The Batson-Spielman-Srivastava barrier function method has had a number of applications to geometry and functional analysis, arose in computer science, and is a very original form of potential function argument. The method was first developed to prove the existence of (and construct in deterministic polynomial time) sparsifiers of graphs that preserve their spectral properties. Such sparsifiers were motivated by algorithmic applications: essentially any algorithm that needs to compute cuts approximately can be sped up by being given as input a sparsified version of the original input.

Beyond sparsifiers however, the method has had numerous applications, many of which are surveyed by Assaf Naor in this paper. Some prominent examples are construction of weighted expander graphs, approximate John decompositions of the identity with fewer points, dimension reduction of subsets/subspaces of $\ell_1^n$, a tight version of Bourgain and Tzafriri's restricted invertibility principle. For all of the above applications, the barrier function method yields essentially tight bounds, gives an efficient deterministic algorithm in addition to an existence proof, and often provides a more elementary proof than prior methods (although not without some hairy calculations).

Fast forward to 2013, and the barrier function method, on steroids, and augmented with the machinery of interlacing polynomials, was used by Marcus, Srivastava, and Spielman, to solve one of the most notorious problems in functional analysis, the Kadison-Singer problem. This problem arises from fundamental questions in mathematical physics, but it goes much further - it is known to be equivalent to dozens of problems all over mathematics. Not to mention that many analysts (including Kadison and Singer) did not even think the problem had a positive resolution (the cited survey by Cassaza et al. speculates on possible counterexamples).

The Batson-Spielman-Srivastava barrier function method has had a number of applications to geometry and functional analysis, arose in computer science, and is a very original form of potential function argument, reminiscent of the method of pessimistic estimators. Moreover, it goes against the conventional wisdom that analyzing the characteristic polynomial of random matrices is intractable, and one is better off looking at matrix moments instead.

The barrier function method was first developed to prove the existence of (and construct in deterministic polynomial time) sparsifiers of graphs that preserve their spectral properties. Such sparsifiers were motivated by algorithmic applications: essentially any algorithm that needs to compute cuts approximately can be sped up by being given as input a sparsified version of the original input.

Beyond sparsifiers however, the method has had numerous applications, many of which are surveyed by Assaf Naor in this paper. Some prominent examples are construction of weighted expander graphs, approximate John decompositions of the identity with fewer points, dimension reduction of subsets/subspaces of $\ell_1^n$, a tight version of Bourgain and Tzafriri's restricted invertibility principle. For all of the above applications, the barrier function method yields essentially tight bounds, gives an efficient deterministic algorithm in addition to an existence proof, and often provides a more elementary proof than prior methods (although not without some hairy calculations).

Fast forward to 2013, and the barrier function method, on steroids, and augmented with the machinery of interlacing polynomials, was used by Marcus, Srivastava, and Spielman, to solve one of the most notorious problems in functional analysis, the Kadison-Singer problem. This problem arises from fundamental questions in mathematical physics, but it goes much further - it is known to be equivalent to dozens of problems all over mathematics. Not to mention that many analysts (including Kadison and Singer) did not even think the problem had a positive resolution (the cited survey by Cassaza et al. speculates on possible counterexamples).