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).

I am  interested in geometric approaches to questions arising in theoretical computer science, specifically the
complexity of matrix multiplication and P versusNP,   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 are examining the relationship between
invariant differential operators on homogeneous varieties (via parabolic Verma modules),
and projective differential geometry. See the survey
Exterior differential systems, Lie algebra cohomology and the rigidity of homogeneous varieties .

I am currently finishing the book Geometry of tensors with applications. From the preface:
"Tensors are ubiquitous in the sciences. They provide a useful way to organize data.
The geometry describing qualitative properties of tensors is   a powerful
tool for extracting information from data sets and a beautiful
subject  in its own right. This book has three intended uses: as a classroom textbook,
a reference work for researchers, and a research manuscript."

Specific projects I am working on:
 
1. Equations for secant varieties of Segre and Veronese varieties, with applications
to the complexity of matrix multiplication (with G. Ottaviani)
2. Rigidity and flexibility of homogeneous varieties,   uniform geometric characterization of
G-modules with a finite number of  orbits, and local differential geometry via BGG resolutions. (with C. Robles and M. Eastwood)
3. The Geometric Complexity Theory (Mulmuley-Sohoni)  approach to VP v.  VNP (with L. Manivel and N. Ressayre)
4. Conjectures of Comon, Eisenbud and Strassen regarding tensors and symmetric tensors (with J. Buczinski)
 
I am also mentoring A. Boralevi in her work on cohomology of homogeneous vector bundles and relations
between parabolic Verma modules and quiver representations, 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.