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Events
Prospects in Geometry 2004

MPI für Mathematik in den Naturwissenschaften Leipzig

G3 10 (Lecture hall)

conference

25.11.04 27.11.04

Prospects in Geometry 2004

The conference is supported by Graduate College "Analysis, Geometry and their Interaction with the Natural Sciences", Leipzig University, Leipzig University.

Speakers

Björn Andreas

Humboldt Universität, Berlin

Paul Biran

Tel-Aviv University

Oscar Garcia-Prada

Consejo Superior de Investigaciones Cientificas, Madrid

Thomas Kappeler

Universität Zürich

Ernst Kuwert

Universität Freiburg

Peter Michor

Universität Wien

Gabriel P. Paternain

University of Cambridge

Tristan Rivière

ETH Zürich

Kang Zuo

Universität Mainz

Program

11:00 - 12:00	Peter Michor (Universität Wien, Austria) From Riemannian geometries on spaces of plane curves to vanishing geodesic distance on spaces of submanifolds and of diffeomorphisms. Joint work with David Mumford.We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of immersions from $S1$ to the plane modulo the group of diffeomorphisms of $S1$, acting as reparameterizations. In particular we investigate the metric for a constant $A> 0$: $$ G^A_c(h,k) := \int_{S1}(1+A\kappa_c(\theta)2)\langle h(\theta),k(\theta)\rangle \|c'(\theta)\|\,d\theta $$ where $\kappa_c$ is the curvature of the curve $c$ and $h,k$ are normal vector fields to $c$. For $A=0$, the geodesic distance between any two distinct curves is 0, while for $A>0$ the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are $\ge 0$, while for curves with high curvature or perturbations of high frequency, the curvatures are $\le 0$. In fact, many results hold in a much more general situation. The $L2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the $L2$-metric. This is in particular true for Burgers' equation.
15:00 - 16:00	Ernst Kuwert (Universität Freiburg, Germany) The Willmore functional from an analytic viewpoint
16:30 - 17:30	Tristan Rivière (ETH Zürich, Switzerland) Uniqueness of tangent cones and regularity issues for calibrated integral currents

09:30 - 10:30	Thomas Kappeler (Universität Zürich, Switzerland) On geodesic exponential maps of the Virasoro group
11:00 - 12:00	Kang Zuo (Universität Mainz, Germany) Complex multiplication, Griffiths-Yukawa couplings and rigidity
15:00 - 16:00	Oscar Garcia-Prada (Consejo Superior de Investigaciones Cientificas, Spain) Representations of surface groups and Higgs bundles
16:30 - 17:30	Björn Andreas (Humboldt Universität, Germany) U(N) Vector Bundles on Elliptic Calabi-Yau Threefolds for String Theory Compactification
19:00 - 20:00	joint Dinner at "Auerbachs Keller" (on the basis of à la carte and individual payment)

09:30 - 10:30	Gabriel P. Paternain (University of Cambridge, United Kingdom) Zero entropy and bounded topology
11:00 - 12:00	Paul Biran (Tel-Aviv University, Israel) Projective Embeddings and Lagrangian Cycles

Prospects in Geometry 2004

Speakers

Björn Andreas

Paul Biran

Oscar Garcia-Prada

Thomas Kappeler

Ernst Kuwert

Peter Michor

Gabriel P. Paternain

Tristan Rivière

Kang Zuo

Program

Organizers

Jürgen Jost

Hans-Bert Rademacher

Matthias Schwarz

Administrative Contact

Antje Vandenberg