Research of Joseph M. Landsberg


My research is in differential geometry (primarily using exterior differential systems techniques),
algebraic geometry (primarily subvarieties of projective space) and their interactions with representation
theory (e.g., the geometry of rational homogeneous varieties).

Recently I have become interested in geometric approaches to questions arising in theoretical computer science
(complexity of matrix multiplication and P?=NP)  and more generally the geometry of
varieties in spaces of tensors. For an introduction to these topics, see the survey
article Geometry and the complexity of matrix multiplication (to appear in Bull. AMS)
 
I have a longstanding interest in the geometry of subvarieties of projective space. For
a survey of recent progress, see  Differential geometry of submanfolds
of projective space.

Very recently, in joint work with  with C. Robles, we have recently advanced the technology of  exterior
differential systems (EDS), by showing that represented Lie groups give rise to a natural
series of EDS that can be resolved using Lie algebra cohomology (as opposed to the sometimes
cumbersome traditional Cartan algorithm). See our article Fubini-Griffiths-Harris rigidity
and Lie algebra cohomology for details.


Specific projects I am working on:
 

1. The Debarre-de Jong conjecture on uniruled hypersurfaces of low degree (with O. Tommasi)
2. Equations for varieties in spaces of tensors, especially secant varieties of homogeneous varieties (various parts with  L. Manivel, G. Ottaviani, and J. Weyman)
3. Rigidity and flexibility of homogeneous varieties (with C. Robles)
4. Questions related to the Hwang-Mok program on Fano varieties and projective geometry (with C. Robles)
5. Hermitian differential geometry and operator theory (with J. Sarkar, based on work of R. Douglas)