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 apply these fields to questions outside
of mathematics, especially questions coming from theoretical computer science and signal processing.
My primary current interest is in geometric approaches to questions arising in theoretical computer science, specifically the
complexity of matrix multiplication, matrix rigidity and P versus NP, and more generally the geometry of
varieties in spaces of tensors. For an introduction to these topics, see my 2012 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 currently working on:
1. Aspects
of Geometric Complexity Theory related to the Chow variety
("split variety") and its secant varieties (with S. Kumar)
2. Lower bounds for the complexity of matrix multiplication (with several people, including TAMU post-doc C. Ikenmeyer)
3. Geometry of the ideals generated by the size k subpermanents
of an n times n matrix (with K. Efremenko, H. Schenck and J. Weyman)
4. Young flattenings beyond shifted partial derivatives for separating permanent from determinant (with K. Efremenko)
5. Equations for secant varieties of Segre varieties (with TAMU post-doc C. Ikenmeyer, M. Michalek and G. Ottaviani)
6. Upper bounds for the complexity of matrix multiplication (with M. Michalek and V. Williams)
My current PhD students are working on the following problems:
1. Cameron Farnsworth: Complexity and algebraic geometry
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 for Hodge theory
4. Fulvio Gesmundo: Matrix rigidity and the geometry of iterated matrix multiplication