In this talk I will cover some recent results and techniques that should have a big impact on the classical theory of complete embedded minimal surfaces in the next decade. It is my hope that many of the outstanding conjectures in the subject will find their solution in the new theory that I will discuss. As this is a GANG seminar, my talk will one in which there is a dialogue between the speaker and the audience, so feel free to interrupt me at any time.

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As a "multi-media" presentation, this talk will be in GANG (1535) or the colloquium room.

Embedded constant mean curvature (CMC) surfaces are critical points for the area functional subject to an enclosed volume constraint. Physically, they correspond to equilibrium fluid droplets and bubbles. Their study impacts many different areas of mathematics and several applications outside mathematics. In mathematics, the equation of prescribed mean curvature is the best-studied quasilinear elliptic partial differential equation (PDE); its features guide the study of many other nonlinear PDEs. In applications, the design of a spacecraft fuel tank, for example, is determined by considerations of what family of CMC surfaces (the fuel-vapor interfaces) occur as the tank is emptied (in zero-gravity). These lectures will focus on finite topology CMC surfaces and their CMC deformation families, i.e. the moduli spaces M(g,k) of CMC surfaces with genus g and k ends. We will discuss the topological, differentiable and geometric properties of M(g,k), and use this information to better understand the constituent CMC surfaces and their deformations.

Considerations of rotational invariance in one-dimensionally modulated systems such as smectics-A, necessitate nonlinearities in the free energy. The presence of these nonlinearities is critical for determining the layer configurations around defects. We generalize our recent construction for finding exact minima of an approximate nonlinear free energy to the full, rotationally invariant smectic free energy. Our construction exhibits the detailed connection between mean curvature, Gaussian curvature and layer spacing. For layers without Gaussian curvature, we reduce the Euler^? Lagrange equation to an equation governing the evolution of a surface. As an example, we determine the layer profile and free energy of an edge dislocation. All physics terms will be happily explained!

CANCELED for Physics talk

Excluding the sphere, the unduloids are the simplest constant mean curvature (CMC) surfaces in R^3 and can be used as building blocks to construct an important class of CMC surfaces. In this talk, I will present the basic properties of the unduloids and properties of the space of unduloids viewed as a manifold (moduli space). In particular, this manifold possesses a natural 2-form which is believed, but not proven, to be a symplectic form.

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DIFFERENT TIME: 2:30. This is joint with QFT. Probably in 1634.

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A calibrated submanifold is a special type of minimal submanifold. Some of the most interesting types of calibrated submanifolds arise inside of manifolds with special holonomy. There are two types of calibrated geometries that arise in 7-manifolds whose holonomy is contained in G_2: the associative 3-folds and the coassociative 4-folds. There are very few examples of coassociative 4-folds in the literature. Bryant and Ionel applied certain symmetry techniques to special Lagrangian geometry (a calibrated geometry associated to SU(n) holonomy) in order to find nice families and explicit examples. I will discuss how these same techniques produce special families of coassociative 4-folds. In doing so an interesting duality between associative cones and coassociative cones arises. Using either the primitive map perspective in the work of Kong, Terng, and Wang or the twistor perspective in Bryant's work leads to methods for constructing examples of coassociative 4-folds.