Southeast Geometry Conference 2011
Titles and Abstracts

Jacob Bernstein (Stanford University.)
- Title: A variational characterization of the catenoid.
- Abstract: We show that a suitable piece of the catenoid is the unique surface of least area within a geometrically natural class of minimal surfaces-- i.e. this property characterizes the catenoid. The proof relies on a techniques involving the Weierstrass representation used by Osserman and Schiffer to show the sharp isoperimetric inequality for minimal annuli. An alternate approach that avoids the Weierstrass representation will also be discussed. This latter approach depends on a conjectural sharp eigenvalue estimate for a geometric operater and has interesting connections with spectral theory. This is joint work with C. Breiner.

Jason Cantarella (University of Georgia.)
- Title: Distance Subspaces of Configuration Space and the Square Peg Problem.
- Abstract: Suppose we are interested in finding "special" configurations of points on a submanifold of Euclidean space. For instance, the "square peg" theorem says that there are an odd number of squares inscribed in a generic (C¹) plane curve. How can we go about this? In this talk, we'll describe this as a subspace problem: we are interested in detecting intersections of certain natural submanifolds of the configuration space of n points in a Euclidean space. We then develop some tools in the differential and algebraic topology of configuration space to handle such problems. The result is a sort of construction kit for square peg-like theorems.

Joe Fu (University of Georgia.)
- Title: Integral geometry of real and complex space forms.
- Abstract: We examine the comparative integral geometry of euclidean space, the sphere and hyperbolic space, in the light of recent algebraic developments in the subject. We then discuss the much more difficult (and in many respects still open) case of the complex space forms: Cⁿ, (under the action of the unitary group), complex projective space and complex hyperbolic space. Much of this represents joint work with A. Bernig and G. Solanes.

Mohammad Ghomi (Georgia Institute of Technology.)
- Title: Directed immersions of closed manifolds.
- Abstract: Given any finite subset X of the sphere Sⁿ, n > 1, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space Rⁿ⁺¹ whose Gauss map misses X. In particular, this answers a question of M. Gromov.

Mike Jablonski (University of Oklahoma.)
- Title: Ricci solitons on solvmanifolds.
- Abstract: Ricci solitons are special solutions to the Ricci flow which evolve by dilation and pullback by diffeomorphisms. While these metrics are interesting in their own right, they often arise from modified limits of the Ricci flow. As such, understanding these metrics is crucial for understanding the longterm behavior of this geometric evolution.
  We will discuss some new results for homogeneous Ricci solitons. Of particular interest is the case of solvmanifolds, i.e., when there exists a transitive solvable group of isometries. We will show that the only examples of Ricci soliton metrics on solvmanifolds (up to isometry) occur on solvable Lie groups, are necessarily simply-connected (when non-flat), and are so-called algebraic solitons.

John Pardon (Princeton University.)
- Title: Gromov's knot distortion.
- Abstract: Gromov defined the distortion of an embedding of S¹ into R³ and asked whether every knot could be embedded with distortion less than 100. There are (many) wild embeddings of S¹ into R³ with finite distortion, and this is one reason why bounding the distortion of a given knot class is hard. I will discuss recent work which shows that there exist knots which require arbitrarily large distortion. For example, torus knots require large distortion (by work of the speaker), as do the (knotted, connected) ramification sets of ramified covers M--> S³ where M is an arithmetic hyperbolic 3-manifold (work of Gromov and Guth). I will also mention some natural conjectures about the distortion, for example that the distortion of the (2,p)-torus knots is unbounded.

Jason Parsley (Wake Forest University.)
- Title: Cohomology reveals when helicity is a diffeomorphism invariant.
- Abstract: We consider the helicity of a vector field, which calculates the average linking number of the field∏s flowlines. Helicity is invariant under certain diffeomorphisms of its domain we seek to understand which ones. Extending to differential (k+1)-forms on domains R^2k+1, we express helicity as a cohomology class. This topological approach allows us to find a general formula for how much helicity changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicity-preserving diffeomorphisms on a given domain, finding new ones on the two-holed solid torus and proving that there are no new ones on the standard solid torus. This approach also leads us to define submanifold helicities: differential (k+1)-forms on n-dimensional subdomains of R^m.

Clay Shonkwiler (Haverford College.)
- Title: The Search for Higher Helicities.
- Abstract: Helicity was introduced in 1958 by Woltjer, who showed that the helicity of a magnetic field remains constant as the field evolves according to the equations of ideal magnetohydrodynamics and that it provides a lower bound for the field energy during such evolution. In fact, helicity is the only known topological invariant of vector fields. Arnol'd suggested that there should be higher helicities which provide lower bounds on the field energy when the first helicity vanishes. The integral which computes helicity is analogous to Gauss's famous integral formula for the linking number, suggesting that geometric integral formulas for higher-order linking invariants may lead to higher helicities. I will present such an integral formula for Milnor's triple linking number, which was discovered by proving a correspondence between the link-homotopy invariants of three-component links and the homotopy invariants of associated maps to a configuration space. This correspondence is of independent interest, as it establishes the n=3 case of a conjecture of Koschorke.

Ray Treinen (Mathematical Sciences Research Institute.)
- Title: Floating Drops: a survey of problems and results.
- Abstract: Consider the equilibrium state of three fluids with different densities. The configuration depends on surface tension values, as well as densities when in a gravitational field. Problems are introduced in various settings, with results and open problems presented. We will focus on the question of symmetry about a vertical line. Under certain conditions both light and heavy drops are shown to have this symmetry.

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