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.

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.)

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.

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 $C_p$.