Georgia Institute of Technology College of Sciences

Abstract:

We present several results on exotic open 4-manifolds and knotted surfaces in 4-manifolds. These include obstructive, constructive, and qualitative results. 

Given a knot diagram, there are certain questions one can ask: Is this a diagram of the unknot? What is the minimal crossing number? Is this knot a connected sum of two smaller knots? These questions are examples of problems in knot theory for which we do not have an efficient algorithm to solve. In this talk, we will see how these computationally hard problems in knot theory can serve as underpinnings for a quantum-secure cryptography system.

Nash's isometric embedding theorem is a fundamental result in differential geometry, establishing that every smooth Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. Nash's original proof relies on a sophisticated iterative scheme to overcome the loss of derivatives arising from the nonlinear isometric embedding equations. In the late 1980s, Matthias Günther introduced an alternative approach that avoids the Nash–Moser iteration scheme and the associated tame estimates, considerably simplifying the argument.

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.

Symplectic geometry has become very important for our understanding of 4-manifolds and exotic structures on them. In this talk, I hope to: 

A fundamental question in contact topology is to understand which symplectic 4-manifolds can have a given contact 3-manifold as their boundary. Such 4-manifolds are called symplectic fillings, and their classification is an important problem in symplectic topology.In this talk, I will introduce the problem of classifying symplectic fillings and discuss how cutting a contact 3-manifold along certain special tori can help us understand its fillings.

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”).

The Conway knot C is an 11-crossing knot that bounds a locally flat disk in B^4. After many decades of unknown status, C was proven to not bound a smoothly embedded disk by Piccirillo. In this talk I will show that in fact, no sum of Conway knots bounds a smoothly embedded disk. This is joint work with Marco Golla (and independent of work of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.)

The dual Garside left canonical form is used to solve the conjugacy problem in braid theory, introduced by Xu (3-braids), Kang-Ko-Lee (4-braids) and Birman-Ko-Lee (n-braids). The fractional Dehn twist coefficient (FDTC) is a measurement of the boundary twist of a surface homeomorphism, introduced by Honda-Kazez-Matic. In contact geometry, it is a powerful tool to detect tightness/overtwistedness of a given contact structure. In this talk, I will give applications of the left canonical form to the computation of the FDTC of 4-braids.