Efficient Congruencing

The current hot topic for me is the new efficient congruencing approach to Vinogradov's mean value theorem, which now delivers the best possible bounds for the number of variables required to deliver optimal estimates, proving the main conjecture in Vinogradov's mean value theorem. You can read about this subject here:

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T. D. Wooley, Vinogradov's mean value theorem via efficient congruencing, Annals of Math. 175 (2012), 1575--1627.

T. D. Wooley, Vinogradov's mean value theorem via efficient congruencing, II, Duke Math. J. 162 (2013), 673--730.

S. T. Parsell, S. M. Prendiville and T. D. Wooley, Near-optimal mean value estimates for multidimensional Weyl sums, Geom. Funct. Anal. 23 (2013), 1962--2024.

T. D. Wooley, Translation invariance, exponential sums and Waring's problem, Proceedings of the International Congress of Mathematicians, August 13--21, 2014, Seoul, Korea, Volume II, Kyung Moon Sa Co. Ltd., Seoul, Korea, 2014, pp. 505--529.

K. Ford and T. D. Wooley, On Vinogradov's mean value theorem: strongly diagonal behaviour via efficient congruencing, Acta Math. 213 (2014), 199--236.

T. D. Wooley, Multigrade efficient congruencing and Vinogradov's mean value theorem, Proc. London Math. Soc. (3) 111 (2015), no. 3, 519--560.

T. D. Wooley, Approximating the main conjecture in Vinogradov's mean value theorem, Mathematika 63 (2017), no. 1, 292--350.

T. D. Wooley, The cubic case of the main conjecture in Vinogradov's mean value theorem, Adv. Math. 294 (2016), 532--561.

T. D. Wooley, Discrete Fourier restriction via efficient congruencing, Internat. Math. Res. Notices 2017 (2017), no. 5, 1342--1389.

T. D. Wooley, Nested efficient congruencing and relatives of Vinogradov's mean value theorem, Proc. London Math. Soc. 118 (2019), no. 4, 942--1016; arxiv:1708.01220.

... and here are some consequences of EFFICIENT CONGRUENCING:

T. D. Wooley, The asymptotic formula in Waring's problem, Internat. Math. Res. Notices (2012), No. 7, 1485--1504.

T. D. Wooley, Rational solutions of pairs of diagonal equations, one cubic and one quadratic, Proc. London Math. Soc. (3) 110 (2015), no. 2, 325--356.

B. Wei and T. D. Wooley, On sums of powers of almost equal primes, Proc. London Math. Soc. (3) 111 (2015), no. 5, 1130--1162.

T. D. Wooley, Mean value estimates for odd cubic Weyl sums, Bull. London Math. Soc. 47 (2015), no. 6, 946--957.

T. D. Wooley, Perturbations of Weyl sums, Internat. Math. Res. Notices 2016 (2016), no. 9, 2632--2646.