Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Spin Geometry (3)"

Time: 14:30

Room: MC 107

We will write down a complete classification of all Clifford algebras of the form $Cl(n,0)$ and $Cl(0,n)$. From here, we will look at a theorem that relates periodicity of Clifford algebras with Bott periodicity in K-theory. Next, we will consider the Pauli-spin and Dirac matrices and see how they generate important algebras such as the real quaternion algebra, the complex quaternion algebra, the real Dirac algebra and higher order Clifford algebras. From here, we will start a discussion about the Dirac operator and Dirac equation, and subsequently introduce important examples such as the Pauli-Dirac and Dirac-Yukawa operators.

Geometry and Topology

Speaker: Paul Smith (University of Washington)

"Graded modules over path algebras of quivers."

Time: 15:30

Room: MC 108

We establish connections between various algebras and module categories that can be associated to a finite directed graph. These connections involve the singularity category for certain finite dimensional algebras, von Neumann regular algebras, Leavitt path algebras, $C^*$-graph algebras, AF algebras, some ideas from symbolic dynamics, and the space of Penrose tilings as a special case.

Analysis Seminar

Speaker: Hadi Seyedinejad (Western)

"Non-open complex analytic maps"

Time: 15:30

Room: MC 108

Fibres of a morphism between complex spaces form a family that encodes much information regarding the behaviour of the morphism. For example, knowing only about the variation of the topological dimension of the fibres suffices to determine whether a mapping is open or not (Remmert Open Mapping Theorem). This has lead us to an efficient algebraic method of testing for openness by means of the blow-up mapping, and successively, to a very efficient method of testing for flatness (joint work with Janusz Adamus). Apart from merely detecting non-openness, I am also trying to study different modes of being non-open for an analytic map, especially in the general setting of maps over a singular target.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: Spin Structures on Manifolds"

Time: 14:30

Room: MC 107

A spin structure on an oriented (Euclidean) vector bundle is a principal spin-bundle that is a non-trivial 2-fold covering of the oriented (orthonormal) frame bundle of $E$, denoted by $P_{SO}(E)$. An (oriented Riemannian) manifold is called spin if its tangent bundle has such a structure. It turns out that the existence of a spin structure has a topological obstruction -- namely, the (vannishing) of the second Stiefel-Whitney class of the manifold. In this talk, we will introduce spin structures and identify the obstruction for its existence in terms of the second Stiefel-Whitney class. Furthermore, some examples will be examined, including showing that for $n$-tori and for compact Riemann surfaces of genus $g$, there are exactly $2^n$ and $2^{2g}$ non-equivalent spin structures, respectively.

Algebra Seminar

Speaker: Claudio Quadrelli (Western and Milano-Bicocca)

"Rigid fields and $p$-Galois groups: easy solutions for different equations"

Time: 14:30

Room: MC 108

Let $p$ be an odd prime. A field $F$ is said to be $p$-rigid if certain conditions on the cyclic algebras constructed over $F$ are satisfied. $p$-rigid fields have been studied through the last decades. In this talk I will present the properties of $p$-rigid fields together with new characterizations of such fields and their $p$-Galois groups (proved in joint work with S. Chebolu and J. Minac). In particular, given a field $F$, it is possible to detect whether $F$ is $p$-rigid simply by small quotients of $G_F(p)$, or by the cohomological dimension of $G_F(p)$, or by the $\mathbb{F}_P$-cohomology ring of $G_F(p)$, where $G_F(p)$ is the maximal pro-$p$ Galois group of $F$. In this case it is also possible to describe completely and explicitly every $p$-extension of $F$ (in a rather nice way) and every $p$-Galois group of $F$. Our results extend (and simplify) some previous results obtained by R. Ware, A. Engler and J. Koenigsmann; and are related to some important work of I. Efrat, A. Topaz, and others; and last but not least, they provide a new point of view upon such topics.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Isospectral Plane Domains that are not Isometric (2)"

Time: 14:30

Room: MC 107

Because our construction is done via Riemannian orbifolds, we will continue from last week by discussing some important theory and examples related to the Riemannian geometry of orbifolds. In fact, we will see some very special examples that show not all orbifolds are constructed via a group action on a manifold. We will then look at some planar isospectral domains that were constructed in 1994 by Buser and Conway as a segue to proving our main theorem about isospectrality and nonisometry of the two plane domains constructed by Gordon, Webb and Wolpert.