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.

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Program image by Jesko Hüttenhain. The image, generated with the help of Mathematica, shows a circuit board superimposed on the algebraic surface with equation x^{3}y + y^{3}z + z^{3}x + 5z = 0.