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Scaling problems and their algorithms have recently found a myriad of applications in different areas of mathematics, such as in extremal combinatorics (quantitative generalizations of the Sylvester-Gallai theorem), invariant theory (the null cone problem for matrix semi-invariants), functional analysis (the Brascamp-Lieb inequalities), non-commutative algebra (the word problem for free-skew fields) and many others.

In this talk, we will first define scaling problems and then discuss some of the structural theory behind them. After that, we will elaborate on the applications of this structural theory to problems in extremal combinatorics, in particular discussing how such scaling problems can give us quantitative generalizations of the Sylvester-Gallai theorem.

The talk is based on joint work with Zeev Dvir, Ankit Garg, and Joseph Solymosi.