Tag - Topology
Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. In fact, even without complete classification statements, the techniques used can suggest constructions of symplectic fillings with interesting topology. These fillings can be used in cut-and-paste operations called star surgery to construct examples of exotic 4-manifolds.
We consider the one-parameter families of transfer operators for geodesic flows on negatively curved manifolds. We show that the spectra of the generators have some "band structure" parallel to the imaginary axis. As a special case of "semi-classical" transfer operator, we see that the eigenvalues concentrate around the imaginary axis with some gap on the both sides.
I will discuss some recent results with Aaron Brown and Zhiren Wang on actions by higher rank lattices on nilmanifolds. I will present the result in the simplest case possible, SL(n,ℤ) acting on 𝕋n, and try to present the ideas of the proof. The result imply existence of invariant measures for SL(n,ℤ) actions on 𝕋n with standard homotopy data as well as global rigidity of Anosov actions on infranilmanifolds and existence of semiconjugacies without assumption on existence of invariant measure.
I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions.
The pentagram map and its analogues act on interesting and complicated spaces. The simplest of them is the classical moduli space M0,n of rational curves of genus 0. These moduli spaces have a rich combinatorial structure related to the notion of "Coxeter frieze pattern" and can be understood as a "cluster manifolds". In this talk, I will explain how to describe the action of the pentagram map (and its analogues) in terms of friezes. The main goal is to understand how this action fits with the cluster algebra structure, and in particular, with the canonical (pre)symplectic form.
For a geometrically finite hyperbolic group with small critical exponent, the spectral
method for counting is not available, as there is no point eigenvalue of the Laplace operator on the L2-spectrum. We will explain counting results for orbits of a big class of thin groups acting on a symmetric variety of the real hyperbolic group, which are obtained via ergodic approach.
We will describe a recent effective counting result for Apollonian circle packings. The main ingredient of this result is an effective equidistribution of closed horospheres in an infinite volume hyperbolic 3-manifold whose fundamental group has critical exponent bigger than one. We will explain how the spectral theory of Lax and Phillips can be used for such equidistribution results.
An elementary homomorphism from a free group to the pure braid group yields interesting connections between braid groups, homotopy theory, and low dimensional topology. This map induces a map on the Lie algebra obtained from the descending central series. Further, this map induces a morphism of simplicial groups. All of these maps are shown to be injective.
Brunnian braids are discussed. The analogous maps of Lie algebras induced on the filtration quotients of the mod-p descending central series is again an injection. Using these facts it turns out that the homotopy groups of this simplicial group, those of the 2-sphere, are isomorphic to natural subquotients of the pure braid group. In addition, the mod-p analogues give a connection between the classical unstable Adams spectral sequence, and the mod-p analogues of Vassiliev invariants of pure braids.
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