Using the theory of FI-Hyper homology, we proved representation stability for the equivariant cohomology of configuration spaces of manifolds with a group action. In particular, this can be applied to the group cohomology of the braid group of S^2 to show it has homological periodicity. For pure braid groups of S^2, we show that their cohomology grows polynomially.
In this talk, I will explore the concept of decorated ribbon graphs, graphene diagrams, and virtual links. Then introduce a novel coloring technique called "color evaluation" and some of its structures in graphene diagrams.
This talk is about a construction on smooth manifolds. The surgery is by Li-Li and was discovered in 2007. This surgery was used by Li-Li to solve the minimal genus problem in various 4-manifolds. My work (unpublished) talks about the case in which the surfaces are transverse (rather than disjoint).
The stem-wise computation of unstable homotopy groups of spheres is a historically coveted problem in algebraic topology which has seen little progress in the past several decades. In this talk we will discuss a novel approach to the problem which combines the composition methods of Toda with the unstable Adams spectral sequence via computer automation. A brief survey will be given of our new results as well as their applications to more tangible problems of a geometric nature.
The cohomology of arithmetic groups has connections to many areas of mathematics such as number theory and diffeomorphism groups. Classifying spaces of congruence subgroups of symplectic groups have an algebro-geometric interpretation as the moduli space of principally polarized abelian varieties with level structures. These congruence subgroups Sp_2n(Z,L) are the kernel of the mod-L reduction map Sp_2n(Z) to Sp_2n(Z / LZ). By work of Borel-Serre, H^i(Sp_2n(Z,L)) vanishes for i > n^2. I will report on lower bounds in the top degree i = n^2.
In a joint work with Lorenzo Ruffoni, we prove that double suspension of any finite 3-dimensional CW complex X is wedge sum of spheres and Moore spaces, which can be ascertained from the integral homology groups of the space X. This was motivated by looking at suspension homotopy type of geometrically finite groups, in connection to a conjecture by Gromov. We find another application in concluding certain closed aspherical 4-manifolds, construction by Davis trick, are spin.
In this talk I will explain a connection between elliptic fibrations of complex surfaces, spherical braid groups, and SL_2 character varieties. The only background assumed will be standard first-year graduate courses.
Motivated by the Mordell-Weil group in number theory, Farb and Looijenga constructed a subgroup of the smooth mapping class group of generic elliptically fibered 4-manifolds. But what happens if we allow all types of singular fibers? The story becomes much richer. In this talk, we consider a specific elliptic surface with general fibers and introduce some beautiful subgroups of its mapping class group.
In this talk we will give a method to construct Heegaard splittings of oriented graph manifolds with orientable bases. A graph manifold is a closed 3-manifold admitting only Seifert-fibered pieces in its Jaco–Shalen–Johansson decomposition.
L. Lusternik and L. Schnirelmann introduced their 'category' (now the so-called LS-Category) in 1934 as a numerical invariant to study (among other things) criitcal points of real-valued functions on manifolds. Since then it's been generalized and used in several different settings. We try to do the same in the context of large-scale geometry, defining the 'Coarse LS-Category' (or c-cat in short).