from finitely…

Extends circle pattern theorem to quasi-simplicial triangulations.

A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …

Study simplicial volume of manifolds from reflection group trick.

This work is motivated by two problems: 1) The approach of manifolds and spaces by triangulations. 2) The complexity growth in sequences of polyhedra. Considering both problems as related, new criteria and methods for approximating smooth manifolds are deduced. When the sequences of polyhedra are obtained by the action…

We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants on smooth surfaces with smooth boundaries. The differences to the smooth case a…

For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (i.e. geometric realization of its si…

Constructs Serre spectral sequence for bounded cohomology.

A new definition for vector fields extends the Jacobi set concept.

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in $\mathbb{R}_{\geq 0}$. In dimension 4 we prove that every non-negat…

Geometric models for representations up to homotopy using simplicial vector bundles.

New $f$-vectors reveal geometric Lefschetz-like decompositions of flag spheres.

We provide an action for gauge theories discretized on simplicial meshes, inspired by finite element methods. The action is discretely gauge invariant and we give a proof of consistency. A discrete Noether's theorem that can be applied to our setting, is also proved.

Branched covers are applied frequently in topology - most prominently in the construction of closed oriented PL d-manifolds. In particular, strong bounds for the number of sheets and the topology of the branching set are known for dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain a simplic…

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in 1982. In order to study its main properties, Gromov himself initiated the dual theory of bounded cohomology, that developed into an active and independent research field. Gromov's theory of bounded cohomology was based on the use of mult…