Titles and Abstracts

Bi-Lipschitz geometry of singularities

Let us consider a subspace $X$ of $R^n$ defined by polynomial equations and fix a point $p$ on $X$. To what does $X$ ressemble near $p$?

When the implicit function theorem applies at $p$, the answer to the above question is clear!

What happens when $p$ is a singular point? A classical result ensures that $X$ is locally topologically conical: for any sufficiently small radius $\epsilon$, the intersection of $X$ with the ball of radius $\epsilon$ around $p$ is homeomorphic to the cone with vertex $p$ and base the intersection of $X$ with the sphere of radius $\epsilon$. Nevertheless, $X$ is generally not metrically conical: there exist parts of $X$ that contract faster than linearly as $\epsilon$ tends to $0$. A natural problem is then to construct classifications of germs up to local bi-Lipschitz homeomorphism.

I will give an introductory course on this very active subject, at the intersection of metric topology and algebraic geometry. There will be plenty of examples and illustrations.

Singularities and surgeries

Singularities are concentrated topology. When an algebraic variety develops a singularity, some part of its topology is mercilessly crushed to a point. Sometimes, one can reconstruct what was crushed, which can give access to topologically interesting regions of the smoothed variety which one might not have noticed without the help of the singularity. Alternatively, passing between smoothings and resolutions, or different smoothings of a singularity can give useful surgery operations relating different manifolds. Moreover, allowing the equations of a variety to vary in a loop encircling a singular variety gives rise to monodromy diffeomorphisms of the variety; sometimes these diffeomorphisms are enough to generate the mapping class group, and sometimes the surgery operations allow you to give different descriptions of the same diffeomorphism, leading to relations in the mapping class group.

We will explore these circles of ideas through concrete examples. Specifically (roughly lecture-by-lecture):

• We will derive the chain relations in the mapping class groups of 2-dimensional manifolds and the associated 4-manifold surgery, and use this to construct an exotic 4-manifold.
• We will introduce rational blow-down and use this to construct another exotic 4-manifold (conjecturally diffeomorphic to the previous one).
• We will introduce techniques from almost toric geometry that allow you to draw 2-dimensional pictures to help visualise these surgeries.
• We will study some singularities that have several different smoothings, and look at the "wormhole surgeries" that you get by swapping them.