Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more wellbehaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples.

The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the socalled "chasm at depth 4" suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation).

Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example, Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms.

Beside algebraic methods, analytic methods have been used for quite some time in theoretical computer science. Very recently, Fourier analysis was essential for proving a variant of the famous Unique Games Conjecture, the so-called 2-2 Conjecture.

Analytic methods can also be used to solve algebraic problems. Garg et al. use analytic scaling algorithms to solve problems from algebra, so-called orbit closure problems, by using geometric invariant theory to establish the link. Central problems, like the tensor border rank or non-commutative identity testing, can be written as such an orbit closure problem.

These new directions will be in the focus of the Dagstuhl Seminar and reflect the developments in the field since the previous Dagstuhl Seminar 18391. Taking the recent exciting developments outlined above into account, we also include analytic methods this time.

This Dagstuhl Seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and analytic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar will play an important role in educating a diverse community about the latest new techniques, spurring further progress.

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Markus Bläser, Valentine Kabanets, Ronen Shaltiel, and Jacobo Torán