The chromatic view of stable homotopy theory uses the algebraic geometry of formal groups to organize calculations and the search for large scale phenomena. One of the guiding principles, due to Hopkins, is the Chromatic Splitting Conjecture, which predicts how to rebuild stable homotopy types from simpler pictures. Recently we have seen that this conjecture is not quite true; we will discuss what goes wrong and how it might be fixed. This is joint work with Agnes Beaudry and Hans-Werner Henn, with the hard part done by Beaudry.

I will present formal proofs of selected results discussed in preceding seminar talks using the Coq library UniMath.

Last week we explained the geometric context of the problem of describing the moduli space of Special Lagrangian deformations of a compact manifold. The analytic details were more or less straight-forward as the elliptic theory of PDEs in a compact manifold is very complete and in some sense canonical. In this second and final part of my talk I will consider SL-submanifolds in $\mathbb{C}^{n}$, therefore the SL-submanifolds to be considered shall be non-compact and/or singular. I will survey the elliptic theories available in singular settings, and I will focus on the case of edge-singularities, the next level of singularities after conical. I will conclude the talk with a theorem describing the moduli space of SL-deformations with boundary conditions for a SL submanifold with edge-singularities.

TBA

For a smooth manifold M with a Lie group G action, we can define equivariant cohomology based on differential forms on M. That is Weil model. This construction is analogous to Borel construction on the level of differential forms. The Cartan model is then derived from the Weil model. The Cartan model provides an explicit way to compute equivariant cohomology. We will introduce both models and prove that they are equivalent.

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.

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.

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.

2-groups, or "categorical groups", are "higher dimensional" algebraic structures which generalize groups. In particular, they can be seen as group objects in categories, and as such they are useful for describing the symmetries of categories. I will describe 2-groups and their actions, and describe how similar generalizations of the transformation groupoids associated to group actions show up in this context. Time permitting, I will outline a motivating example for this work coming from the geometry of gerbes. Based on joint work with Roger Picken (IST, Lisbon).