K-theory of Continuous Deformations of C＊-algebras

We study K-theory of continuous deformations of C＊-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.

Equivariant eta forms and equivariant differential K-theory

In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also derived.Numerical experiments validate the theoretical analysis.

Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds

Comparing to the construction of stringy cohomology ring of equivariant sta-ble almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds,the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold.The authors show that for a finite group G and a G-equivariant stable almost complex manifold X,the G-invariant part of the stringy cohomology ring of(X,G)is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold[X/G].Similar result holds when G is a torus and the action is locally free.Moreover,for a compact presentable stable almost complex orbifold,they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring.

Quantum K-theory and q-difference Equations

This is a set of lecture notes for the first author’s lectures on the difference equations in2019 at the Institute of Advanced Study for Mathematics at Zhejiang University.We focus on explicit computations and examples.The convergence of local solutions is discussed.

The Gap Labelling Integrated Density of States for a Quasi Crystal Universe Is Identical to the Observed 4.5 Percent Ordinary Energy Density of the Cosmos

Condense matter methods and mathematical models used in solving problems in solid state physics are transformed to high energy quantum cosmology in order to estimate the magnitude of the missing dark energy of the universe. Looking at the problem from this novel viewpoint was rewarded by a rather unexpected result, namely that the gap labelling method of integrated density of states for three dimensional icosahedral quasicrystals is identical to the previously measured and theoretically concluded ordinary energy density of the universe, namely a mere 4.5 percent of Einstein’s energy density, i.e. E(O) = mc2/22 where E is the energy, m is the mass and c is the speed of light. Consequently we conclude that the missing dark energy density must be E(D) = 1 - E(O) = mc2(21/22) in agreement with all known cosmological measurements and observations. This result could also be interpreted as a strong evidence for the self similarity of the geometry of spacetime, which is an expression of its basic fractal nature.

LENTICULAR NONCOMMUTATIVE TORI

All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than 1.

The Coarse l^(p)-Novikov Conjecture and Banach Spaces with Property(H)

In this paper,for 1<p<∞,the authors show that the coarse l^(p)-Novikov conjecture holds for metric spaces with bounded geometry which are coarsely embeddable into a Banach space with Kasparov-Yu’s Property(H).

A note on quantum odometers

We discuss various aspects of noncommutative geometry of smooth subalgebras of Bunce-Deddens-Toeplitz algebras.

Verlinde/Grassmannian Correspondence and Rank 2 δ-Wall-Crossing

Motivated by Witten’s work,we propose a K-theoretic Verlinde/Grassmannian correspondence which relates the GL Verlinde numbers to the K-theoretic quasimap invariants of the Grassmannian.We recover these two types of invariants by imposing different stability conditions on the gauged linear sigma model associated with the Grassmannian.We construct two families of stability conditions connecting the two theories and prove two wall-crossing results.We confirm the Verlinde/Grassmannian correspondence in the rank two case.

On the classification of certain real rank zero C^(*)-algebras

In this paper,we give a classification of real rank zero C^(*)-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras C.

Some G-M-type Banach spaces and K-groups of operator algebras on them

By providing several new varieties of G-M-type Banachspaces according to decomposable and compoundable properties, this paper discusses the operator structures of thesespaces and the K-theory of the algebra of the operators on these G-M-type Banach spaces throughcalculation of the K-groups of the operator ideals contained in the class of Riesz operators.

ISOPARAMETRIC HYPERSURFACES WITH FOUR DISTINCT PRINCIPAL CURVATURES

I. PREPARATION AND LEMMASWe start recalling theMain Theorem of Abresch. Given an isoparametric hypersurface in Sn+1 with g= 4, then the pair （m, m+ ）-w. r. g, we may assume that m ≤m+, satisfies one of the three conditions below:

The Whitehead Groups of Semigroups

Let S be a semigroup with identity and zero,（S）the category of finitely generated projective S-systems.In this paper,the Whitehead group K1S of S is defined to be K0Ω（S）, where Ω（S）is the Loop category of（S）,and the structure of K1S is given.

Nonexistence of Almost Complex Structures on S^(2m)×S^(2n)

Let X be a finite CW complex, and let ξ be a real vector bundle over X. We say that ξ has a complex structure if it is isomorphic to the real bundle r(ω)underlying some complex vector bundle ω over X. Let M be a closed connected smooth manifold. We say that M has an almost structure if its tangent bundle has a complex structure.

Persistence Approximation Property for Maximal Roe Algebras

Persistence approximation property was introduced by Hervé Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let X be a discrete metric space with bounded geometry, assume that X admits a fibred coarse embedding into Hilbert space and X is coarsely uniformly contractible, then Cmax*(X) has persistence approximation property. The authors also give an application of the quantitative K-theory to the maximal coarse Baum-Connes conjecture.

Classification of topological trivial matter with non-trivial defects

In this paper, we apply the K-theory to classify topological trivial fermionic phases which, nonetheless,host symmetry-protected non-trivial defects. An important implication of our work is that the existence of Majorana zero mode in the vortex core is neither a necessary nor a sufficient condition for the superconductor in question being topologically non-trivial.

DECOMPOSITION OF STEINBERG GROUPS

Let A be an associative ring with identity, St_n(A) the Steinberg group of dimension n over A. The stable Steinberg group is the direct limit of the St_n(A) in a natural way. Several authors studied various normal forms for elements of St(A) and St_n(A) (cf.[ 3—7]).

SOME REMARKS ON DECOMPOSITION OF STEINBERG GROUP

Sharpe once gave a beautiful 4-term decomposition LPLU for infinite Steinberg group; recently Li Fu-an has given a sufficient and necessary condition for 4-term decomposition LWLU of finite Steinberg group over commutative rings. The present note extends the result of [2] to non-commutative rings and studies the decomposition of a finite Steinberg group over local rings.

CENERALIZED COHOMOLOGY OPERATIONS AND THE FINITENESS OF MULTIPLICATIVE ψ_k-MODULES

In this paper we prove that for a fixed l, the number of multiplicative ψk-module M=Z/p[y1,…,y1]/（Z/p[y1,…,y1]）p+1with degyi=2bi, i=1,…, l, is finite, and for a fixed l, the unstable polynomial Ap-al-gebras P（B）=Z/p[x1,…,x1] which are realizable are finite.

On the Structures of K2(Z[G]), G a Finite Abelian p-Group

Let G be a finite abelian p-group,Г the maximal Z-order of Z[G]. We prove that the 2-primary torsion subgroups of K2Z[G]) and K2(Г)are isomorphic when p ≡ 3, 5, 7 (mod 8, and K2(Z[G])■zZ[1/p] is isomorphic to K2(Г)■zZ[1/p] when p = 2,3,5,7. As an application, we give the structure of K2(Z[G]) for G a cyclic p-group or an elementary abelian p-group.