Geometry and Topology

Series
Time
for
Location
Speaker
Organizer

Planar contact manifolds have been intensively studied to understand several aspects of 3-dimensional contact geometry. In this talk, we define "iterated planar contact manifolds", a higher-dimensional analog of planar contact manifolds, by using topological tools such as "open book decompositions" and "Lefschetz fibrations”. We provide some history on existing low-dimensional results regarding Reeb dynamics, symplectic fillings/caps of contact manifolds and explain some generalization of those results to higher dimensions via iterated planar structure.

Series
Time
for
Location
Speaker
Organizer
In 1957, Smale proved a striking result: we can turn a sphere inside out without any singularity. Gromov in his thesis, proved a generalized version of this theorem, which had been the starting point of the h-principle. In this talk, we will prove Gromov's theorem and see applications of it.
Series
Time
for
Location
Speaker
Organizer
We show that for any connected sum of lens spaces L there exists a connected sum of lens spaces X such that X is rational homology cobordant to L and if Y is rational homology cobordant to X, then there is an injection from H_1(X; Z) to H_1(Y; Z). Moreover, as a connected sum of lens spaces, X is uniquely determined up to orientation preserving diffeomorphism. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p.
Series
Time
for
Location
Speaker
Organizer
This is the second lecture of the series on h-principle. We will introduce jet bundle and it's various properties. This played a big role in the devloping modern geometry and topology. And using this we will prove Whitney embedding theorem. Only basic knowledge of calculus is required.
Series
Time
for
Location
Speaker
Organizer
Take a branched covering map of the sphere over itself so that the forward orbit of each critical point is finite. Such maps are called Thurston maps. Examples include polynomials with well-chosen coefficients acting on the complex plane, as well as twists of these by mapping classes. Two basic problems are classifying Thurston maps up to equivalence and finding the equivalence class of a Thurston map that has been twisted.
Series
Time
for
Location
Speaker
Organizer
Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g ≥ 2. Given a finite subgroup H < Mod(Sg), let Fix(H) denote the set of fixed points induced by the action of H on the Teichmuller space Teich(Sg). In this talk, we give an explicit description of Fix(H), when H is cyclic, thereby providing a complete solution to the Modular Nielsen Realization Problem for this case. Among other applications of these realizations, we derive an intriguing correlation between finite order maps and the filling systems of surfaces.
Series
Time
for
Location
Speaker
Organizer
It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically/Stein fillable when r is in (0,1].
Series
Time
for
Location
Speaker
Organizer
Understanding contact structures on hyperbolic 3-manifolds is one of the major open problems in the area of contact topology. As a first step, we try to classify tight contact structures on a specific hyperbolic 3-manifold. In this talk, we will review the previous classification results and classify tight contact structures on the Weeks manifold, which has the smallest hyperbolic volume. Finally, we will discuss how to generalize this method to classify tight contact structures on some other hyperbolic 3-manifolds.
Series
Time
for
Location
Speaker
Organizer
The theorem of Dehn-Nielsen-Baer says the extended mapping class group is isomorphic to the outer automorphism group of the fundamental group of a surface. This theorem is a beautiful example of the interconnection between purely topological and purely algebraic concepts. This talk will discuss the background of the theorem and give a sketch of the proof.
Series
Time
for
Location
Speaker
Organizer
We will discuss a celebrated theorem of Sharkovsky: whenever a continuous self-map of the interval contains a point of period 3, it also contains a point of period n , for every natural number n.

Pages

Subscribe to RSS - Geometry and Topology