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Workshop Programme

for period 18 - 22 December 2006

Trends in Noncommutative Geometry

18 - 22 December 2006

Timetable

Monday 18 December
08:30-09:00 Registration
09:00-10:00 Connes, A (IHES)
  QG and RH, an analogy Sem 1
 

We shall start from the role of KMS states in two different contexts, first in arithmetic then in the electroweak phase transition in particle physics. We shall then build a dictionary between the set-up of noncommutative geometry leading to the spectral realization of the zeros of the Riemann zeta function and a tentative set-up to encode the spectral model in Lorentzian signature, allowing to relate to cosmological models. This is joint work with Matilde Marcolli.

 
10:00-11:00 Wassermann, A (Aix-Marseille 2)
  Operator algebras and elliptic cohomology Sem 1
11:00-11:30 Coffee
11:30-12:30 Plymen, R (Manchester)
  Affine Hecke algebras, Langlands duality and geometric structure Sem 1
 

I will begin with Langlands duality, the origin (in the local Langlands conjecture) of the Langlands-Deligne-Lusztig parameters (t,u,\rho), and the concept of an L-packet.

I will then move on to affine Hecke algebras H(W,q). Each parameter t is the central character of an induced H(W,q)-module.

I'll define the extended quotient, and describe a (conjectural) geometric structure for the set of simple H(W,q)-modules. This conjecture was inspired by the theorem of Baum & Nistor for the periodic cyclic homology of H(W,q).

This will be illustrated in some detail by the affine Hecke algebra attached to the exceptional group G_2.

[Joint work with Anne-Marie Aubert and Paul Baum]

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12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Zilber, B (Oxford)
  Zariski geometries: from classical to quantum Sem 1
 

The notion of a Zariski geometry has been developed in Model Theory in the search for ``logically perfect'' structures. Zariski geometries are abstract topological structures with a dimension, satisfying certain assumptions. Algebraic varieties over algebraically closed fields and compact complex manifolds are Zariski geometries and for some time it was thought that all Zariski geometries are of this kind. In fact the classical examples are only ``limit'' cases of the general pattern, where one necessarily comes to a coordinatisation by non-commutative algebras. Conversely, we show in particular that any quantum algebra at roots of unity of certain type coordinatises a geometric object which is a Zariski geometry. This includes quantum groups.

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15:00-15:30 Tea and Poster Session
15:30-16:00 Plazas, J (MPI, Bonn)
  Homogeneous coordinate rings for noncommutative tori and modular functions Sem 1
 

We analyze the homogeneous coordinate rings of real multiplication noncommutative tori defined by Polishchuk. Our aim is to understand how these rings give rise to an arithmetic structure on the noncommutative torus. We start by giving an explicit presentation of these rings in terms of their natural generators. The appearance of theta functions in these computations provides information about the rationality of the homogeneous coordinate rings. We use the modularity of these rings to obtain algebras which do no depend on the choice of a complex structure on the noncommutative torus. These algebras are obtained by an averaging process over (limiting) modular symbols.

 
16:00-17:30 Cartier, P (IHES)
  Nonlinear transformations in Lagrangians and a graphical calculus Sem 1
 

We describe first a general graphical method to represent nonlinear transformations , and apply it to the case of the invariant transformations of tensors of a certain kind . We interpret then the Lie algebra as well as the Hopf algebra of functions of such a group of nonlinear transformations . Finally we show how to connect these constructions with the Hopf algebra introduced by Connes and Kreimer in Renormalization Theory . As a bonus , we settle a vexing question of normalization connected with the number of symmetries of a Feynman diagram , and we recover a theorem of Connes and Kreimer about the renormalization group .

 
17:30-18:30 Welcome Wine Reception
18:45-19:30 Dinner at Wolfson Court (Residents only)
Tuesday 19 December
09:00-10:00 Consani, K (Johns Hopkins)
  Motives and noncommutative geometry Sem 1
 

The talk will survey on recent developments and applications of the theory of motives in geometry, physics and arithmetic.

 
10:00-11:00 Ardakov, K (Sheffield)
  Iwasawa algebras Sem 1
 

Commutative regular local rings play an important role in classical algebraic geometry and are precisely the commutative local Noetherian rings of finite global dimension. Iwasawa algebras form a natural class of complete semilocal Noetherian rings with good homological properties, which are noncommutative in general. These algebras have applications in number theory and have connections to Lie theory, but their algebraic structure is still rather mysterious. I will present an overview of the known ring-theoretic properties of Iwasawa algebras.

 
11:00-11:30 Coffee
11:30-12:30 Christensen, E (Copenhagen)
  A study of fractals based on non commutative methods Sem 1
 

A fractal set in a Euclidian space is by nature non smooth, and the concept of a differentiable function on it makes no sense. The methods from noncommutative geometry are designed to describe smooth structures in a language based on operators on Hilbert spaces, modules over self-adjoint algebras and cohomological invariants. This language may also be used in the case where the algebra is a subalgebra of the continuous functions on a compact fractal set.

Connes' book "Noncommutative Geometry" contains several results along this line. One of the spectral triples Connes constructs is a a countable sum two-dimensional modules. We have made a general study of this in the general setting of a compact metric space, and obtained some results of general nature.

We have then looked at Sierpinski's Gasket, and it is clear that sums of two dimensional modules is not the right sort of module here, the reason being that such a module can not detect all the holes in the gasket. We have then constructed a spectral triple for the Sierpinski Gasket, based on all the holes, i. e. an infinite sum of modules based on all the triangles in the gasket. It turns out that this spectral triple measures the geodesic distance on the gasket, it gives the Minkowski dimension of the gasket, it induces a multiple of the self-similar Hausdorff measure via the Dixmier trace and it detects the holes, by a non trivial pairing with the K-theory.

 
12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Larsen, N (Oslo)
  Hecke algebras and the finite part of the Connes-Marcolli C*-algebra Sem 1
 

In a joint work with M. Laca and S. Neshveyev, we study the C*-algebra of a Hecke pair arising from the ring inclusion of the 2x2 integer matrices in the rational ones. This algebra is included in the multiplier algebra of the C*-algebra considered by Connes and Marcolli in their work on quantum statistical mechanics of Q-lattices. We uncover structural properties of our Hecke algebra which somewhat surprinsingly mirror the one dimensional case of the C*-algebra of Bost and Connes associated to the inclusion of the integers in the rationals; there is a symmetry group given by an action of the finite integral ideles, and the fixed-point algebra decomposes as a tensor product over the primes. In classifying KMS-states, we are again guided by the one-dimensional case, but have to deal with difficulties intrinsic to our two-dimensional set-up.

 
15:00-15:30 Tea and Poster Session
15:30-16:30 Neshveyev, S (Oslo)
  Phase transition in the Connes-Marcolli GL2-system Sem 1
 

Connes and Marcolli have recently introduced a GL2-system, a two dimensional analogue of the Bost-Connes system associated with rational numbers. They analyzed the space of KMS-states of the system for inverse temperatures outside the critical region (1,2]. We complete their analysis by showing that for each inverse temperature in the critical region there exists a unique KMS-state. This uniqueness turned out to be related to equidistribution of Hecke points on the modular curve. On the way we develop some general tools for analyzing KMS-states on crossed product of abelian algebras by Hecke algebras. (Joint work with M. Laca and N.S. Larsen.)

 
16:30-17:00 Kellendonk, J (Claude Bernard Lyon 1)
  Topological boundary maps in physics, general theory and applications Sem 1
 

$K$-theory boundary maps and their duals in cyclic cohomology are used to relate topological invariants of bulk and boundary in physical systems. We explain in particular an application to potential scattering thus providing a topological explanation of Levinson's theorem.

 
17:00-18:00 Poster Session
18:45-19:30 Dinner at Wolfson Court (Residents only)
Wednesday 20 December
09:00-10:00 Hunton, J (Leicester)
  Aperiodically ordered patterns Sem 1
 

Aperiodically ordered patterns, i.e., non-repeating yet highly structured tilings such as the Penrose tiling(s), have proved to be a source of interaction between noncommutative geometry, dynamics and topology. Associated K-theoretic invariants have given both geometric and physical information about the underlying patterns and their realisation as models for so-called 'quasicrystals'. This talk will present both background and some recent results and perspectives aiming to understand further the properties of aperiodic patterns, their K-theory and related topology.

 
10:00-11:00 Monod, N (Geneva)
  An invitation to bounded cohomology Sem 1
 

Historically, bounded cohomology has its roots in the cohomology of Banach algebras (after B.E. Johnson). M. Gromov turned it into a powerful and versatile tool in geometry and group theory. Lately, bounded cohomology has found applications in non-commutative measure theory, being a central tool for certain orbit equivalence rigidity results.

This lecture will present a leisurly and non-specialized introduction to bounded cohomology. We will highlight its connection to topics such as amenability, characteristic classes, quasification, rigidity, orbit equivalence.

 
11:00-11:30 Coffee
11:30-12:30 Voigt, C (Wilhelms-Universitat Munster)
  The Baum-Connes conjecture for the dual of SU$_q$(2) Sem 1
 

We describe the proof of an analogue of the Baum-Connes conjecture for the dual of the quantum SU(2) group of Woronowicz. Following the work of Meyer and Nest, the formulation of the conjecture is based on the use of triangulated categories arising from equivariant Kasparov theory. The main ingredient in the proof is an explicit analysis of the equivariant K-theory and K-homology of the standard Podles sphere. This involves the study of equivariant Fredholm modules representing twisted Dirac operators as well as Hopf-Galois theory and the equivariant Chern character.

 
12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Rogalski, D (UCSD)
  Noncommutative projective surfaces Sem 1
 

We discuss recent work, joint with Toby Stafford, which describes a large class of noncommutative surfaces in terms of blowing up. Specifically, let A be a connected graded noetherian algebra, generated in degree 1, and suppose that the graded quotient ring Q(A) is of the form k(X)[t, t^-1; sigma] for some projective surface X with automorphism sigma. Then we prove that A can be written as a naive blowup of a projective surface Y birational to X. This enables one to obtain a deep understanding of the structure of such algebras.

 
15:00-15:30 Tea and Poster Session
15:30-16:00 Soltan, P (Warsaw)
  On quantum groups from multiplicative unitaries Sem 1
 

I will describe the theory of quantum groups defined by (modular) multiplicative unitary operators. In this approach to quantum groups we do not assume existence of Haar measures. Moreover there can be many different multiplicative unitaries giving rise to the same quantum group. Nevertheless, I will show that all the important objects of the theory are independent of the choice of the multiplicative unitary operator. The most important of these objects is the ultraweak topology on the C*-algebra describing our quantum group. In classical case this topology determines the class of the Haar measure.

 
16:00-16:30 Kasprzak, P (Warsaw)
  Quantum Lorentz groups obtained by Rieffel deformation Sem 1
 

The aim of this talk is to describe two examples of Quantum Lorentz Groups obtained by Rieffel Deformation. In particular, a description in terms of noncommutative coordinates on them will be given. If time permits, we will present a refinement of Rieffel Deformation, based on Landstad theory of crossed product. The benefits of this approach are: simple proofs of invariance of K-theory and preservation of nuclearity under the deformation.

 
16:30-17:00 Kauffman, LH (Illinois at Chicago)
  Non-Commutative worlds Sem 1
 

This talk shows how forms of gauge theory, Hamiltonian mechanics, quantum mechanics and genreal relativity arise from a non-commutative framework for calculus and differential geometry. Discrete calculus is seen to fit into this pattern by a reformulation in terms of commutators. Differential geometry begins here, not with the concept of parallel translation, but with the concept of a physical trajectory and algebra related to the Jacobi identity that governs that trajectory. We discuss how Jacobi identity controls the Levi-Connection in this context. We generalize the Feynman-Dyson derivation of electormagnetism in a non-commutative context, and we show how natural constraints on non-commutative derivations give rise to fourth-order, generalizations of Einstein's equations for general relativity (joint work with Anthony Deakin and Clive Kilmister).

Related Links

 
17:00-18:00 Poster Session
19:30-18:00 Conference Dinner at Clare College (Great Hall) Bar open from 21:00 - 23:30pm
Thursday 21 December
09:00-10:00 Karoubi, M (Paris 7-Denis Diderot)
  Twisted K-theory old and new Sem 1
 

Twisted K-theory has been invented by P. Donovan and the author more than 35 years ago. The motivation at that time was to give a satisfactory Thom isomorphism (and Poincaré duality theorem) in K-theory.

In this talk, we revisit the subject in the light of new developments (J. Rosenberg, M.-F. Atiyah and G. Segal...). We also sketch a proof of the Thom isomorphism theorem in this setting and give explicit computations in the equivariant case, in relation with the "Chern character" for finite groups introduced by Baum, Connes and Slominska.

 
10:00-11:00 Gordon, I (Edinburgh)
  Hecke algebras and quiver varieties Sem 1
 

Rational Cherednik algebras were introduced five years ago as degenerations of Cherednik's double affine Hecke algebras. They have a very rich representation theory which has found applications in many areas including combinatorics, invariant theory, algebraic integrable systems, algebraic symplectic geometry, Lie theory.

In this talk we will discuss new connections between representations of Cherednik algebras (category O), the combinatorics Hecke algebras of finite Coxeter groups, and the geometry of Nakajima quiver varieties. This is a small first step towards a (noncommutative) geometric picture of the representation theory of these algebras.

 
11:00-11:30 Coffee
11:30-12:30 Retakh, V (Rutgers)
  Noncommutative loops over Lie algebras and Lie groups Sem 1
 

ABSTRACT

In this talk I will introduce noncommutative loops over Lie algebras as a tool for studying algebraic groups over noncommutative rings.

Given a Lie algebra $g$ sitting inside an associative algebra $A$ and any associative algebra $F$, the $F$-loop algebra is the Lie subalgebra of tensor product $F\otimes A$ generated by $F \otimes g$.

For a large class of Lie algebras $g$, including semisimple ones, an explicit description of all $F$-loop algebras will be presented. This description has a striking resemblance to the commutator expansions of $F$ used by M. Kapranov in his approach to noncommutative geometry.

I will also define and study Lie groups associated with $F$-loop algebras.

This is a joint paper with A. Berenstein (Univ. of Oregon).

 
12:30-13:30 Lunch at Wolfson Court
14:00-14:30 Zhang, R (Sydney)
  Noncommutative vector bundles on quantum homogeneous spaces and representations of quantum groups Sem 1
 

Quantum group equivariant vector bundles on quantum homogeneous spaces are classified. An analogue of Dolbeault cohomology is presented for such noncommutative vector bundles. A quantum version of the Bott-Borel-Weil theorem is discussed within this noncommutative geometric framework, realizing representations of quantized universal enveloping algebras on the cohomology groups.

 
14:30-15:00 Pagani, C (Copenhagen)
  A noncommutative family of instantons Sem 1
 

We construct $\theta$-deformations $A(SL_\theta(2,\mathbb{H}))$, $A(Sp_\theta(2))$ of the corresponding classical groups.

Starting from the basic instanton on a noncommutative four-sphere $S^4_\theta$, we construct a noncommutative family of instantons parametrized by the quantum quotient of $A(SL_\theta(2,\mathbb{H}))$ by $A(Sp_\theta(2))$.

Based on joint work with Giovanni Landi (University of Trieste), Cesare Reina (SISSA, Trieste), Walter van Suijlekom (MPI, Bonn).

 
15:00-15:30 Tea and Poster Session
15:30-16:15 Bocklandt, R (Antwerp)
  Graded 3-dimensional Calabi Yau algebras Sem 1
 

We prove that Graded Calabi Yau Algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.

Related Links

 
16:15-17:00 Berger, R (Saint-Etienne)
  Poincare-Birkhoff-Witt deformations of Calabi-Yau algebras Sem 1
 

It is a joint work with Rachel Taillefer. Recently, Bocklandt proved a conjecture by Van den Bergh in its graded version, stating that a graded quiver algebra A (with relations) which is Calabi-Yau of dimension 3 is defined from a homogeneous potential W. Our main result is the following: if we add to W any potential of smaller degree, we get a Calabi-Yau algebra which is a Poincaré-Birkhoff-Witt (PBW) deformation of A, and the so-obtained PBW deformations are characterised among all the PBW deformations of A. This main result and some examples will be presented. An N-version of the PBW theorem due to Ginzburg and myself will be used.

Related Links

 
17:00-18:00 Poster Session
18:45-19:30 Dinner at Wolfson Court (Residents only)
Friday 22 December
09:00-10:00 Woronowicz, SL (Warsaw)
  Unbounded elements in C-star algebras Sem 1
 

In many situations we deal with unbounded operators that are in a sense related to operator algebras. This is the case in quantum mechanics, where (unbounded) Hamiltonian is related to the algebra of observables and in locally compact (non-compact) quantum groups, where matrix elements of finite-dimensional representations are related to the C-star algebra of "functions on the group". Similarly infinitesimal generators of a Lie group are related to the algebra C-star of the group. In all these cases the unboundnes is the only feature that prevents us to to include the operators to the algebra. Instead we say that the operators are affiliated with the algebra. The aim of the talk is to define the affiliation relation in the C-star algebra context and show a number of applications and properties.

 
10:00-11:00 Hannabus, K (Oxford)
  Noncommutative and nonassociative T-duality for principal bundles Sem 1
 

T-duality, which arose in string theory, provides a procedure for linking pairs of principal bundles over the same base. However, some dual bundles are not geometric, but can still be described by noncommutative or even nonassociative algebras. The latter also requires some generalisations of standard C*-algebra ideas.

Related Links

 
11:00-11:30 Coffee and Poster Session
11:30-12:30 Stafford, JT (Michigan)
  Sklyanin algebras and Hilbert schemes of points Sem 1
 

This talk describes joint work with Tom Nevins. Noncommutative projective planes have been classified by Artin, Tate and Van den Bergh. In this talk we will show how one can construct (genuine commutative) projective moduli spaces for torsion-free sheaves on these noncommutative projective planes that are analogous to (indeed, deformations of) the moduli spaces of sheaves over the usual commutative projective plane P^2.

The generic noncommutative plane corresponds to the Sklyanin algebra S constructed from an automorphism sigma of infinite order on an elliptic curve E inside P^2. In this case, the moduli space of line bundles over S with the appropriate invariants provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P^2-E. This is an ``elliptic'' analogue of earlier work of Berest, Wilson and others that showed that when S is the homogenised Weyl algebra the corresponding moduli space is Calogero-Moser space.

 
12:30-13:30 Lunch at Wolfson Court
14:00-18:00 Free Afternoon / Short Presentations
18:45-19:30 Dinner at Wolfson Court (Residents only)

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