at the
American Institute of Mathematics, Palo Alto, California
organized by
John Mallet-Paret, Roger Nussbaum, and Hans-Otto Walther
This workshop, sponsored by AIM and the NSF, will focus on low (Hausdorff) dimensional structures in differential delay equations with state dependent time lags. The study of such structures seems to offer the best hope of progress in the broader field. We mention some examples below.
Periodic solutions of differential delay equations often play a central role in understanding the dynamics of the equations, but even proving existence of such solutions may be nontrivial. Related problems include uniqueness and stability questions, regularity (real analyticity) of solutions, and limiting shapes of periodic solutions under singular limits. Nontrivial examples are already provided by equations such as ax'(t) = f(x(t), x(t-r)), where a > 0 and r := r(x(t)), and f and r are given functions.
The local theory of invariant manifolds is reasonably well-developed for differential delay equations with constant time lags, but extending the theory to the state dependent case presents difficulties, e.g., in proving higher order differentiability of invariant manifolds. Related problems are also present for compact attractors, which are typically known to have finite Hausdorff dimension in the constant time lag case, and suspected so for the variable time lag case.
The presence of a Morse decomposition for the maximal attractor occurs for certain classes of systems with a single delay, or a cyclic structure, with a signed feedback. Many such systems, under generic conditions, are of Morse-Smale type and much effort has gone into studying their global structure. Broadening these results to systems with multiple delays is a major open question of much significance, both for the theory and scientific applications.
One goal of the workshop will be to bring together established figures in the field with younger researchers who may become the next generation of leaders. Since many of the equations which have been extensively studied have their origin in simplified models from applications, another objective is to facilitate conversations among researchers with widely varying degrees of interest in applications. We expect significant and fruitful advances in the field to emerge from these interactions.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
For more information email workshops@aimath.org
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