Almost flat vector bundle is a geometric counterpart of the higher index theory. It was introduced by Connes-Gromov-Moscovici for the purpose of proving the Novikov conjecture for a large class of groups. Its central concept is the almost monodromy correspondence, i.e., the rough one-to-one correspondence between almost flat vector bundles and quasi-representations of the fundamental group. There is an attracting question in this topic originally considered by Gromov: is any element of the K-group of the classifying space of a discrete group represented by an almost flat vector bundle? Dadarlat reveals that this problem is related with the quasi-diagonality (or MF property) of group C*-algebras. In this talk we introduce a relative version of the index theory of almost flat bundles inspired by a recent work of Chang-Weinberger-Yu, by focusing on Gromov's question in the relative setting.

When a Riemannian manifold $M$ is equipped with some additional structure, we can sometimes deform the metric using the structure. For example if $M$ has a fiber bundle structure, we can consider a deformation, called the adiabatic limit, which shrinks the metric in the fiber directions and does not change in the base direction. By studying the behavior of spectrum of geometric operators (for example Laplacians) in such deformations, we can sometimes get information on geometric invariants on $M$. This type of deformations can be nicely understood as deformations of Lie groupoids. In this talk, I will explain how to analyze this adiabatic-type deformation using groupoids and their C*-algebras. As an application, I consider a problem in geometric quantization. It is known that, when a symplectic manifold with a Lagrangian fibration and a prequantizing line bundle, the Riemann-Roch number is the sum of the number of Bohr-Sommerfeld fibers and contributions from singular fibers. I will give a proof of this fact by using groupoids, and looking at what appears in the "limit".

We discuss the Rohlin property for cocycle actions of amenable C*-tensor categories on von Neumann algebras. Then we study the classification problems of centrally free actions and of amenable subfactors.

In 1998, Putnam proved that there exists a 6-term exact sequence of K-groups associated with a certain pair of groupoid C*-algebras. In this talk, we will discuss analogous results for the homology groups of totally disconnected etale groupoids, and observe that there exists a long exact sequence of homology groups.

We would like to talk about the homotpy groups of the Cuntz Toeplitz algebras. The Cuntz Toeplitz algebra is an universal C*-algebra generated by isometries with mutually orthogonal ranges whose quotient is the Cuntz algebra. We also want to talk about computations of the homotopy sets of the automorphism groups of the Cuntz algebra by M. Dadarlat and M. Izumi.

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The main subject of the talk is group actions on the Cantor space called self-similar group actions. We discuss Cuntz--Pimsner algebras constructed from self-similar group actions by Nekrashevych. We see that there exist KMS states for canonical gauge actions on the Cuntz--Pimsner algebras. The KMS states are given by the Bernoulli measure. In some cases, we can also observe the uniqueness of KMS states by proving the uniqueness of tracial states on the gauge invariant subalgebras. Moreover, we consider the double commutants of the Cuntz--Pimsner algebras on the GNS spaces of the KMS states and determine their types.

Arveson have provided the notion of product system. He associated an E$_0$-semigroup on type I factor with an product system and classified E$_0$-semigroups up to cocycle conjugacy. In this talk, we develop the dilation theory and the classification theory of E$_0$-semigroups in terms of W$^*$-bimodules, reflected by Bhat-Skeide's work. First, we provide a notion of product system of W$^*$- bimodules and obtain a one-to-one correspondence between a CP$_0$-semigroup and a pair of a product system of W$^*$-bimodules and a unit. The correspondence gives a new construction of dilations and a classification of E$_0$-semigroups on a general von Neumann algebra by product system of W$^*$-bimodules.

I will talk about my recent work on tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g. $\mathrm{SL}_n(\mathbf{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization. This is joint work with Amine Marrakchi.

Let $A$ be a simple separable nuclear C$^*$-algebra with a unique tracial state and no unbounded traces, and let $\alpha$ be an outer action of a finite group $G$ on $A$. In this talk, we show that if $\alpha$ is strongly outer, then $\alpha\otimes \mathrm{id}$ on $A\otimes\mathcal{W}$ has the Rohlin property, where $\mathcal{W}$ is a certain stably projectionless C$^*$-algebra having trivial $K$-groups and a unique tracial state and no unbounded traces.