Georgia Institute of Technology College of Sciences

For any smooth manifold, the topology of its tangent bundle determines much about how vector fields behave on it. On a closed, connected, oriented surface, every tangent vector field must vanish somewhere, unless the surface is a torus. The hairy ball theorem gives the familiar example: you can’t comb a hairy sphere without making a cowlick.

The situation changes dramatically in dimension 3. Not only can we always find nonvanishing vector fields on closed, oriented 3-manifolds, but their tangent bundles are entirely trivial! In this talk, we’ll prove this remarkable fact using obstruction theory and introduce several ideas from algebraic topology that help us understand vector fields on manifolds.

On a given Riemann surface, we may talk about various projective structures that are compatible with the given complex structure. These structures are equivalent to what are known as $sl_2$ opers, or in terms of Faltings’ paper that we mainly follow, as permissible connections. Then there are several interesting questions we may ask about the monodromies of these objects, and the one we focus on today is whether the monodromy group is conjugate into $PSL_2(\mathbb R)$ in $PSL_2(\mathbb C)$ (“the real monodromy property”). In this talk, we illustrate (literally!) Faltings’ idea on this matter by some pictures and examples, and explain how uniformization and accessory problem fit into this theory. If time permitted, we will also see how this shows up in the recent Analytic Langlands program proposed by Etingof, Frenkel and Kazhdan.

Heegaard Floer homology is a modern algebraic package that has seen utility in many areas of low dimensional topology. In this introductory talk, we will give a birds eye view of its construction and properties, and present a highlight reel on the problems it helps solve. As many of our beloved faculty find use for Floer homology in their research, this talk intends to ease the newcomer into the vernacular they may hear in a Seminar talk, and at the same time, convince them that it is a beautiful and rich field of study. We do not assume knowledge in low dimensional topology. It is also rumoured that some Dunkin Donuts will be in attendance.

In his 1956 paper "On manifolds homeomorphic to the 7-sphere'', John Milnor constructed some examples of manifolds that are homeomorphic, but not diffeomorphic, to the standard unit sphere. They are now called exotic 7-spheres. This example established that the differential structure of a manifold can carry information not given by its topological structure. Thus, Milnor founded differential topology as a stand-alone field. On my first reading of the paper, I thought that many of the choices Milnor made on his road to constructing the first exotic spheres seemed rather strange and arbitrary. Why 7-spheres? Why look for them among $S^3$-bundles over $S^4$ with structure group $SO(4)$? And what's the motivation behind his complicated mod 7-valued lambda invariant that detects exotica in these examples? Fortunately, Milnor answered some of these questions in his essay "Classification of $(n-1)$-connected $2n$-dimensional manifolds and the discovery of exotic spheres''. This talk is an attempt to understand this essay.