Homotopy Theory

Speaker: (Western)

"Organizational meeting"

Time: 13:00

Room: MC 107

Colloquium

Speaker: Pawel Gladki (Uniwersytet Śląski)

"Witt equivalence of function fields over global fields (joint Algebra Seminar-Colloquium)"

Time: 15:30

Room: MC 107

In this talk we investigate the Witt equivalence of certain types of fields. In particular, we show that for two Witt equivalent function fields over global fields there is a natural bijection between certain Abhyankar valuations of these fields, that corresponds to Witt equivalence of respective residue fields. We also examine to what extent this result carries over to Abhyankar valuations with finite residue field.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western University)

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

Time: 11:00

Room: MC 106

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.

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.

Analysis Seminar

Speaker: Purvi Gupta (University of Michigan)

"Asymptotic Estimates for Volume Approximations of Pseudoconvex Domains"

Time: 14:30

Room: MC 107

Several results in convex geometry establish asymptotic estimates for the gap between a convex domain and approximating polyhedra of increasing complexity. Is it possible to do the same for approximations of pseudoconvex domains by analytic polyhedra? In this talk, I will discuss a class of polyhedral objects in strongly pseudoconvex domains that allow for such estimates. Connections with both the Fefferman hypersurface measure and a tiling problem on the Heisenberg group will be discussed. I will also indicate how our formulation suggests a way to discuss volume approximations of more general pseudoconvex domains.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western University)

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

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: Javad Rastegari (Western)

"Application of Lie groups to differential equations"

Time: 13:00

Room: MC 106

The theory of Lie groups originated from Sophus Lie's work on symmetry groups of differential equations. Those are transformation groups acting on the space of independent and dependent variables of a differential equation, which transform a solution to another solution.

Our focus is on 1-parameter connected Lie groups and we use the generating vector field as a powerful tool for the calculations. Once we obtain a symmetry group for a differential equation, we are able to construct new solutions from already known ones.

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Toda brackets in stable stems"

Time: 14:00

Room: MC 107

We will use (primary and) secondary cohomology operations to describe the structure of the stable stems in low dimensions and compute a few Toda brackets in the stable stems.

Colloquium

Speaker: Dan Isaksen (Wayne State University)

"Higher compositions in algebra and topology"

Time: 15:30

Room: MC107

I will give a general introduction to the subject of Massey products and Toda brackets, suitable for a general mathematical audience. I will describe some of their many uses in algebra and topology, and I will present some new results about the general theory of fourfold Massey products.

Noncommutative Geometry

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

"NCG Learning Seminar: Clifford algebras and their representations II"

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.

Algebra Seminar

Speaker: Francesco Sala (Western)

"Geometric representations of affine Kac-Moody algebras via quiver varieties"

Time: 14:30

Room: MC 107

Nakajima constructed highest weight representations of A-type affine Kac-Moody algebras by using the (equivariant) cohomology of cyclic quiver varieties. In the present talk I will describe a geometric realization of level one highest weight representations of A-type affine Kac-Moody algebras by using moduli spaces of sheaves on a 2-dimensional root toric stack over the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}_k$. If time permits, I will explain the conjectural relation between these two geometric constructions.

Geometry and Topology

Speaker: Caroline Junkins (Western)

"The Tits algebras and the gamma-filtration of a twisted flag variety"

Time: 15:30

Room: MC 107

For an algebraic group G over an arbitrary field F, the geometry of projective homogeneous G-varieties has yet to be fully classified. A effective tool used towards such a classification is the cohomological invariant given by the set of Tits algebras of G. A result of Panin provides a connection from the Tits algebras of G to the Grothendieck group of G, and in particular to its associated gamma-filtration. In this talk, we use the Tits algebras of G to construct a torsion element in the gamma-filtration of a flag variety twisted by means of a PGO-torsor. This generalizes a construction in the HSpin case previously obtained by Zainoulline.

Noncommutative Geometry

Speaker: Ali Fathi (Western University (Phd Candidate))

"Regularized traces of elliptic operators"

Time: 15:00

Room: MC 107

I will explain the construction of Kontsevich-Vishik canonical trace on non-integer order classical pseudodifferential operators. This construction has it roots in the old methods of extracting a finite part from a divergent sum or integral (infra-red and ultra-violet divergence), used by mathematicians and physicists.

If time permits I will explain some of the results on generalizations of this construction to noncommutative setting.

Pizza Seminar

Speaker: John Malik (Western)

"Generalizing to the division algebras"

Time: 17:00

Room: MC 108

We survey the real normed division algebras (the real numbers, the complex numbers, the quaternions, and the octonions) and discuss how a result of M. Franz over the complex numbers was effectively generalized to all four of these number systems in the summer of 2014. The recommended background for this talk is second year linear algebra.

Graduate Seminar

Speaker: Allen O'Hara (Western)

"Bruhat Decompositions and Generating Functions"

Time: 13:00

Room: MC 106

Algebraic groups are a well studied object that arise when one has an algebraic variety with a group structure compatible with the variety. In the same vein algebraic monoids are varieties with a monoid structure imposed on them. An interesting thing happens to certain algebraic groups and algebraic monoids called the Bruhat deomposition, which provides a wealth of knowledge about the groups/monoids in terms of double cosets. We'll take a look at two collections of algebraic monoids and their Bruhat decompositions, and determine generating functions for the "sizes" of their Bruhat decompositions.

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Toda brackets in stable stems, part 2"

Time: 14:00

Room: MC 107

We will use (primary and) secondary cohomology operations to describe the structure of the stable stems in low dimensions and compute a few Toda brackets in the stable stems.

Noncommutative Geometry

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

"NCG Learning Seminar: Spin Manifolds and Dirac Operators"

Time: 11:00

Room: MC 106

In this talk I will define spin structure and spin manifolds and give some examples. I will also quickly review some notions of Riemannian geometry like connections, curvature; in particular, the Levi-Civita connection. Then I will define the spin connection and using that I will introduce the Dirac operator.

Algebra Seminar

Speaker: Andrei Negut (Columbia)

"Stable bases for cyclic quiver varieties"

Time: 14:30

Room: MC 107

We will outline a certain program for Nakajima quiver varieties, in the cyclic quiver example. The picture includes two algebras that act on the K-theory of these varieties: one is the original picture by Nakajima, rephrased in terms of shuffle algebras, and the other one is the Maulik-Okounkov quantum toroidal algebra. The connection between the two is provided by the action of certain operators in the so-called "stable basis", and we will present formulas for this action. These formulas can be perceived as a generalization of Lascoux-Leclerc-Thibon ribbon tableau Pieri rules.