arXiv subjects

Guoliang Yu

Publications and source records attributed to Guoliang Yu.

At least 37 records · Page 2Linked to original sources

arXiv2023

The UCT for $C^*$-algebras with finite complexity

A $C^*$-algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's $KK$-theory to a commutative $C^*$-algebra. This paper is motivated by the problem of establishing the range of validity of the UCT, and in particular, whether the UCT holds for all nuclear $C^*$-algebras. We introduce the idea of a $C^*$-algebra that "decomposes" over a class $\mathcal{C}$ of $C^*$-algebras. Roughly, this means that locally, there are approximately central elements that approximately cut the $C^*$-algebra into two $C^*$-subalgebras from $\mathcal{C}$ that have well-behaved intersection. We show that if a $C^*$-algebra decomposes over the class of nuclear, UCT $C^*$-algebras, then it satisfies the UCT. The argument is based on controlled $KK$-theory, as introduced by the authors in earlier work. Nuclearity is used via Kasparov's Hilbert module version of Voiculescu's theorem, and Haagerup's theorem that nuclear $C^*$-algebras are amenable We say that a $C^*$-algebra has finite complexity if it is in the smallest class of $C^*$-algebras containing the finite-dimensional $C^*$-algebras, and closed under decomposability; our main result implies that all $C^*$-algebras in this class satisfy the UCT. The class of $C^*$-algebras with finite complexity is large, and comes with an ordinal-number invariant measuring the complexity level. We conjecture that a $C^*$-algebra of finite nuclear dimension and real rank zero has finite complexity; this (and several other related conjectures) would imply the UCT for all separable nuclear $C^*$-algebras. We also give new local formulations of the UCT, and some other necessary and sufficient conditions for the UCT to hold for all nuclear $C^*$-algebras.

math.OA↗

arXiv2023

A proof of Gromov's cube inequality on scalar curvature

Gromov proved a cube inequality on the bound of distances between opposite faces of a cube equipped with a positive scalar curvature metric in dimension $\leq 8$ using minimal surface method. He conjectured that the cube inequality also holds in dimension $\geq 9$. In this paper, we prove Gromov's cube inequality in all dimensions with the optimal constant via Dirac operator method. In fact, our proof yields a strengthened version of Gromov's cube inequality, which does not seem to be accessible by minimal surface method.

math.DG↗

arXiv2023

Unconventional Ferroelectricity in Violation with Neumann's Principle

The physical properties of crystals are governed by their symmetry according to Neumann's principle. However, we present a case that contradicts this principle wherein the polarization is not invariant under its symmetry. We term this phenomenon as unconventional ferroelectricity in violation of Neumann's principle (UFVNP). Our group theory analysis reveals that 33 symmorphic space groups have the potential for UFVNP, with 26 of these symmorphic space groups belonging to non-polar groups. Notably, the polarization component in UFVNP materials is quantized. Our theory can explain the experimentally proven in-plane polarization of the monolayer α-In2Se3, which has C3v symmetry. Additionally, we employ first-principles calculations to demonstrate the existence of UFVNP in Td phase AgBr, which was not initially anticipated to exhibit polarization. Thus, UFVNP plays an integral role in characterizing and exploring the possible applications of ferroelectrics, significantly expanding the range of available materials for study.

cond-mat.mtrl-sci↗

arXiv2023

Bilayer Stacking Ferrovalley without Breaking Time-Reversal Symmetry

Non-volatile manipulation of valley polarization in solids has long been desired for valleytronics applications but remains challenging. Here, we propose a novel strategy for non-volatile manipulating valleys through bilayer stacking, which enables spontaneous valley polarization without breaking time-reversal symmetry. We call this noval physics as bilayer stacking ferrovalley (BSFV). The group theory analysis reveals that the two-dimensional (2D) valley materials with hexagonal and square lattices can host BSFV. By searching the 2D material database, we discovered 14 monolayer 2D materials with direct gaps that are candidates for realizing BSFV. Further first-principles calculations demonstrate that BSFV exists in RhCl3 and InI bilayers. The bilayer stacking breaks their three- and four-fold rotation symmetry, resulting in 39 and 326 meV valley polarization, respectively. More interestingly, the valley polarization in our systems can be switched by interlayer sliding. Our study opens up a new direction for designing ferrovalley materials and thus greatly enriches the platform for the research of valleytronics.

cond-mat.mtrl-sci↗

arXiv2023

The coarse Baum-Connes conjecture for certain relative expanders

Let $\left( 1\to N_m\to G_m\to Q_m\to 1 \right)_{m\in \mathbb{N}}$ be a sequence of extensions of finite groups such that their coarse disjoint unions have bounded geometry. In this paper, we show that if the coarse disjoint unions of $\left( N_m \right)_{m\in \mathbb{N}} $ and $\left( Q_m \right)_{m\in \mathbb{N}} $ are coarsely embeddable into Hilbert space, then the coarse Baum-Connes conjecture holds for the coarse disjoint union of $\left( G_m \right)_{m\in \mathbb{N}}$. As an application, the coarse Baum-Connes conjecture holds for the relative expanders constructed by G. Arzhantseva and R. Tessera, and the special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This enlarges the class of metric spaces known to satisfy the coarse Baum-Connes conjecture. In particular, it solves an open problem raised by G. Arzhantseva and R. Tessera on the coarse Baum-Connes conjecture for relative expanders.

math.KT↗

arXiv2023

$K$-theory of relative group $C^*$-algebras and the relative Novikov conjecture

The relative Novikov conjecture states that the relative higher signatures of manifolds with boundary are invariant under orientation-preserving homotopy equivalences of pairs. In this paper, we study the relative Baum-Connes assembly map for any pair of groups and apply it to solve the relative Novikov conjecture when the groups satisfy certain geometric conditions.

math.OA↗

arXiv2023

On Gromov's dihedral extremality and rigidity conjectures

In this paper, we develop a new index theory for manifolds with polyhedral boundary. As an application, we prove Gromov's dihedral extremality conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles between two compact manifolds with polyhedral boundary in all dimensions. We also prove Gromov's dihedral rigidity conjecture for a class of positively curved manifolds with polyhedral boundary in all dimensions.

math.DG↗

arXiv2023

Covering complexity, scalar curvature, and quantitative $K$-theory

We establish a relationship between a certain notion of covering complexity of a Riemannian spin manifold and positive lower bounds on its scalar curvature. This makes use of a pairing between quantitative operator $K$-theory and Lipschitz topological $K$-theory, combined with an earlier vanishing theorem for the quantitative higher index.

math.KT↗

arXiv2023

On Gromov's compactness question regarding positive scalar curvature

In this paper, we give both positive and negative answers to Gromov's compactness question regarding positive scalar curvature metrics on noncompact manifolds. First we construct examples that give a negative answer to Gromov's compactness question. These examples are based on the non-vanishing of certain index theoretic invariants that arise at the infinity of the given underlying manifold. This is a $ \sideset{}{^1}\varprojlim$ phenomenon and naturally leads one to conjecture that Gromov's compactness question has a positive answer provided that these $ \sideset{}{^1}\varprojlim$ invariants also vanish. We prove this is indeed the case for a class of $1$-tame manifolds.

math.DG↗

arXiv2022

Dynamical complexity and controlled operator K-theory

In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the $K$-theory of the associated crossed product $C^*$-algebra by splitting it up into simpler pieces and using the methods of controlled $K$-theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. We have tried to keep the paper as self-contained as possible: we hope the main part will be accessible to someone with the equivalent of a first course in operator $K$-theory. In particular, we do not assume prior knowledge of controlled $K$-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant $K$-theory to set up.

math.KT↗

arXiv2022

Quantitative K-theory, positive scalar curvature, and band width

We develop two connections between the quantitative framework of operator $K$-theory for geometric $C^*$-algebras and the problem of positive scalar curvature. First, we introduce a quantitative notion of higher index and use it to give a refinement of the well-known obstruction of Rosenberg to positive scalar curvature on closed spin manifolds coming from the higher index of the Dirac operator. We show that on a manifold with uniformly positive scalar curvature, the propagation at which the index of the Dirac operator vanishes is related inversely to the curvature lower bound. Second, we give an approach, using related techniques, to Gromov's band width conjecture, which has been the subject of recent work by Zeidler and Cecchini from a different point of view.

math.KT↗

arXiv2022

$l^1$-higher index, $l^1$-higher rho invariant and cyclic cohomology

In this paper, we study $l^1$-higher index theory and its pairing with cyclic cohomology for both closed manifolds and compact manifolds with boundary. We first give a sufficient geometric condition for the vanishing of the $l^1$-higher indices of Dirac-type operators on closed manifolds. This leads us to define an $l^1$-version of higher rho invariants. We prove a product formula for these $l^1$-higher rho invariants. A main novelty of our product formula is that it works in the general Banach algebra setting, in particular, the $l^1$-setting. On compact spin manifolds with boundary, we also give a sufficient geometric condition for Dirac operators to have well-defined $l^1$-higher indices. More precisely, we show that, on a compact spin manifold $M$ with boundary equipped with a Riemannian metric which has product structure near the boundary, if the scalar curvature on the boundary is sufficiently large, then the $l^1$-higher index of its Dirac operator $D_M$ is well-defined and lies in the $K$-theory of the $l^1$-algebra of the fundamental group. As an immediate corollary, we see that if the Bost conjecture holds for the fundamental group of $M$, then the $C^\ast$-algebraic higher index of $D_M$ lies in the image of the Baum-Connes assembly map. By pairing the above $K$-theoretic $l^1$-index results with cyclic cocycles, we prove an $l^1$-version of the higher Atiyah-Patodi-Singer index theorem for manifolds with boundary. A key ingredient of its proof is the product formula for $l^1$-higher rho invariants mentioned above.

math.KT↗

arXiv2021

Interface engineering of ferroelectricity in thin films of thiophosphate ABP 2 X 6 (A = Cu, Ag; B = In, Bi, Cr, V; and X = S, Se)

Two-dimensional ferroelectrics (FEs) are promising in the miniaturization of memory devices with ultra-high-density data storage and low power consumption. However, many thiophosphate monolayers, i.e., analogs of CuInP$_2$S$_6$ and referred to as ABP$_2$X$_6$, lose ferroelectricity and instead exhibit an antiferroelectric (AFE) or paraelectric ordering. We propose to tune the AFE ABP$_2$X$_6$ monolayers into the FE ordering through interface engineering. The mechanism is that there are couplings between the charge polarizations of the ABP$_2$X$_6$ monolayers and the local dipoles as well as the induced electronic polarizations in the substrate which have a tendency to stabilize the FE ordering. We further perform first-principles calculations for CuInP$_2$Se$_6$ and CuCrP$_2$S$_6$ monolayers and their van der Waals heterostructures. We find that an AFE CuInP$_2$Se$_6$ monolayer becomes FE as interfaced with graphene, MoS$_2$, and h-BN monolayers. In contrast, the CuCrP$_2$S$_6$ monolayer remains AFE since there is a large energy difference between the AFE and FE phases. Interfacing it with a MoTe$_2$ monolayer induces a metal-insulator transition for the heterostructure, whereas interfacing with a polar surface MgO(111) can drive it into FE. The interfacing effect can also be used to manipulate the FE properties of ABP$_2$X$_6$ multilayers. We further find that the AFE-to-FE transition is electrically switchable in these systems. In particular, it is accompanied by an indirect-direct band-gap transition for the CuInP$_2$Se$_6$ monolayer. Our study offers an effective approach to tune the FE and electronic properties of ABP$_2$X$_6$ thin films for applications in electronics and optoelectronics.

cond-mat.mtrl-sci↗

arXiv2021

Higher rho invariant and delocalized eta invariant at infinity

In this paper, we introduce several new secondary invariants for Dirac operators on a complete Riemannian manifold with a uniform positive scalar curvature metric outside a compact set and use these secondary invariants to establish a higher index theorem for the Dirac operators. We apply our theory to study the secondary invariants for a manifold with corner with positive scalar curvature metric on each boundary face.

math.KT↗

arXiv2021

On the range of the relative higher index and the higher rho-invariant for positive scalar curvature

Let $M$ be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on $M$ forms an abelian group $P(M)$ after fixing a positive scalar curvature metric. The group $P(M)$ measures the size of the space of positive scalar curvature metrics on $M$. Weinberger and Yu gave a lower bound of the rank of $P(M)$ in terms of the number of torsion elements of $π_1(M)$. In this paper, we give a sharper lower bound of the rank of $P(M)$ by studying the image of the relative higher index map from $P(M)$ to the real K-theory of the group $\mathrm{C}^\ast$-algebra $\mathrm{C}^\ast_{\mathrm{r}}(π_1(M))$. We show that it rationally contains the image of the Baum-Connes assembly map up to a certain homological degree depending on the dimension of $M$. At the same time we obtain lower bounds for the positive scalar curvature bordism group by applying the higher rho-invariant.

math.KT↗

arXiv2021

Approximations of delocalized eta invariants by their finite analogues

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose $M$ is a closed smooth spin manifold and $\widetilde M$ is a $Γ$-regular covering space of $M$. Let $\langle α\rangle$ be the conjugacy class of a non-identity element $α\in Γ$. Suppose $\{Γ_i\}$ is a sequence of finite-index normal subgroups of $Γ$ that distinguishes $\langle α\rangle$. Let $π_{Γ_i}$ be the quotient map from $Γ$ to $Γ/Γ_i$ and $\langle π_{Γ_i}(α) \rangle$ the conjugacy class of $π_{Γ_i}(α)$ in $Γ/Γ_i$. If the scalar curvature on $M$ is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of $\widetilde M$ at the conjugacy class $\langle α\rangle$ is equal to the limit of the delocalized eta invariants for the Dirac operators of $M_{Γ_i}$ at the conjugacy class $\langle π_{Γ_i}(α) \rangle$, where $M_{Γ_i}= \widetilde M/Γ_i$ is the finite-sheeted covering space of $M$ determined by $Γ_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{Γ_i}$ at the conjugacy class $\langle π_{Γ_i}(α) \rangle$ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for $Γ$.

math.KT↗

arXiv2021

Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension

Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topological $K$-theory of finite dimensional simplicial complexes.

math.DG↗

arXiv2021

The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite\-/dimensional symmetric spaces. Thus our main theorem can be considered as an infinite-dimensional analogue of Kasparov's theorem on the Novikov conjecture for groups acting properly and isometrically on complete, simply connected and non-positively curved manifolds. As a consequence, we show that the Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This result is inspired by Connes' theorem that the Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs classes of groups of diffeormorphisms.

math.KT↗