Research of Joseph M. Landsberg

My current research is primarily in theoretical computer science, more specifically, applying algebraic
geometry and representation theory to central questions such as determining the complexity of matrix
multiplication and the separation of algebraic complexity classes (e.g. Geometric Complexity Theory).

I have broad mathematical interests including: 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), and the application of these areas to
questions outside of mathematics.

 

Projects I am currently working on:
 
1. Deriving matrix multiplication algorithms via geometry (with G. Ballard, N. Ryder, B. Liu (TAMU grad. student))
2.  Lower bounds for the complexity of matrix multiplication (with M. Michalek and  TAMU post-doc C. Ikenmeyer)
3. Establishing geometric foundations for the study of Valiant's conjecture on permanent v. determinant and its variants (with
K. Efremenko, H. Schenck, J. Weyman, N. Ressayre and TAMU post-doc C. Ikenmeyer)
 

My current PhD students are working on the following problems:

1. Cameron Farnsworth: Waring border rank of the determinant and permanent
2. Youghui Guan: the ideal of the Chow variety and its secant varieties
3. Curtis Porter: Exterior differential systems and its applications,  in particular CR geometry
4. Fulvio Gesmundo: Matrix rigidity and the geometry of iterated matrix multiplication

My current Master's student Bingjin Liu is working on deriving matrix multiplication algorithms via geometry.