Mathematics

For every ∞ -category C , there is a homotopy n -category h n C and a canonical functor γ n :C→ h n C . We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Based on the idea of the homotopy n -category, we introduce the notion of an n -derivator and study the main examples arising from ∞ -categories. Following the work of Maltsiniotis and Garkusha, we define K -theory for ∞ -derivators and prove that the canonical comparison map from the Waldhausen K -theory of C to the K -theory of the associated n -derivator D (n) C is (n+1) -connected. We also prove that this comparison map identifies derivator K -theory of ∞ -derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy n -category, we define also a K -theory space K( h n C,can) associated to h n C . We prove that the canonical comparison map from the Waldhausen K -theory of C to K( h n C,can) is n -connected.

We give a survey on higher invariants in noncommutative geometry and their applications to differential geometry and topology.

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.

In this paper we reduce the generalized Hilbert's third problem about Dehn invariants and scissors congruence classes to the injectivity of certain Chern--Simons invariants. We also establish a version of a conjecture of Goncharov relating scissors congruence groups of polytopes and the algebraic K -theory of C .

We formulate the (co)bar construction theory of dg K -(co)rings and the calculus theory of the Hochschild homology and cohomology of dg K -rings. As applications, we compare the Hochschild (co)homologies of a complete typical dg K -ring and its Koszul dual. Moreover, we show that the Koszul dual of a finite dimensional complete typical d -symmetric dg K -ring is a d -Calabi-Yau dg algebra whose Hochschild cohomology is a Batalin-Vilkovisky algebra. Furthermore, we prove that the Hochschild cohomologies of a finite dimensional complete typical d -symmetric dg K -ring and its Koszul dual are isomorphic as Batalin-Vilkovisky algebras. In conclusion, we found a connection between the Batalin-Vilkovisky algebra structures on the Hochschild cohomologies of d -Calabi-Yau dg algebras and d -symmetric dg K -rings.

We study the quotient space obtained by the flip action on the quantum n-tori. The Hochschild, cyclic and periodic cyclic homology are calculated.

Given a central arrangement of lines A in a 2 -dimensional vector space V over a field of characteristic zero, we study the algebra D(A) of differential operators on V which are logarithmic along A . Among other things we determine the Hochschild cohomology of D(A) as a Gerstenhaber algebra, establish a connection between that cohomology and the de Rham cohomology of the complement M(A) of the arrangement, determine the isomorphism group of D(A) and classify the algebras of that form up to isomorphism.

We compute the Hochschild cohomology ring of the algebras A=k?�X,Y??( X a ,XY?�qYX, Y a ) over a field k where a?? and where q?�k is a primitive a -th root of unity. We find the the dimension of HH n (A) and show that it is independent of a . We compute explicitly the ring structure of the even part of the Hochschild cohomology modulo homogeneous nilpotent elements.

We describe the Hochschild cohomology ring for a family of self-injective algebras of tree class E 6 in terms of generators and relations. Together with the results of the previous paper, this gives a complete description of the Hochschild cohomology ring for a self-injective algebras of tree class E 6 .

The Hochschild cohomology rings for self-injective algebras of tree classes E 7 and E 8 with finite representation type was described in terms of generators and relations.