Recently, using the desingularization technique, a new family of complete properly embedded self-shrinkers asymptotic to cones in three dimensional Euclidean space has been constructed by Kapouleas-Kleene-Moeller and independently by Nguyen.

In this talk, we present the uniqueness of self-shrinking ends asymptotic to any given cone in general Euclidean space. The feature of our uniqueness result is that we do not require the control on the boundaries of self-shrinking ends or the rate of convergence to cones at infinity. As applications, we show that, there do not exist complete properly embedded self-shrinkers other than hyperplanes having ends asymptotic to rotationally symmetric cones.

Nonlinear dynamical systems arising in biological, physical and chemical sciences are often subject to random influences, which are also known as noise. Stochastic differential equations are appropriate models for some of these systems. The noise in these stochastic differential equations may be Gaussian or non-Gaussian in nature. Non-Gaussianity of the noise manifests as nonlocality at some macroscopic level. In addition, randomness may have delicate, or even profound, impact on the overall evolution of dynamical systems. The speaker will present an overview of some available theoretical and numerical techniques for analyzing stochastic dynamical systems, especially escape probability, mean exit time, invariant manifolds, bifurcation and quantifying the impact of uncertainty. The differences in dynamics under Gaussian and non-Gaussian noises are highlighted, theoretically or numerically.

Skew products over subshifts of finite type naturally appear when one attempts to apply the methods of classical dynamical systems to random dynamical systems. There is also a close connection between these skew products and partially hyperbolic dynamical systems on smooth manifolds.

Even for the fiber dimension equal to one, we are far from understanding what typical skew products look like. During the last 30 years there appeared several papers studying the skew products with a circle fiber. I will talk about the case when the fiber is an interval, and fiber maps are orientation-preserving diffeomorphisms.

In the work joint with V. Kleptsyn, we developed a theorem which gives us a complete* description of the dynamics of typical step skew products (fiber map depends only on a single symbol in the base sequence). We also obtained a similar result for generic skew products using an additional assumption of partial-hyperbolic nature.

*except some subset which projects onto zero measure set in the base

A very useful technique in homological algebra, is the Basic Peturbation Lemma which tells how cohomology of a complex changes when a differential is perturbed. It has numerous applications in algebra, topology, and computational methods in algebra; some of which will be reviewed in the talk.