Computer science is permeated with canonical hard problems that, a priori, have a fundamentally combinatorial structure, such as the graph coloring problem, the Steiner tree problem, the Hamiltonian cycle problem, the k-clique problem, and so forth. Accordingly, it would perhaps be quite reasonable to expect that currently the asymptotically fastest solution techniques would rely on the canonical combinatorial algorithms toolbox, such as carefully tailored combinatorial (backtrack/branching) search and case-by-case analysis, combined with, say, advanced data structures.However, this is not the case.Indeed, currently the fastest known exact/parameterized algorithms for each of the aforementioned problems (and beyond) rely on a mixed bag of advanced _algebraic_ techniques ranging from fast matrix multiplication to sieving e.g. via Möbius inversion and polynomial identity testing. This, in essence, signals that the development of systematic algorithmic principles and tools to cope with exponential-sized combinatorial spaces associated with hard search and enumeration problems is rather in its infancy. The proposed project aims to improve our understanding how such spaces can be systematically transformed and filtered using advanced algebraic and combinatorial techniques.The results of the project are of foremost interest in fundamental research in improving our understanding of computation, but potential exists also for breakthroughs that affect the computing practice, for example in connection with specific canonical tasks such as matrix multiplication or frontier applications such as motif problems in bioinformatics.