# Topics in Noncommutative Geometry: 3era Escuela de Invierno Luis Santaló-CIMPA Research School

# Titles and abstracts of the Lectures

*Courses*

## Henrique Bursztyn (IMPA, Brazil) - Stefan Waldmann (Freiburg, Germany)

### Introduction to deformation quantization.

The course will give an introduction to deformation quantization, which is a mathematical formulation of the quantization problem in physics resorting to Gerstenhaber’s deformation theory of associative algebras. The ﬁrst part of the course will be elementary, recalling the basic notions of quantization, Poisson structures and presenting the deformation quantization problem. The second part of the course will survey constructions and the classiﬁcation of star products on symplectic manifolds, also mentioning Kontsevich’s classiﬁcation in the case of general Poisson manifolds. As a last part, if time permits, I will discuss Morita equivalence of algebras and the problem of classifying Morita equivalent deformation quantizations.

**Bursztyn's slides 1, 2.**

## Victor Ginzburg (Chicago, USA)

### Noncommutative geometry and Calabi-Yau algebra.

We give an elementary introduction to some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the CY algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential. Numerical invariants, like ranks of cyclic homology groups, are expected to be given by ‘matrix integrals’ over representation varieties. We discuss examples of CY algebras involving quivers, 3-dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3-manifolds and Chern-Simons, and quantizations of Del Pezzo surfaces.

- V. Ginzburg, Calabi-Yau algebras. arXiv:math/0612139v3

## Victor Kac (MIT, USA)

### Vertex algebras and Poisson vertex algebras

In my course I will discuss the foundations of the theory of vertex algebras and Poisson vertex algebras, the latter being the quaiclassical limits of the former. I will explain how the Poisson vertex algebras are used to construct integrable infinite-dimensional Hamiltonian systems.

Lecture 1. Lie conformal algebras and formal distribution calculus

Lecture 2. Local quantum fields and vertex algebras

Lecture 3. Local Poisson brackets and Poisson vertex algebras

Lecture 4. The variational complex

Lecture 5. Poisson vertex algebras and integrable Hamiltonian PDE

- V.G. Kac, Vertex algebras for beginners, University Lecture Series, 10,

Second edition, 1998. - A. De Sole, V.G. Kac, Finite vs affine W-algebras, Japan. J. Math

1(2006),137-261 arXiv:math-ph/0511055 - A. Barakat, A. De Sole, V.G. Kac, Poisson vertex algebras in the theory of

Hamiltonian equations, Japan. J. Math 4(2009),141-252 arXiv:0907.1275

## Max Karoubi (Jussieu, France)

### Various forms of Bott periodicity theorem and applications

1) *Proof of Bott periodicity via index maps.* The equivariant case. One of the tricks to prove Bott periodicity is to deﬁne a reverse map of the Bott map (using elliptic operators) and to prove that both compositions are isomorphisms. We illustrate this principle by enlightning examples in the equivariant case. In particular, we compute the equivariant K-theory of the Thom space of a vector bundle (real or complex) in a purely algebraic way.

2)* Applications of Bott periodicity.* These applications are manifold. We select two famous applications due to Adams (spheres with an H-space structure, vector ﬁelds on spheres) and another one in the theory of C*-algebras.

3)* Relations between Algebraic K-theory and topological K-theory.* Algebraic K-theory is a more subtle invariant for rings than topological K- theory. The comparizon between the two theories gives interesting invariants linked with regulator maps and cyclic homology.

4)* Fredholm operators and twisted K-theory.* Twisted K-theory was developed in the early stages of K-theory and has found recently applications in Physics. We revisit the subject in order to extend the Thom isomorphism and deﬁne new cohomology operations.

5)* The hermitian approach to Bott periodicity. Applications to the computation of the homology of discrete groups.* K-theory may be viewed as a deep study of projective modules and the general linear group. In this section, we extend the theory to other classical groupes like the symplectic group. We then view Bott periodicity from this angle. As applications we redeﬁne invariants of quadratic forms and compute a large part of the homology of various discrete groups.

**Karoubi-Villamayor's papers 1, 2.**

**Notes on twisted bundles**

## Ralf Meyer (Göttingen, Germany)

### Actions of higher categories on C*-algebras

The standard way to describe symmetries in non-commutative geometry are actions of topological groups of groupoids. But these are themselves classical objects. Higher category theory generalises categories to 2-categories by replacing the set of arrows by a groupoid of arrows and 2-arrows. This is a natural way to describe symmetry "groups" that are themselves non-commutative. Since taking groupoid C*-algebras is not functorial, it is genuinely different from the quantum group point of view.

I will explain how 2-categories act on C*-algebras, give several examples and relate it to existing notions of generalised group actions such as Fell bundles. With my coworkers Alcides Buss and Chenchang Zhu, I am currently trying to extend as much as possible from the group case to these more general kinds of symmetries (crossed products, proper actions on spaces and C*-algebras, universal proper actions, assembly maps, equivariant bivariant K-theory). My lectures should culminate with my latest results in this areas.

## Henri Moscovici (Ohio State, USA)

### Index theory and characteristic classes of noncommutative spaces

This series of ﬁve expository lectures will be devoted to the methods that allow the explicit calculation of the characteristic classes of noncommutative spaces. The ﬁrst two lectures will cover the developments that led to the non- commutative version of the local index formula [1]; the third and the fourth will highlight its applications to the transverse geometry of foliations, which led in turn to the detection of a novel form of transverse symmetry and to the Hopf- cyclic cohomology [2], as well as to the resurgence of the classical pseudogroups of Lie and Cartan morphed into Hopf algebras [4]; the fifth lecture will discuss the extension of the theory of characteristic classes to twisted spectral triples.

[1] A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2; www.alainconnes.org/en/downloads.php

[2] A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the trans- verse index theorem, Comm. Math. Phys. 198 (1998), no. 1; arxiv.org/abs/math/9806109

[3] A. Connes and H. Moscovici, Type III and spectral triples, Traces in Ge- ometry, Number Theory and Quantum Fields, Aspects of Mathematics E 38, Vieweg Verlag 2008, pp. 5771; arxiv.org/abs/math/0609703

[4] H. Moscovici and B. Rangipour, Hopf algebras of primitive Lie pseu- dogroups and Hopf cyclic cohomology; arxiv.org/abs/0803.1320

## Holger Reich (Düsseldorf, Germany)

### An introduction to algebraic K-theory

The 5 hour course should give a ﬁrst introduction to algebraic K-theory and lead up to the formulation of the Farrell-Jones conjecture. The ﬁrst three lectures introduce the functors K0 and K1 with an emphasis on group rings. They should be accessible for students who are familiar with basic notions from algebra. The most important applications to topology, i.e. Walls ﬁniteness obstruction and the h-cobordism theorem will be mentioned but only for mo- tivational purposes. The remaining two lectures should on the one hand give an introduction to higher algebraic K-theory and on the other hand develop the language that is necessary in order to formulate the Farrell-Jones conjecture about the algebraic K-theory of a group ring. Here some background from homotopy theory (homotopy groups) is needed.

Detailed description in key words:

Lecture 1: group completion, review of projective modules, projective modules and vector bundles, deﬁnition of K0 and basic properties, group rings and basic representation theory of ﬁnite groups

Lecture 2: traces, Hattori stallings rank and the Bass conjecture, the passage from the integral to the rational group ring, ﬁrst examples of functors over the orbit category.

Lecture 3: deﬁnition of K1 , Whitehead Lemma, the Farrell-Jones conjectures in low dimensions, review of manifolds and basic homotopy theoretic notions, Walls ﬁniteness obstruction (statement), the h-cobordism theorem (statement).

Lecture 4: group completion for topological monoids, deﬁnition of higher K-theory, fundamental theorems.

Lecture 5: functors over the orbit category, homological algebra and spectra over the orbit category, formulation of the Farrell-Jones conjectures

## Jonathan Rosenberg (Maryland, USA)

### Examples and applications of noncommutative geometry and K-theory

1. Introduction to Kasparov's KK theory

2. K-theory and KK-theory of crossed products (the theorems of Connes and Pimsner-Voiculescu, origins of the Baum-Connes conjecture).

3. The universal coefficient theorem for KK and some of its applications.

4. A fundamental example in noncommutative geometry: topology and geometry of the irrational rotation algebra (noncommutative 2-torus).

5. Applications of the irrational rotation algebra in number theory and physics.

##### Lecture Notes.

## Boris Tsygan (Northwestern, USA)

### Introduction to noncommutative calculus

I will start with explaining that much of the standard differential calculus can be deﬁned in purely algebraic terms starting from the ring of functions. Then I will show how to develop noncommutative differential calculus, namely how to extend the standard calculus from rings of functions to arbitrary rings, commutative or not. Some parts of this program can be made quite elementary, others require more sophisticated algebraic apparatus. I will define Hochschild and cyclic complexes of rings and show how they generalize standard calculus of differential forms and multivectors. Then I will develop more advanced noncommutative calculus. The new phenomenon is that this can be done but involves a highly nontrivial choice of geometrical nature. I will finish by some applications, mainly to index theorems. I expect the ﬁrst half of my lectures to be suitable for a student that has some general mathematical background.

## Andrzej Zuk (Paris VII, France)

### Automata groups

The class of automata groups contains several remarkable countable groups. Their study has led to the solution of a number of important problems in group theory. Its recent applications have extended to the fields of algebra, geometry, analysis and probability.

**Slides**

*Single Lectures*

## María Paula Gómez-Aparicio (Paris XI, France)

### Twisting the Baum-Connes conjecture by a non unitary representation

Let G be a locally compact group and ρ a non-unitary

ﬁnite dimensional representation of G. We consider tensor products of

ρ by some unitary representations of G in order to deﬁne two Banach

group algebras analogous to the group C

^{*}-algebras, C

^{∗}(G) and C*

_{r}(G).

We then calculate the K-theory of such group algebras for a large class

of groups satisfying the Baum-Connes conjecture by constructing a

twisted version of the Baum-Connes map. This shows that these twisted

group algebras behave in the same way as the C

^{∗}-algebras, C

^{∗}(G) and

C*

_{r}(G), at the level of K-theory.

## Ruy Exel (UFSC, Florianópolis, Brazil)

### Partial representations of groups

I will describe the notion of a partial representation of a

group and will show some of its connnections to the theory of crossed

products of C*-algebras by partial group actions. Time permiting I

will also give an application of this theory to the study of

Cuntz-Krieger algebras.

## Eduardo Hoefel (UFPR, Curitiba, Brazil)

### The Spectral Sequence of Kontsevich's Compactification doesn't

collapse at $E^2$

The coefficients in Kontsevich's formality map involve integrals over certain compact

manifolds with corners $\overline C_{p,q}(\mathbb{H})$. We show that

the Spectral Sequence induced by its boundary strata doesn't collapse at $E^2$ in general,

as opposed to the spectral sequence of $\overline C_{n}(\mathbb{C})$ (the Axelrod-Singer

compactification of the moduli space of points in the complex plane) which always collapse at $E^2$.

## Bram Mesland (Utrecht, Netherlands)

**Morphisms of spectral triples and smoothness in KK-theory**

It is well known that the appropriate notion of morphism for noncommutative rings leads to a category of bimodules, with composition given by the tensor product. This notion adapts to $C^{*}$-algebras by considering $C^{*}$-bimodules, also known as $C^{*}$ correspondences. $C^{*}$-algebras can be viewed as noncommutative locally compact Hausdorff spaces. Spectral triples are the corresponding notion of noncommutative manifold. They are also the cycles for $K$-homology. In order to construct a bimodule category for these objects, it is necessary to develop a framework of smooth algebras and a notion of differentiable $C^{*}$-module. The morphisms in the resulting category are unbounded $KK$-cycles, with some additional structure. Composition of morphisms leads to an algebraic formula for the Kasparov product in $KK$-theory.

## Rubén Sánchez García (Düsseldorf, Germany)

### Equivariant K-homology for hyperbolic reflection groups

Let P be a finite volume geodesic polyhedron in hyperbolic 3-dimensional space with interior angles between incident faces of the form \pi/n, n \ge 2 an integer. Reflections on the faces generate a Coxeter group \Gamma_P. This group acts by isometries on hyperbolic space with fundamental domain P. We present computations of the associated equivariant K-homology in terms of the geometry of P. This coincides via the Baum-Connes isomorphism with the K-theory of the reduced C^*-algebra of \Gamma_P. This is work in progress with Jean-François Lafont and Ivonne Ortiz.

## Travis Schedler (MIT, USA)

### Noncommutative crepant resolutions

The geometry of singular varieties, such as a vector space modulo a finite group action, is often uncovered by considering a

resolution, i.e., a smooth variety with a proper, birational map to the original variety. For surfaces, there even exists a minimal resolution, and for higher-dimensional varieties, there sometimes exists a replacement of minimal, called "crepant." However, these resolutions can be complicated---even when the original variety is affine, they will have a complicated global structure. Noncommutative geometry offers an alternative: a notion of "noncommutative crepant resolution" of a singular affine variety Spec R, which is a noncommutative algebra A over R which is finite as a module (the replacement of proper), and which is Azumaya over the smooth locus of Spec R (the replacement of birational), and obeys some crepancy and

smoothness conditions. By a result of Van den Bergh, these exist whenever an ordinary crepant resolution exists, and the two are derived equivalent.

I will introduce these notions and fundamental results of Van den

Bergh, Stafford, Iyama, Wemyss, and others, and state some results and

conjectures that are work in progress with Bocklandt, Broomhead, and

Davison.

## Gonçalo Tabuada (U. Nova, Lisbon, Portugal)

### Non-commutative motives

In this talk I will describe the construction of the category of

non-commutative motives in Drinfeld-Kontsevich's non-commutative algebraic

geometry program. In the process, I will present the first conceptual

characterization of Quillen's higher K-theory since Quillen's foundational

work in the 70's. As an application, I will show how these results allow

us to obtain for free the higher Chern character from K-theory to cyclic

homology.

## Roberto Trinchero (Instituto Balseiro, Argentina)

**Examples of non-integer dimensional geometries.**

**Examples of non-integer dimensional geometries.**

Within the general framework of non-commutative geometry two

examples of spectral triples with non-integer dimension spectrum are

considered. These triples involve commutative $C^{\star} $-algebras.

The first example has complex dimension spectrum and trivial

differential algebra. The other is a parameter dependent deformation

of the canonical spectral triple over $S^{1}$. It's dimension spectrum

includes real non-integer values. It has a non-trivial differential

algebra and in contrast with the one dimensional case there are no

junk forms for a non-vanishing deformation parameter. The distance on

this space depends non-trivially on this parameter.

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