Geometry Seminar Spring 2012

Organizers: Jacob Bernstein (jbern@math.*), Brian Clarke (bfclarke@*) and Yanir Rubinstein (yanir@math.*)

Time: Wednesdays at 4 PM

Location: 383N

(*=stanford.edu)

Speaker: Stephen Preston (Boulder)

Title: The geometry of whips and chains

Abstract:

I will discuss some partial differential equations that arise as geodesic equations on the manifold of planar curves parametrized by arc length (viewed as a submanifold of the flat space of all planar curves). The equations have very different properties depending on whether we use a metric defined by the $L^2$ norm or the $H^1$ norm. In the $L^2$ case we obtain a geometric wave equation (where the tension in the whip is determined by a nonlocal equation analogous to the pressure in an ideal fluid), which can be well-approximated by a finite-dimensional model (a chain with rigid links). In the $H^1$ case we get a smooth ODE on an infinite-dimensional manifold and global existence of solutions. I will explain various aspects of infinite-dimensional geometry on the way, and the talk should be accessible to students with some familiarity with basic ODEs, the wave equation, and basic submanifold Riemannian geometry.

Speaker: TBA

Title:

Abstract:

TBA

Speaker: Steffen Rohde (Univ. of Washington/MSRI)

Title: SLE and random maps

Abstract:

In the first part of my talk, I will discuss a selection of highlights around the Schramm-Loewner Evolution SLE, aiming to give an introduction for the non-specialist. In the second part, I will discuss connections to other (random) objects of current interest (circle homeomorphisms, trees, and planar maps), as well as speculation about the conformal geometry of the "Brownian map".

Speaker: Jean-Michael Bismut (Orsay)

Title: The hypoelliptic Laplacian

Abstract:

If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian L_b is an operator acting on the total space X of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian (when b-->0) and the geodesic flow (when b-->1). Up to lower order terms, L_b is a weighted sum of the harmonic oscillator along the fibre TX and of the generator of the geodesic flow. One expects that, in this deformation, there are conserved quantities. In the talk, I will explain the underlying algebraic, analytic and probabilistic aspects of its construction, and outline some of the applications obtained so far.

Speaker: Peter Petersen (UCLA)

Title: Existence and uniqueness of warped product Einstein manifolds.

Abstract:

We shall consider the classical problem of trying to find warped product metrics with a fixed base that are also Einstein. There are by now many such examples and some of them are quite surprising. We'll indicate a new construction where the base is an arbitrary algebraic Ricci soliton. We'll also explain when such constructions are forced to be unique as well as when they are not unique.

Speaker: Jie Qing (UCSC)

Title: Hypersurfaces in hyperbolic space and conformal metrics on domains in sphere

Abstract:

In this talk I will introduce a global correspondence between properly immersed horospherically convex hyper surfaces in hyperbolic space and complete conformal metrics on subdomains in the boundary at infinity of hyperbolic space. I will discuss when a horospherically convex hypersurface is proper and when the hyperbolic Gauss map is injective. These are expected to be useful to the understandings of both elliptic problems of Weingarten hypersurfaces in hyperbolic space and elliptic problems of complete conformal metrics on subdomains in sphere.

Speaker: Mike Wolf (Rice)

Title: Polynomial Pick forms for affine spheres and real projective polygons

Abstract:

(Joint work with David Dumas.) Convex real projective structures on surfaces, corresponding to discrete surface group representations into SL(3, R), have associated to them affine spheres which project to the convex hull of their universal covers. Such an affine sphere is determined by its Pick (cubic) differential and an associated Blaschke metric. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons; along the way, a strong estimate on asymptotics is found. We will carefully describe the background material.

Speaker: Jonathan Dahl (UC Berkeley)

Title: The multi-marginal optimal transportation problem and minimal networks

Abstract:

I will discuss the multi-marginal generalization of the optimal transportation problem, as well as the related minimal network problem in $L^2$ Wasserstein space. For the minimal network problem, I will also show methods for obtaining pointwise entropy and $L^\infty$ estimates.

Speaker: Davi Maximo (UT)

Title:On the blow-up of four dimensional Ricci flow singularities

Abstract:

In this talk we prove that the gradient shrinking soliton metric constructed by Feldman-Ilmanen-Knopf on the tautological line bundle over CP^1 is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four dimensional manifolds do not necessarily have non-negative Ricci curvature.

Speaker: TBA

Title:

Abstract:

TBA

Speaker: Stephen Preston (Boulder)

Title: The geometry of whips and chains

Abstract:

I will discuss some partial differential equations that arise as geodesic equations on the manifold of planar curves parametrized by arc length (viewed as a submanifold of the flat space of all planar curves). The equations have very different properties depending on whether we use a metric defined by the $L^2$ norm or the $H^1$ norm. In the $L^2$ case we obtain a geometric wave equation (where the tension in the whip is determined by a nonlocal equation analogous to the pressure in an ideal fluid), which can be well-approximated by a finite-dimensional model (a chain with rigid links). In the $H^1$ case we get a smooth ODE on an infinite-dimensional manifold and global existence of solutions. I will explain various aspects of infinite-dimensional geometry on the way, and the talk should be accessible to students with some familiarity with basic ODEs, the wave equation, and basic submanifold Riemannian geometry.

Speaker: Boris Botvinnik (Oregon)

Title: Concordance and isotopy of metrics with positive scalar curvature

Abstract:

Two positive scalar curvature metrics g0, g1 on a manifold M are called psc-isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics g0, g1 of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant, i.e., there exists a metric g of positive scalar curvature on the cylinder M × I which extends the metrics g0 on M × {0} and g1 on M × {1} and is a product metric near the boundary. We sketch a proof that if psc-metrics g0, g1 on M are psc-concordant, then there exists a diffeomorphism φ : M × I → M × I with φ|M×{0} = Id (a pseudoisotopy) such that the metrics g0 and (φ|M×{1})*g1 are psc-isotopic. In particular, for a simply connected manifold M with dim M ≥ 5, psc-metrics g0, g1 are psc-isotopic if and only if they are psc-concordant. To prove these results, we employ a combination of relevant methods: surgery tools related to Gromov-Lawson construction, classic results on isotopy and pseudo-isotopy of diffeomorphisms, standard geometric analysis related to the conformal Laplacian, and the Ricci flow.

Speaker: Bay Area Differential Geometry Seminar

Title:

Abstract:

See the webpage for details.

Speaker: Fernando Marques (IMPA)

Title:

Abstract:

TBA