The shadowing problem is related to the following question: under which condition, for any pseudotrajectory (approximate trajectory) of a vector field there exists a close trajectory? We study $C^1$-interiors of sets of vector fields with various shadowing properties. In the case of discrete dynamical systems generated by diffeomorphisms, such interiors were proved to coincide with the set of structurally stable diffeomorphisms for most general shadowing properties.

We prove that the $C^1$-interior of the set of vector fields with Oriented shadowing property contains not only structurally stable vector fields. Also, we have found additional assumptions under which the $C^1$-interiors of sets of vector fields with Lipschitz, Oriented and Orbit shadowing properties contain only structurally stable vector fields.

Some of these results were obtained together with my advisor S.Yu.Pilyugin.

I will discuss the reliability of large networks of coupled oscillators in response to fluctuating inputs. The networks considered are quite generic. In this talk, I view them as idealized models from neuroscience and borrow some of the associated language. Reliability is the opposite of trial-to-trial variability; a system is reliable if a signal elicits identical responses upon repeated presentations. I will address the problem on two levels: neuronal reliability, which concerns the behavior of individual neurons (or oscillators) embedded in the network, and pooled-response reliability, which measures total outputs from subpopulations. The effects of network structure, cell heterogeneity and noise on reliability will be discussed. Our findings are based largely on dynamical systems ideas (with a slight statistical mechanics flavor) and are supported by simulations. This is joint work with Kevin Lin and Eric Shea-Brown.

This is joint work with Richard Ketchersid.

Schipperus introduced the countable-finite game in the early 1990s. It is
an infinite game played between two players relative to a set S. In the
presence of choice, it is obvious that player II has a winning strategy
for all S, and it is natural to ask whether choice can be dispensed with.

AD+ is a technical strengthening of AD introduced by Hugh Woodin. It is
open whether AD+ actually follows from AD. All known models of AD come
from certain canonical models produced by the derived model construction.

In these canonical models, we show that every set either embeds the reals
or else is well-orderable.

From this we deduce that, except for the case when S is countable, the
countable-finite game on S is undetermined in these models.