Seminars and Colloquia by Series

Euler equation with fixed or free boundaries - from a Lagrangian point of view

Series
School of Mathematics Colloquium
Time
Thursday, November 6, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Chongchun Zeng – School of Mathematics, Georgia Tech
In this talk, we discuss 1.) the nonlinear instability and unstable manifolds of steady solutions of the Euler equation with fixed domains and 2.) the evolution of free (inviscid) fluid surfaces, which may involve vorticity, gravity, surface tension, or magnetic fields. These problems can be formulated in a Lagrangian formulation on infinite dimensional manifolds of volume preserving diffeomorphisms with an invariant Lie group action. In this setting, the physical pressure turns out to come from the combination of the gravity, surface tension, and the Lagrangian multiplier. The vorticity is naturally related to an invariant group action. In the absence of surface tension, the well-known Rayleigh-Taylor and Kelvin-Helmholtz instabilities appear naturally related to the signs of the curvatures of those infinite dimensional manifolds. Based on these considerations, we obtain 1.) the existence of unstable manifolds and L^2 nonlinear instability in the cases of the fixed domains and 2.) in the free boundary cases, the local well-posedness with surface tension in a rather uniform energy method. In particular, for the cases without surface tension which do not involve hydrodynamical instabilities, we obtain the local existence of solutions by taking the vanishing surface tension limit.

Topology of Riemannian submanifolds with prescribed boundary

Series
School of Mathematics Colloquium
Time
Thursday, October 16, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mohammad Ghomi – School of Mathematics, Georgia Tech
We prove that a smooth compact submanifold of codimension $2$ immersed in $R^n$, $n>2$, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman. These results consist of joint works with Stephanie Alexander and Jeremy Wong, and Robert Greene.

Geometry and Topology in Fluid Mechanics

Series
School of Mathematics Colloquium
Time
Thursday, October 2, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John Etnyre – School of Mathematics, Georgia Tech
Describe the trajectories of particles floating in a liquid. This is a surprisingly difficult problem and attempts to understand it have involved many diverse techniques. In the 60's Arold, Marsden, Ebin and others began to introduce topological techniques into the study of fluid flows. In this talk we will discuss some of these ideas and see how they naturally lead to the introduction of contact geometry into the study of fluid flows. We then consider some of the results one can obtain from this contact geometry perspective. For example we will show that for a sufficiently smooth steady ideal fluid flowing in the three sphere there is always some particle whose trajectory is a closed loop that bounds an embedded disk, and that (generically) certain steady Euler flows are (linearly) unstable.

Geometry and Topology in Fluid Mechanics

Series
School of Mathematics Colloquium
Time
Thursday, October 2, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John Etnyre – School of Mathematics, Georgia Tech
Describe the trajectories of particles floating in a liquid. This is a surprisingly difficult problem and attempts to understand it have involved many diverse techniques. In the 60's Arold, Marsden, Ebin and others began to introduce topological techniques into the study of fluid flows. In this talk we will discuss some of these ideas and see how they naturally lead to the introduction of contact geometry into the study of fluid flows. We then consider some of the results one can obtain from this contact geometry perspective. For example we will show that for a sufficiently smooth steady ideal fluid flowing in the three sphere there is always some particle whose trajectory is a closed loop that bounds an embedded disk, and that (generically) certain steady Euler flows are (linearly) unstable.

Dynamical networks - interplay of topology, interactions and local dynamics

Series
School of Mathematics Colloquium
Time
Thursday, September 18, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Leonid Bunimovich – School of Mathematics, Georgia Tech
It has been found about ten years ago that most of the real networks are not random ones in the Erdos-Renyi sense but have different topology (structure of the graph of interactions between the elements of a network). This finding generated a steady flux of papers analyzing structural aspects of networks.  However, real networks are rather dynamical ones where the elements (cells, genes, agents, etc) are interacting dynamical systems. Recently a general approach to the studies of dynamical networks with arbitrary topology was developed. This approach is based on a symbolic dynamics and is in a sense similar to the one introduced by Sinai and the speaker for Lattice Dynamical Systems, where the graph of interactions is a lattice. The new approach allows to analyse a combined effect of all three features which characterize a dynamical network (topology, dynamics of elements of the network and interactions between these elements) on its evolution. The networks are of the most general type, e.g. the local systems and interactions need not to be homogeneous, nor restrictions are imposed on a structure of the graph of interactions. Sufficient conditions on stability of dynamical networks are obtained. It is demonstrated that some subnetworks can evolve regularly while the others evolve chaotically. This approach is a very natural one and thus gives a hope that in many other problems (some will be discussed) on dynamical networks a progress could be expected.

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