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The space of Euclidean/spherical/hyperbolic polyhedral metrics on a surface with cone points is the set of all equivalence classes of Euclidean/spherical/hyperbolic polyhedral metrics on the surface. The decorated Teichmüller space is the set of all decorated hyperbolic metric on the surface minus the cone points up to isometry isotopic to the identity map. The Teichmüller space of a surface with boundary is the space of all equivalence classes of hyperbolic metrics with geodesic boundary on the surface minus the cone points. The five spaces of metrics are C^1-diffeomorphic to each other.
The 2-D Surface Quasi-Geostrophic Equations have been under
investigation due to some interesting ties between this 2-D model and the
3-D Euler equations. In this talk we will explore a blow-up condition on
solutions to the 2D Quasi-Geostrophic Equations that is analogous to the
well known Beale-Kato-Majda blow-up condition for solutions to 3D Euler.
The dynamics of fluids, i.e. liquids and gases, is an important part of the continuum mechanics. This lecture is devoted to the qualitative analysis of mathematical models of motion of a viscous incompressible fluid around a compact body B, translating and rotating in the fluid with given time-independent translational and angular velocities u∞ and ω. The translation can be considered, without the loss of generality, to be parallel to the x3-axis.
Following a strategy developed by Athreya and Cheung, we compute the gap distribution of the slopes of saddle connections on the octagon by translating the problem to a question about return times of the horocycle flow to an appropriate Poincare Section. This same strategy was used by Athreya, Chaika, and Lelievre to compute the gap distribution on the Golden L. The octagon is the first example of this type of computation where the Veech group has two cusps.
We describe and characterize an extension to the classical path coupling method referred to as aggregate path coupling. In conjunction with large deviations estimates, we use this aggregate path coupling method to prove rapid mixing of Glauber dynamics for a large class of statistical mechanical models, including models that exhibit discontinuous phase transitions which have traditionally been more difficult to analyze rigorously.
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....