Algorithms and Complexity in Algebraic Geometry

Aug. 21–Dec. 19, 2014

Polynomial equations are ubiquitous in the mathematical sciences. The study of their solutions is the domain of algebraic geometry. Recently, there has been an explosion of activity, as computer scientists, physicists, applied mathematicians and engineers have realized the potential utility of modern algebraic geometry. This has brought forth an increased focus on quantitive and algorithmic questions. The semester will emphasize connections to geometric complexity theory. In this novel framework, fundamental lower bound questions can be rephrased and approached via representation theory and algebraic geometry. This applies to an arithmetic version of P versus NP as well as to multilinear algebra problems such as tensor rank and the complexity of matrix multiplication.

View the textbook written by J. M. Landsberg in connection with this program.

Organizers:

Peter Bürgisser (Technical University of Berlin; chair), Joseph M. Landsberg (Texas A&M University), Ketan Mulmuley (University of Chicago), Bernd Sturmfels (UC Berkeley)

Long-Term Participants (including Organizers):

Hirotachi Abo (University of Idaho), Saugata Basu (Purdue University), Alessandra Bernardi (University of Bologna), Grigoriy Blekherman (Georgia Institute of Technology), Jaroslaw Buczynski (Polish Academy of Sciences), Peter Bürgisser (Technical University of Berlin; chair), Matthias Christandl (University of Copenhagen), Jan Draisma (Technische Universiteit Eindhoven), David Eisenbud (Mathematical Sciences Research Institute and UC Berkeley), Ioannis Emiris (University of Athens), Elizabeth Gross (San Jose State University), Jonathan Hauenstein (University of Notre Dame), Serkan Hosten (San Francisco State University), Neeraj Kayal (Microsoft Research India), Teresa Krick (Universidad de Buenos Aires), Joseph M. Landsberg (Texas A&M University), Wei Li (Chinese Academy of Sciences), Lek-Heng Lim (University of Chicago), Ricky Liu (University of Michigan), Satya Lokam (Microsoft Research India), Gregorio Malajovich (Universidade Federal do Rio de Janeiro), Laurent Manivel (University of Montreal), Dhagash Mehta (North Carolina State University), Ketan Mulmuley (University of Chicago), Luke Oeding (Auburn University), Giorgio Ottaviani (University of Florence), Pablo Parrilo (Massachusetts Institute of Technology), Youming Qiao (University of Technology, Sydney), Jose Rodriguez (North Carolina State University), Steven Sam (UC Berkeley, Miller Institute), Michael Shub (City University of New York), Frank Sottile (Texas A&M University), Bernd Sturmfels (UC Berkeley), Leslie Valiant (Harvard University), Jerzy Weyman (University of Connecticut), Ryan Williams (Stanford University), Virginia Vassilevska Williams (Stanford University), Ke Ye (University of Chicago)

Research Fellows:

Klim Efremenko (University of Chicago), Michael Forbes (Massachusetts Institute of Technology; Google Research Fellow), María Isabel Herrero (Universidad de Buenos Aires), Christian Ikenmeyer (Texas A&M University), Kaie Kubjas (Aalto University; Microsoft Research Fellow), Mateusz Michalek (Polish Academy of Sciences), Matthew Niemerg (Colorado State University), Benjamin Rossman (National Institute of Informatics), Cynthia Vinzant (North Carolina State University)

Visiting Graduate Students and Postdocs:

Paul Breiding (Technical University of Berlin), Cameron Farnsworth (Texas A&M University), Fulvio Gesmundo (Texas A&M University), Yonghui Guan (Texas A&M University), Jesko Hüttenhain (Technical University of Berlin), Fotis Iliopoulos (UC Berkeley), Mario Kummer (University of Konstanz), Simone Naldi (LAAS-CNRS, Toulouse), Elina Robeva (UC Berkeley), Zvi Rosen (UC Berkeley), Ben Weitz (UC Berkeley)

Workshops

Algebraic Geometry Boot Camp

Tuesday, Sep. 2–Friday, Sep. 5, 2014

Organizers:

Joseph M. Landsberg (Texas A&M University)

Geometric Complexity Theory

Monday, Sep. 15–Friday, Sep. 19, 2014

Organizers:

Ketan Mulmuley (University of Chicago; chair), Peter Bürgisser (Technical University of Berlin), Laurent Manivel (University of Montreal), Leslie Valiant (Harvard University)

Solving Polynomial Equations

Monday, Oct. 13–Thursday, Oct. 16, 2014

Organizers:

Frank Sottile (Texas A&M University; chair), Jonathan Hauenstein (University of Notre Dame), Pascal Koiran (École Normale Supérieure de Lyon)

Tensors in Computer Science and Geometry

Monday, Nov. 10–Friday, Nov. 14, 2014

Organizers:

Lek-Heng Lim (University of Chicago; chair), Henry Cohn (Microsoft Research New England), Jason Morton (Pennsylvania State University), Giorgio Ottaviani (University of Florence)

Symbolic and Numerical Methods for Tensors and Representation Theory

Monday, Nov. 17–Thursday, Nov. 20, 2014

Organizers:

Alessandra Bernardi (University of Bologna), Anton Leykin (Georgia Institute of Technology), Luke Oeding (Auburn University), Claudiu Raicu (University of Notre Dame), Cynthia Vinzant (North Carolina State University)

Algorithms and Complexity in Algebraic Geometry Reunion

Monday, Dec. 14–Wednesday, Dec. 16, 2015

Organizers:

Peter Bürgisser (Technical University of Berlin), Joseph M. Landsberg (Texas A&M University), Ketan Mulmuley (University of Chicago), Bernd Sturmfels (UC Berkeley)

Program image by Jesko Hüttenhain. The image, generated with the help of Mathematica, shows a circuit board superimposed on the algebraic surface with equation x³y + y³z + z³x + 5z = 0.