Low-dimensional topology studies the shape of 3-dimensional and 4-dimensional spaces. Surprisingly, higher dimensional spaces are easier to understand as there is room to perform a certain topological trick. Hence dimensions 3 and 4 are the most difficult but also physically the most relevant.
Our Universe is 3-dimensional. We can only see a small segment of it limited by the range of our telescopes. This segment looks like 3-dimensional coordinate space, but globally it might have a more complicated shape. To better understand what we mean by that, imagine we are tiny short-sighted observers living in a 2-dimensional universe that is the surface of a doughnut. What we would see is indistinguishable from the 2-dimensional coordinate plane no matter where we stand, and we would find it difficult to decide whether our world is the plane, a sphere, or a doughnut. In case of the Earth, people observed its curvature to deduce it has the shape of a sphere.
If we also consider time, we obtain the concept of the 4-dimensional space-time, whose shape is a mystery. More generally, an n-manifold for any non-negative integer n is a space that locally looks like n-dimensional coordinate space. Topology is the study of the properties of such objects which are unchanged by continuous deformations, as if they were made out of rubber, but not in the local geometry such as its curvature. From the point of view of topology, the surface of the Earth and a sphere have the same shape: one can be continuously deformed into the other without tearing or puncturing it. To distinguish two spaces, topologists develop ingenious invariants that are unchanged by deformation. For example, the number of holes of a surface is a numerical invariant.
In dimension 3, the existence of knots becomes central due to the lack of room. There are many different ways one can tie a knot in a circular piece of rope. In fact, any 3- and 4-manifold can be described by a collection of such knots, each labeled by a number. 3-manifolds have been well understood using ideas from geometry, similarly to the way our predecessors realized the Earth is round by noticing its curvature.
However, the classification of 4-manifolds up to smooth deformations is still little understood. Here ideas from theoretical physics have been revolutionary. The most important open question is the smooth 4-dimensional Poincaré conjecture (SPC4): This states that if a 4-manifold can be continuously deformed into the standard 4-sphere, then this can also be done smoothly. Many experts believe this to be false, and to find a counterexample, it would suffice to show that a certain knot in 3-space is not the boundary of a disk in 4-space.
The first part of the project aims to develop tools sensitive enough to decide what kind of surfaces a given knot bounds in 4-dimensions, and use this to find a counterexample to SPC4. Surprisingly, this might also shed light on the action of certain enzymes called recombinases on the DNA which separates the copy of the DNA strand when a cell divides. In another direction, I am planning to use certain algebraic objects related to theoretical physics to define new invariants of 3- and 4-manifolds. Finally, in dimension 3, I will explore the relationship of a deep invariant of 3-manifolds and the geometries they possess.
