Algebraic K-theory of Fredholm modules and KK-theory

Bounded and unbounded Fredholm modules for quantum projective spaces

We construct explicit generators of the K-theory and K-homology of the coordinate algebra of `functions' on quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and spectral triples of any positive real dimension.

建立在 $n$-立方体上的自相似集的组合 Fredholm 模

A combinatorial Fredholm module on self-similar sets built on $n$-cubes

本文在 Cantor 尘、Sierpinski 地毯等自相似集上构造了高维 Fredholm 模，推广了 Connes 的组合构造，并计算了其诱导算子的 Dixmier 迹。

Hilbert C*-模的 Fredholm 复形

Fredholm complexes of Hilbert C*-modules

本文研究以正则算子为微分的 Hilbert C*-模上链复形，刻画 Fredholm 性质，定义 K-理论值指标并证明其扰动稳定性及 Hodge 分解下的表达式。

Finitely summable Fredholm modules for boundary actions of hyperbolic groups

We construct a family of odd, finitely summable Fredholm modules over the crossed product C*-algebra $C(\bd \G)\rtimes \G$ associated to the action of a non-elementary hyperbolic group $\G$ on its Gromov boundary $\bd \G$. These Fredholm modules all represent the same, distinguished class in K-homology, namely that of the `boundary extension' of $C(\bd \G)\rtimes \G$ associated to the Gromov compactification of $\G$, and is typically nonzero. Their summability is closely related to the Hausdorff dimension of the boundary. We use these results to compute the Connes-Chern character of the bounda…

Fredholm modules over certain group C*-algebras

Motivated by the search for new examples of "noncommutative manifolds", we study the noncommutative geometry of the group C*-algebras of various discrete groups. The examples we consier are the infinite dihedral group ${\bf Z} \times_σ {\bf Z}_2$ and the semidirect product ${\bf Z} \times_σ {\bf Z}$. We present a unified treatment of the K-homology and cyclic cohomology of these algebras.

Fredholm Modules for Quantum Euclidean Spheres

The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres $S_q^{N-1}$. We also construct the corresponding Chern characters in cyclic homology and cohomology and com…

On finitely summable Fredholm modules from Smale spaces

We prove that all K-homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K-d…

Equivariant Fredholm modules for the full quantum flag manifold of $SU_q(3)$

We introduce $C^*$-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct $SL_q(3,\mathbb{C})$-equivariant Fredholm modules for the full quantum flag manifold $X_q = SU_q(3)/T$ of $SU_q(3)$, based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold $ X_q $ satisfies Poincaré duality in equivariant $ KK $-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to $SU_q(3)$.

Spectral flow in Fredholm modules, eta invariants and the JLO cocycle

We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$. The framework is that of {\it odd unbounded} $θ$-{\it summable} {\it Breuer-Fredholm modules} for a unital Banach *-algebra, $\mathcal A$. In the type $II_{\infty}$ case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known. We borrow Ezra Getzler's idea (suggested by I. M. Singer) of considering spectral flow (and eta i…

The Chern Character of θ-summable Fredholm Modules over dg Algebras and Localization on Loop Space

We introduce the notion of a {\vartheta}-summable Fredholm module over a locally convex dg algebra Ω and construct its Chern character as a cocycle on the entire cyclic complex of Ω, extending the construction of Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this Chern character, we prove an index theorem involving an abstract version of a Bismut-Chern character constructed by Getzler, Jones and Petrack in the context of loop spaces. Our theory leads to a rigorous construction of the path integral for N=1/2 supersymmetry which satisfies a Duistermaat-Heckman type lo…

实Hilbert $C^*$-模中的解析指标理论与谱流

Analytic index theory and spectral flow in real Hilbert $C^*$-modules

本文在实Hilbert $C^*$-模中定义并研究Clifford反线性斜伴随Fredholm算子和自伴奇Fredholm算子的解析指标与谱流，利用Van Daele $K$-理论统一处理八种实$K$-群，并证明谱流的Robbin-Salamon定理。

Spectral triples and finite summability on Cuntz-Krieger algebras

We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $θ$-summable although their bounded transform, when constructed using the sign-function, will already on the level of analytic $K$-cycles …

Algebraic K-theory of finitely projective modules on $\mathbb{E}_{\infty}$-rings

In this paper, we study the K-theory on higher modules in spectral algebraic geometry. We relate the K-theory of an $\infty$-category of finitely generated projective modules on certain $\mathbb{E}_{\infty}$-rings with the K-theory of an ordinary category of finitely generated projective modules on ordinary rings.

The asymptotic Connes-Moscovici characteristic map and the index cocycles

We show that the (even and odd) index cocycles for theta-summable Fredholm modules are in the image of the Connes-Moscovici characteristic map. To show this, we first define a new range of asymptotic cohomologies, and then we extend the Connes-Moscovici characteristic map to our setting. The ordinary periodic cyclic cohomology and the entire cyclic cohomology appear as two instances of this setup. We then construct an asymptotic characteristic class, defined independently from the underlying Fredholm module. Paired with the $K$-theory, the image of this class under the characteristic map yield…

Quantum weighted projective and lens spaces

We generalize to quantum weighted projective spaces in any dimension previous results of us on K-theory and K-homology of quantum projective spaces `tout court'. For a class of such spaces, we explicitly construct families of Fredholm modules, both bounded and unbounded (that is spectral triples), and prove that they are linearly independent in the K-homology of the corresponding C*-algebra. We also show that the quantum weighted projective spaces are base spaces of quantum principal circle bundles whose total spaces are quantum lens spaces. We construct finitely generated projective modules a…

通过非交换几何棱镜观察经典调和分析

Classical harmonic analysis viewed through the prism of noncommutative geometry

本文通过非交换几何框架，将经典调和分析与L^p空间分析结合，提出新的K-理论群和索引定理，揭示Hilbert变换和Riesz变换的作用。

Smale空间KK-对偶中基本类的几何表示元

A geometric representative for the fundamental class in KK-duality of Smale spaces

该研究针对非交换几何中Smale空间KK-对偶的基本类表示问题，通过推广Bowen马尔可夫划分构造受控利普希茨常数的动力单位分解，结合非周期点采样得到非交换惠特尼嵌入定理类似物，最终构造出表示基本类的θ-可和Fredholm模。

Projective modules over the real algebraic sphere of dimension 3

We show that all the projective modules over the coordinate ring of the real algebraic sphere of dimension 3 are free

Higher algebraic $K$-theory of finitely generated torsion modules over principal ideal domains

The main purpose of this paper is computing higher algebraic $K$-theory of Koszul complexes over principal ideal domains. The second purpose of this paper is giving examples of comparison techniques on algebraic $K$-theory for Waldhausen categories without the factorization axiom.

Algebraic Cobordism as a module over the Lazard ring

In this paper we study the structure of the Algebraic Cobordism ring of a variety as a module over the Lazard ring, and show that it has relations in positive codimensions. We actually prove the stronger graded version. This extends the result of M.Levine-F.Morel claiming that this module has generators in non-negative codimensions. As an application we compute the Algebraic Cobordism ring of a curve. The main tool is Symmetric Operations in Algebraic Cobordism.

On the Algebraic Classification of Module Spectra

Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra ($S$-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring $π_*R$ has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number $N \geq 4$, we prove that the homotopy category of $R$-modules is equivalent to the derived category of the homotopy ring $π_*R$. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of $R$-modules. T…

K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems

This is the integral text of my thesis. The first part is an expanded version of "Riemann-Roch theorems for Deligne-Mumford stacks", where I deal with Artin stacks over general bases. In the second part, I prove some Riemann-Roch statment for D-modules on Deligne-Mumford stacks, and I also consider the problem of algebraization of analytic stacks.