Geometry and Topology

Speaker:

"No lecture"

Time: 15:30

Room: MC 108

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Chern-Weil's approach to Chern classes for vector bundles III"

Time: 14:00

Room: MC 106

I will start from the definition of vector bundles over a manifold, basic operations on vector bundles, connections and curvature, Chern-Weil's approach to Chern classes of vector bundles, and basic properties of Chern classes.

Colloquium

Speaker: Dror Bar-Natan (Toronto)

"Homomorphic expansions and w-knots"

Time: 15:30

Room: MC 108

Even though little known, the notion of a "homomorphic expansion" is extremely general; it makes sense in the context of practically any algebraic structure, be it a group, or a group homomorphism, or a quandle, or a planar algebra, or a circuit algebra with unzip operations, or whatever.

Even though little known, w-knots make a cool generalization of ordinary knots. They contain ordinary knots and are contained in 2-knots in 4-space and are easier than the latter. They are a quotient of "virtual knots" and are easier then those.

My talk will be about these two notions, homomorphic expansions and w-knots, and about what happens when the two are put together. Lie algebras arise, and Lie groups, and the Kashiwara-Vergne statement, which is one of the deeper statements about the relationship between Lie groups and Lie algebras.

There are also u-knots, and v-knots, and f-knots, and other things which are not knots at all, and there are equally nifty things to say about homomorphic expansions for all those. But not today.

(For more information and handouts, click on the title above.)

Stacks Seminar

Speaker: Peter Oman (Western)

"Toposes and Groupoids"

Time: 11:30

Room: MC 107

We will show how localic groupoids model a generalized notion of 'topological space' or topos. This talk will introduce toposes, monadic descent, and give an overview of extended Grothendieck-Galois theory developed by A. Joyal and M. Tierney.

Stable Homotopy

Speaker: Sam Isaacson (Western)

"The algebraic Whitehead conjecture"

Time: 13:30

Room: MC 106

Algebra Seminar

Speaker: David Wehlau (Queen's )

"Invariants for the modular cyclic group of prime order via classical invariant theory"

Time: 14:30

Room: MC 108

Let $F$ be any field of characteristic $p$ and let $C_p$ denote the cyclic group of order $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,\dots,V_p$ of $C_p$ defined over $F$. It is also well-known that there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $\textrm{SL}_2(\mathbb{C})$ given by the action on the space $R_d$ of $d$ forms. In this talk I will describe my recent result which reduces the computation of the ring of $C_p$-invariants of a $C_p$-representation $V=\oplus_{i=1}^k V_{n_i+1}$ to the computation of the classical ring of invariants (or covariants) $C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\textrm{SL}_2(\mathbb{C})}$.

This allows us to compute for the first time the ring of invariants for many representations of $C_p$.