[1] His fast spherical filters were (and remain) critical in the construction of the most efficient three-dimensional fast multipole methods (FMMs) for solving the Helmholtz equation and Maxwell's equations.

Other well-known work of his includes contributions to computational methods for time-domain wave propagation,[2][3][4] quadratures for singular integrals,[5][6] and multiwavelets.

[8] He was awarded a Bronze Medal from the U.S. Department of Commerce in 1997 for joint work on processing antenna measurements corrupted by errors in the positions of probes.

[9] Alpert received his Ph.D. from Yale University in 1990, under the supervision of Vladimir Rokhlin.

[10] Alpert worked as a casualty actuary early in his career, and was a Hans Lewy postdoctoral fellow at Lawrence Berkeley National Laboratory and U.C.