Topology

Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g., K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss ongoing work with Keegan Boyle trying to understand the equivariant 4-genus.

A triangulation of a surface is a way to divide it into a finite number of triangles. Let us pick a random triangulation uniformly among all those with a fixed size and genus. What can be said about the behaviour of these random geometric objects when the size gets large? We will investigate three different regimes: the planar case, the regime where the genus is not constrained, and the one where the genus is proportional to the size. Based on joint works with Baptiste Louf, Nicolas Curien and Bram Petri.

A central theme of this workshop is the fact that arithmetic and topological structures become best behaved “in the limit”. The Grothendieck ring of varieties (or stacks) gives an algebro-geometric means of discovering, proving, or suggesting such phenomena.

In the first lecture of this minicourse, Ravi Vakil will introduce the ring, and describe how it can be used to prove or suggest such stabilization in several settings.

In the second lecture of the minicourse, Aaron Landesman will use these ideas to describe a stability of the space of low degree covers (up to degree 5) of the projective line (joint work with Vakil and Wood). The results are cognate to Bhargava’s number field counts, the philosophy of Ellenberg-Venkatesh-Westerland, and Anand Patel’s fever dream.

This is the second lecture in a two part series: part 1.

The discriminant of a space of functions is the closed subset consisting of the functions which are singular in some sense. For instance, for complex polynomials in one variable the discriminant is the locus of polynomials with multiple roots. In this special case, it is known by work of Arnol'd that the cohomology of the complement of the discriminant stabilizes when the degree of the polynomials grows, in the sense that the k-th cohomology group of the space of polynomials without multiple roots is independent of the degree of the polynomials considered. A more general set-up is to consider the space of non-singular sections of a very ample line bundle on a fixed non-singular variety. In this case, Vakil and Wood proved a stabilization behaviour for the class of complements of discriminants in the Grothendieck group of varieties. In this talk, I will discuss a topological approach for obtaining the cohomological counterpart of Vakil and Wood's result and describe stable cohomology explicitly for the space of complex homogeneous polynomials in a fixed number of variables and for spaces of smooth divisors on an algebraic curve.

The cohomology of the space of degree d holomorphic maps from the complex projective line to a sufficiently nice algebraic variety is expected to stabilize as d goes to infinity. The limit is expected to be the cohomology of the double loop space, i.e. the space of degree d continuous maps from the sphere to that variety. This was shown for projective space by Segal, and there has been further subsequent work. In joint work with Tim Browning, we give a new approach to the problem for smooth affine hypersurfaces of low degree (which should also work for projective hypersurfaces, complete intersections, and/or higher genus curves), based on methods from analytic number theory. We take an argument of Birch that solves the number-theoretic analogue of this problem and translate it, step by step, into the language of ell-adic sheaf theory using the sheaf-function dictionary. This produces a spectral sequence that computes the cohomology, whose degeneration would imply that the rational compactly-supported cohomology matches that of the double loop space.