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A group with a cyclically symmetric presentation admits an automorphism of finite order called the shift. In this talk we look at cyclically presented groups which admit a certain decomposition, and relate the shift dynamics for the group to that of its decomposition. Topological methods are used to identify fixed points for powers of the shift.
I will talk about a probabilistic cascade structure that can be naturally associated with certain partial differential equations and how it can be used to study well-posedness questions.
In the context of the still unsolved uniqueness problem for the 3D Navier-Stokes equations, our aim will be to see how the explosion properties of such cascades help establish a connection between the uniqueness of symmetry-preserving (self-similar) solutions and the uniqueness of the general problem.
Based on the joint work with N. Michalowski, E. Thomann, and E. Waymire.
This is joint work with Rob Costa and Patrick Dynes. We prove that if an n x n matrix defined over the field of p-adic numbers satisfies a certain congruence property, then it has a strictly maximal eigenvalue in Q_p and that iteration of the (normalized) matrix converges to a projection operator onto the corresponding eigenspace. This result may be viewed as an analogue of the Perron-Frobenius theorem for positive real matrices.
In August, Radu Dascaliuc (PI) was awarded National Science Foundation grant DMS-1516487, “Collaborative research: Turbulent cascades and dissipation in the 3D Navier-Stokes model.” The main theme of the project is a rigorous study of various manifestations of turbulence in three-dimensional fluid flows modeled by the Navier-Stokes equations. This is considered both from the perspective of the mathematical theory of turbulence, and as a physical mechanism underlying possible blow-ups (singularities) of the solutions of the system. This is a three-year grant in the amount of $79,000....