Abstract: Existence of massive gauge bosons breaks down the local gauge invariance of Yang-Maills Lagrangians. In this lecture we shall look at one method to deal with this situation through the introduction of Higgs fields.

Abstract: This talk will be a nontechnical introduction to the interactions between homotopy theory and 2-category theory, focusing particularly on homotopical perspectives on 2-monads. No prior experience with 2-categories will be necessary.

Abstract: A classical result, usually ascribed to Noether and Lefschetz, states that a very general surface $X$ in complex projective 3-space $\mathbb{P}^3$ has Picard number 1 (i.e., the group of isomorphism classes of line bundles on $X$ is $\mathbb{Z}$). The problem of finding an estimate of the codimension of the loci in the moduli spaces of surfaces in $\mathbb{P}^3$ whose points correspond to surfaces with a bigger Picard group is unsuspectedly complicated.

In a first part of my talk I will review these classical results. Then I will sketch a generalization to surfaces in normal toric 3-folds.