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).

My primary current interest is in geometric approaches to questions arising in theoretical computer science, specifically the
complexity of matrix multiplication and P versus 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 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 recently finished  the book Geometry of tensors with applications, AMS   GSM 128. 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.  The Geometric Complexity Theory (Mulmuley-Sohoni)  approach to VP v.  VNP (with several people, including
TAMU post-doc Harlan Kadish)
2.  The complexity of matrix multiplication (with several people, including TAMU post-doc Jon Hauenstein)
4.  Equations for secant varieties of Segre and Veronese varieties (with G. Ottaviani)
 
 

My PhD students are working on the following problems:
Yang Qi: weak defectivity and Kruskal's theorem
Ming Yang: secant vareities of subspace varieties that arise in signal processing
Ke Ye: geometry of the permanent related to Valiant's conjecture
Curtis Porter: Exterior differential systems and its applications