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 (Bull. AMS)
 
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 and the survey
Exterior differential systems, Lie algebra cohomology and the rigidity of homogeneous varieties .


Specific projects I am working on:
 
1. Equations for varieties in spaces of tensors, especially secant varieties of homogeneous varieties (various parts with  L. Manivel, G. Ottaviani, and J. Weyman)
2. Rigidity and flexibility of homogeneous varieties, and uniform geometric characterization of
G-modules with a finite number of  orbits (with C. Robles)
3. The Mulmuley-Sohoni approach to VP v.s. VNP (with P. Buergisser, L. Manivel and J. Weyman)
4. Conjectures of Comon and Strassen regarding tensors and symmetric tensors (with J. Buczinski, L. Oeding and K. Ye)
5. Relating Valiant's holographic algorithms in complexity theory to the  geometry of spinors (with J. Morton)

I am also mentoring A. Boralevi in her work on cohomology of homogeneous vector bundles,
J. Buczinski in his work on the LeBrun-Salamon conjecture and D. The in his work extending Bryant's
results on the rigidity of Schubert varieties using Lie algebra cohomology techniques. I continue to
mentor Z. Teitler in his work on rank and border rank.