PhD Thesis Defence

Speaker: Asghar Ghorbanpour (Western)

"Rationality of spectral action for Robertson-walker metrics and geometry of determinant line bundle for the nonocmmutative two torus"

Time: 13:30

Room: MC 108

In nonocmmutative geometry, the geometry of a space is given via a spectral

triple $(\mathcal{A,H},D)$.

In this approach the geometric information is encoded in the spectrum of $D$.

To extract this spectral information, one should study the spectral action $\Tr f(D/\Lambda)$.

This function is very closely related to classical spectral functions such as the heat trace $\Tr (e^{-tD^2})$ and the spectral zeta function $\Tr(|D|^{-s})$.

The main focus of this talk is on the methods and tools that can be used to extract the spectral information.

Applying the pseudodifferential calculus and the heat trace techniques, in addition to computing the newer terms of the spectral action, we prove the rationality of this spectral action, which was conjectured by Chamseddine and Connes.

In the second part of the talk, we define the canonical trace for Connes' pseudodifferential calculus on the noncommutative torus and use it to compute the curvature of the determinant line bundle for the noncommutative torus.

At the end, the Euler-Maclaurin summation formula will be used to compute the spectral action of a Dirac operator (with torsion) on the Berger spheres $\mathbb{S}^3(T)$.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western University)

"Curvature of the determinant line bundle for noncommutative tori II"

Time: 15:00

Room: MC 107

In this series of talks we will review Quillen's celebrated determinant line bundle construction on the space of Fredholm operators and study the geometry of this line bundle over the space of Cauchy-Riemann operators on a Riemann surface. Quillen defines a Hermitian metric using zeta regularized determinants on this line bundle and computes its curvature. This computation is then used to define a holomorphic determinant for Cauchy-Riemann operators. It is fairly easy to see that one cannot define a determinant function which is both holomorphic and gauge invariant (conformal anomaly).

Then we will move to a noncommutative setting and review our recent work, with A. Fathi and A. Ghorbanpour, in which we studied the curvature of the determinant line bundle over a space of Dirac operators on the noncommutative two torus. We developed the tools that are needed in our computation of the curvature, including an algebra of logarithmic pseudodifferential symbols and a Konstsevich-Vishik type trace on this algebra. These talks will move slowly and the idea is to develop the necessary tools for further study of the determinant line bundle in noncommutative geometry.

Graduate Seminar

Speaker: Mayada Shahada (Western)

"Multiplicatively collapsing and rewritable algebras"

Time: 13:00

Room: MC 106

A semigroup S is called n-collapsing if, for every a_1,....., a_n in S, there exist functions f \neq g (depending on a_1,....., a_n), such that:

a_{f(1)} \cdots a_{f(n)} = a_{g(1)} \cdots a_{g(n)};

it is called collapsing if it is n-collapsing, for some n. More specifically, S is called n-rewritable if f and g can be taken to be permutations; S is called rewritable if it is n-rewritable for some n.

Semple and Shalev extended Zelmanov's solution of the restricted Burnside problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent.

In this talk, we consider when the multiplicative semigroup of an associative algebra is collapsing; in particular, we will see that the following conditions are equivalent, for all unital algebras A over an infinite field:

(1) The multiplicative semigroup of A is collapsing. (2) A satisfies a multiplicative semigroup identity. (3) A satisfies an Engel identity.

We deduce that, if the multiplicative semigroup of A is rewritable, then A must be commutative

Homotopy Theory

Speaker: Martin Frankland (Western)

"Secondary cohomology operations"

Time: 14:00

Room: MC 107

The Steenrod algebra of (stable) primary operations in mod p cohomology has a rich and fruitful history in homotopy theory, notably with the Adams spectral sequence. Secondary cohomology operations detect additional information not seen by primary operations. We will introduce secondary operations and discuss some of their properties. Then we will present sample calculations and applications from classical homotopy theory, such as the Peterson-Stein formulas and some homotopy groups of spheres.

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western University (Phd Candidate))

"NCG Learning Seminar: Clifford algebras and their representations"

Time: 11:00

Room: MC 106

We will introduce the Clifford algebra associated to a vector space equipped a quadratic form. As an important case, then we give a description of the Clifford algebra for $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the standard quadratic form, as a subalgebra of matrix algebras and prove the periodicity theorem for these algebras. Moreover, representation theory of the Clifford algebras will be discussed.