Georgia Institute of Technology College of Sciences

Poincare Conjecture, undoubtedly, is the most influential and challenging problem in the world of Geometry and Topology. Over a century, it has left it’s mark on developing the rich theory around it. In this talk I will give a brief history of the development of Topology and then I will focus on the Exotic behavior of manifolds. In the last part of the talk, I will concentrate more on the theory of 4-manifolds.

Two knots are concordant to each other if they cobound an annulus in the product of S^3. We will discuss a few basic facts about knot concordance and look at J. Levine’s classical result on the knot concordance group.

 How can we recognize a map given certain combinatorial data? The Alexander method gives us the answer for self-homeomorphisms of finite-type surfaces. We can determine a homeomorphism of a surface (up to isotopy) based on how it acts on a finite number of curves. However, is there a way to apply this concept to recognize maps on other spaces? The answer is yes for a special class of maps, post-critically finite quadratic polynomials on the complex plane (Belk-Lanier-Margalit-Winarski). 

            In this talk, we will discuss Belk-Lanier-Margalit-Winarski’s methods, as well zome difficulties we face when trying to extend their methods to other settings.

Every closed 3-manifold admits foliations, where the leaves are surfaces. For a given 3-manifold, which surfaces can appear as leaves? Kerékjártó and Richards gave a classification up to homeomorphism of noncompact surfaces, which includes surfaces with infinite genus and infinitely many punctures. In their 1985 paper "Every surface is a leaf", Cantwell--Conlon prove that for every orientable noncompact surface L and every closed 3-manifold M, M has a foliation where L appears as a leaf. We will discuss their paper and construction and the surrounding context.