Department of Mathematics

The University of Western Ontario

Quantum Heisenberg Manifolds were first defined by M. Rieffel in 1989 as example of quantization of Heisenberg Manifolds along a Poisson bracket.(A typical Heisenberg Manifold is the quotient of Heisenberg group by a uniform lattice).They are interesting for several reasons, one being just because they are tractable examples of noncommutative manifolds.This means that , like the related but simpler noncommutative tori, Q-Heisenberg manifolds provide a nice setting in which to explore noncommutative geometry. In these series of talks I will explore the different features of the noncommutative geometry on Q-Heisenberg manifolds. We introduce a class of spectral triples on Q-Heisenberg manifold, we introduce the space of L^2 -forms and then we characterize torsion less/Unitary connections. In addition, for a concrete family of unitary connections we compute Ricci curvature and scalar curvature.

The noncommutative two torus $A_theta$ equipped with a general complex structure and Weyl conformal factor, is a noncommutative Riemannian manifold where the metric information is encoded in the Dirac operator $D$ of a spectral triple over this C*-algebra. In a recent joint work with M. Khalkhali, we computed a local expression for the scalar curvature of $A_theta$. This was achieved by finding an explicit formula for the value at the origin of the analytic continuation of the spectra zeta function $\Zeta_a(s) := Trace (a|D|^{-s}) (Real(s) >> 0)$ as a linear functional in $a \in A_theta$ . This local expression was also computed by Connes and Moscovici independently. In this talk, I will explain how they have used this local formula and variational methods to compute the determinant of the Laplacian D2 on $A_theta$.

Studying the Ricci flow of a smooth, closed manifold M equipped with a Riemannian metric g involves the process of allowing the metric g to evolve over time under the PDE g_{t} = -2Ric(g). Ricci flow was, in fact, the main tool used by Perelman to prove the Poincare conjecture. The purpose of this talk will be to discuss what is Ricci flow, to present where it comes from and to provide examples of Ricci flow of certain manifolds. Our discussion will then lead into an analysis of a paper written by Bhuyain and Marcolli, who constructed a version of Ricci flow for noncommutative two-tori. The Ricci flow is a fundamental tool used to understand the geometry and topology of manifolds, and understanding it well will help us understand how we can classify other noncommutative spaces such as noncommutative tori in higher dimensions.

Studying the Ricci flow of a smooth, closed manifold M equipped with a Riemannian metric g involves the process of allowing the metric g to evolve over time under the PDE g_{t} = -2Ric(g). Ricci flow was, in fact, the main tool used by Perelman to prove the Poincare conjecture. The purpose of this talk will be to discuss what is Ricci flow, to present where it comes from and to provide examples of Ricci flow of certain manifolds. Our discussion will then lead into an analysis of a paper written by Bhuyain and Marcolli, who constructed a version of Ricci flow for noncommutative two-tori. The Ricci flow is a fundamental tool used to understand the geometry and topology of manifolds, and understanding it well will help us understand how we can classify other noncommutative spaces such as noncommutative tori in higher dimensions.

An Einstein manifold is a smooth manifold whose Ricci tensor is proportional to the metric. Many homogeneous spaces can be realized as Einstein manifolds, and have been widely studied for general existence and nonexistence of Einstein metrics. In this talk we will give examples of homogeneous and Einstein manifolds and discuss some of the general underlying theory related to these spaces. We will also briefly discuss how this can be extended to the noncommutative case. Finally, we will show that if we are given a closed, connected, one-dimensional subgroup H of SU(3) that has no nonzero fixed points, then SU(3)/H admits an SU(3)-invariant Riemannian structure of strictly positive curvature. This result was first proven in 1975 by Aloff and Wallach, and it was here that the famous Aloff-Wallach spaces were introduced.

We show that the category of coefficients for Hopf cyclic cohomology has two proper subcategories where one of them is the category of stable anti Yetter-Drinfeld modules. Generalizations of suitable coefficients for Hopf cyclic cohomology are introduced. The notion of stable anti Yetter-Drinfeld modules is extended based on underlying symmetries. We show that the new introduced categories for coefficients of Hopf cyclic cohomology and the category of stable anti-Yetter-Drinfeld modules are all different. (This is joint work with Bahram. Rangipour and Dan. Kucerovsky )

An Einstein manifold is a smooth manifold whose Ricci tensor is proportional to the metric. Many homogeneous spaces can be realized as Einstein manifolds, and have been widely studied for general existence and nonexistence of Einstein metrics. In this talk we will give examples of homogeneous and Einstein manifolds and discuss some of the general underlying theory related to these spaces. We will also briefly discuss how this can be extended to the noncommutative case. Finally, we will show that if we are given a closed, connected, one-dimensional subgroup H of SU(3) that has no nonzero fixed points, then SU(3)/H admits an SU(3)-invariant Riemannian structure of strictly positive curvature. This result was first proven in 1975 by Aloff and Wallach, and it was here that the famous Aloff-Wallach spaces were introduced.

Gauss–Bonnet theorem or Gauss–Bonnet formula is one of the star attractions of modern differential geometry. It states that for a compact oriented manifold M, the "curvatura integra" over M is equal to a multiple of Euler Characteristic of M. We shall give an "extrinsic" proof for M as an embedded submanifold (actually an even dimensional hyper surface)of a Euclidean space. The proof is heavily based on the Poincare-Hopf index theorem which states that the sum of indexes of a smooth vector field over M is equal to the Euler characteristic of M.

Since this year we shall be busy with curavture in noncommutative geometry, I thought I should start with the most fundamental classical incarnation of this notion: Gauss' theory of curvature for surfaces, and what it can teach us. All are welcome! When Gauss, in his celebrated paper of 1827, {\it Disquisitiones generales circa superficies curvas} {(\it General investigations of curved surfaces)} after a long series of calculations eventually showed that the extrinsically defined curvature of a surface can be expressed entirely in terms of its intrinsic metric (= the first fundamental form), he got so excited that he called the obvious corollary of this result Theorema Egregium (The Remarkable Theorem). Gauss's formidable curvature formula, and the closely related {\it Gauss-Bonnet theorem} is the foundation stone for all of differential geometry, as it was later shown by Riemann in 1859 that the curvature of higher dimensional manifolds can be understood purely in terms of curvatures of its two dimensional submanifolds. Theorema Egregium can also be regarded as the infinitesimal form of, and in fact is equivalent to, the celebrated Gauss-Bonnet Theorem. This paper of Gauss is the single most important work in the entire history of differential geometry.

Geometric operators defined on a compact Riemannian manifold, e.g. Laplacian, Dirac, provide a framework in which we can investigate some geometric properties while we are completely working with algebra of operators on Hilbert spaces and commutators and spectral analysis of operators. In this setting we will have objects called spectral triples introduced by Alain Connes, which will play role of differential calculus on our (noncommutative) spaces. A spectral triple is a triple (A,H,D) in which A is an involutive algebra (plays role of $C^\infty (M)) and H is Hilbert space on which A acts continuously (it is analogous of the space of the sections of vector bundle which D acts on) and D is an operator (it is our first order elliptic differential operator) which has some properties. This talk is the first session of a series of talks in which we will investigate different properties and examples and objects related to spectral triples. The talk will start with definition of spectral triples and we shall go through classical examples to show where the ideas come from and at the end a spectral triple defined on NC-torus will be discussed.

Since this year we shall be busy with curavture in noncommutative geometry, I thought I should start with the most fundamental classical incarnation of this notion: Gauss' theory of curvature for surfaces, and what it can teach us. All are welcome! When Gauss, in his celebrated paper of 1827, {\it Disquisitiones generales circa superficies curvas} {(\it General investigations of curved surfaces)} after a long series of calculations eventually showed that the extrinsically defined curvature of a surface can be expressed entirely in terms of its intrinsic metric (= the first fundamental form), he got so excited that he called the obvious corollary of this result Theorema Egregium (The Remarkable Theorem). Gauss's formidable curvature formula, and the closely related {\it Gauss-Bonnet theorem} is the foundation stone for all of differential geometry, as it was later shown by Riemann in 1859 that the curvature of higher dimensional manifolds can be understood purely in terms of curvatures of its two dimensional submanifolds. Theorema Egregium can also be regarded as the infinitesimal form of, and in fact is equivalent to, the celebrated Gauss-Bonnet Theorem. This paper of Gauss is the single most important work in the entire history of differential geometry.