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.
In 1925, Heisenberg introduced non-commutativity of coordinates, now known as quantization, to explain the spectral lines of atoms. In topology, finding quantizations of (symplectic or more generally Poisson) spaces can reveal more intricate structures on them. In this talk, we will introduce the main ingredients of quantization. As a concrete example, we will discuss the SL2-character variety, which is closely related to the Teichmüller space, and the skein algebra as its quantization.
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.
Finding fillings of contact structures is a question that has been studied extensively over the last few decades. In this talk I will discuss some motivations for studying this question, and then visit a few ideas involved in the earliest results, due to Eliashberg and McDuff, that paved the way for a lot of current research in this direction.
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).
An infinite-type surface is a surface whose fundamental group is not finitely generated. These surfaces are “big,” having either infinite genus or infinitely many punctures. Recently, it was shown that mapping class groups of these infinite-type surfaces have a wealth of subgroups; for example, there are infinitely many surfaces whose mapping class group contains every countable group as a subgroup.
For a manifold M, the (generalized) Nielsen realization problem asks if the surjection Diff(M) → π_0 Diff(M) is split, where Diff(M) is the diffeomorphism group. When M is a surface, this question was posed by Thurston in Kirby's problem list and was addressed by Morita. I will discuss some more recent work on Nielsen realization problems with connections to flat fiber bundles, K3 surfaces, and smooth structures on hyperbolic manifolds.
Computing, understanding the behavior of concordance invariants obtained from knot Floer homology theories is quite central to the study of the concordance group and low-dimensional topology in general. In this talk, I will describe the method that allows us to compute the concordance invariant epsilon using combinatorial knot Floer homology and talk about some computational results. This is a joint work with S. Dey.
I will compare and contrast a selection of popular equivalence relations on 4 manifolds, and explain some recent progress on classification results.
The speaker will hold online office hours from 3:00-4:00 pm for interested graduate students and postdocs.
All 3-manifolds can be described as surgery on links in the three-sphere by the celebrated theorem of Lickorish and Wallace. Motivated by the L-space conjecture, it is interesting to understand what surgery manifolds are L-spaces, which have the simplest possible Floer homology such as lens spaces. In this talk, we concentrate on surgeries on a family of links, which are called L-space links, and show possible L-space surgeries on such links. We also give some link detection results in terms of the surgeries.
