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Alexander Engel

Publications and source records attributed to Alexander Engel.

36 records · Page 2Linked to original sources
arXiv2020

Homotopy theory with bornological coarse spaces

We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the classification of coarse homology theories and the construction of examples. We show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The example of coarse K-homology will be discussed in detail.

math.AT↗
arXiv2019

Quantum Algorithm for the Vlasov Equation

The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $\mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $\mathcal{O}(\text{polylog}(N_v) t/δ)$ where $δ$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/δ$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.

quant-ph↗
arXiv2019

Polynomially weighted $\ell^p$-completions and group homology

We introduce polynomially weighted $\ell^p$-norms on the bar complex of a finitely generated group. We prove that, for groups of polynomial or exponential growth, the homology of the completed complex does not depend on the value of $p$ in the range $(1,\infty)$.

math.GR↗
arXiv2019

Wrong way maps in uniformly finite homology and homology of groups

Given a non-compact Riemannian manifold M and a submanifold N of codimension q, we will construct under certain assumptions on both M and N a wrong way map in uniformly finite homology. Using an equivariant version of the construction and applying it to universal covers, we will construct wrong way maps in homology of groups. As applications we discuss obstructions against positive scalar curvature metrics and against inessentialness.

math.GT↗
arXiv2018

Homotopy theory with marked additive categories

We construct combinatorial model category structures on the categories of (marked) categories and (marked) pre-additive categories, and we characterize (marked) additive categories as fibrant objects in a Bousfield localization of pre-additive categories. These model category structures are used to present the corresponding $\infty$-categories obtained by inverting equivalences. We apply these results to explicitly calculate various limits and colimits in these $\infty$-categories.

math.AT↗
arXiv2018

Burghelea conjecture and asymptotic dimension of groups

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum-Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counter-example was known. We construct a finitely presentable (even type $F_\infty$) counter-example based on Thompson's group F. We construct as well a finitely generated counter-example with finite decomposition complexity.

math.GT↗
arXiv2018

Index theory of uniform pseudodifferential operators

We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow as a corollary from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the uniform estimates that are present in the definition of our class of pseudodifferential operators which is more general than similar classes defined by other authors. We will revisit Spakula's uniform K-homology and show that multigraded elliptic uniform pseudodifferential operators naturally define classes in it. For this we will investigate uniform K-homology more closely, e.g., construct the external product and show invariance under weak homotopies. The latter will be used to refine and extend Spakula's results about the rough Baum-Connes assembly map. We will identify the dual theory of uniform K-homology. We will give a simple definition of uniform K-theory for all metric spaces and in the case of manifolds of bounded geometry we will give an interpretation of it via vector bundles of bounded geometry. Using a version of Mayer-Vietoris induction that is adapted to our needs, we will prove Poincare duality between uniform K-theory and uniform K-homology for spin-c manifolds of bounded geometry. We will construct Chern characters from uniform K-theory to bounded de Rham cohomology and from uniform K-homology to uniform de Rham homology. Using the adapted Mayer-Vietoris induction we will also show that these Chern characters induce isomorphisms modulo torsion.

math.DG↗
arXiv2018

Equivariant coarse homotopy theory and coarse algebraic $\boldsymbol{K}$-homology

We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the category of equivariant coarse motivic spectra. As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic $K$-homology. Moreover, we discuss the cone functor, its relation with equivariant homology theories in equivariant topology, and assembly and forget-control maps. This is a preparation for applications in subsequent papers aiming at split-injectivity results for the Farrell-Jones assembly map.

math.KT↗
arXiv2018

Transfers in coarse homology

We enlarge the category of bornological coarse spaces by adding transfer morphisms and introduce the notion of an equivariant coarse homology theory with transfers. We then show that equivariant coarse algebraic $K$-homology and equivariant coarse ordinary homology can be extended to equivariant coarse homology theories with transfers. In the case of a finite group we observe that equivariant coarse homology theories with transfers provide Mackey functors. We express standard constructions with Mackey functors in terms of coarse geometry, and we demonstrate the usage of transfers in order to prove injectivity results about assembly maps.

math.KT↗
arXiv2018

Uniform K-theory, and Poincare duality for uniform K-homology

We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K-homology and prove Poincare duality between uniform K-theory and uniform K-homology on spin-c manifolds of bounded geometry.

math.KT↗
arXiv2018

Index theorems for uniformly elliptic operators

We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. The generalization will follow from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the uniform estimates present in the definition of uniform pseudodifferential operators.

math.DG↗
arXiv2018

Rough index theory on spaces of polynomial growth and contractibility

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such manifolds, and we will furthermore get higher-codimensional index obstructions to metrics of positive scalar curvature on closed manifolds with virtually nilpotent fundamental groups. We will give a computation of the homology of (a dense, smooth subalgebra of) the uniform Roe algebra of manifolds of polynomial volume growth.

math.DG↗
arXiv2014

Indices of pseudodifferential operators on open manifolds

We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-homology and show that our elliptic pseudodifferential operators naturally define classes there. Furthermore, we use the uniform coarse assembly map to relate these classes to the index classes of these operators in the K-theory of the uniform Roe algebra. Our investigation of uniform K-homology goes on with constructing the external product for it and deducing homotopy invariance. The next major result is the identification of the dual theory of uniform K-homology: uniform K-theory. We give a simple definition of uniform K-theory for all metric spaces and in the case of manifolds of bounded geometry we give an interpretation via vector bundles of bounded geometry over the manifold. This opens up the door for Chern-Weil theory and we define a Chern character map from uniform K-theory of a manifold to its bounded de Rham cohomology. We introduce a type of Mayer-Vietoris argument for these uniform (co-)homology theories which enables us to show firstly, that the Chern character induces an isomorphism modulo torsion, and secondly, that we have Poincare duality between uniform K-theory and uniform K-homology if the manifold is spin-c. Poincare duality together with the relation of uniform K-homology to the index theorem of Roe mentioned above directly leads to a generalization of the index theorem to elliptic pseudodiffential operators. Finally, using homotopy invariance of uniform K-homology we derive important results about the uniform coarse Baum-Connes conjecture establishing it equally important as the usual coarse Baum-Connes conjecture.

math.DG↗
arXiv2013

Isospectral Alexandrov Spaces

We construct the first non-trivial examples of compact non-isometric Alexandrov spaces which are isospectral with respect to the Laplacian and not isometric to Riemannian orbifolds. This construction generalizes independent earlier results by the authors based on Schueth's version of the torus method.

math.DG↗
arXiv2013

Isospectral Submersion Metrics

We construct continuous families of pairwise isospectral metrics on various Riemannian manifolds (e.g., Lie groups, projective spaces and products of these with tori) which arise as quotients of other manifolds. This is done by developing a general principle which guarantees that the torus method can be used to simultaneously construct isospectral metrics on a manifold and a quotient of it. Furthermore, a suffient condition will be given such that the constructed metrics are submersion metrics.

math.DG↗