Recent Trends X: Frederick Manners awarded EMS Prize 2024

23 Oct 2025

Frederick Manners has been awarded one of the European Mathematical Society prizes in recognition of his outstanding contributions in arithmetic combinatorics and related areas. Manners, who is currently an Associate Professor at the University of California, San Diego, has been particularly praised for his work in the area known as higher-order Fourier analysis. We will focus here on two of his main contributions.

A first highlight is the quantitatively effective proof that Manners gave in 2018 of the inverse theorem for Gowers norms (or $U^d$ norms) on cyclic groups $\mathbb{Z}_N$ (the so-called “integer setting”) [5]. These norms were introduced by Gowers in his famous work on Szemerédi’s theorem in 2000 [1]. Since then, they have become central tools in arithmetic combinatorics, especially useful in relation to detecting and counting various kinds of linear configurations in subsets of abelian groups. These norms have also led to a generalization of classical Fourier analysis. In the classical theory, a function on a compact abelian group is decomposed into fundamental harmonics (the Fourier characters) whose underlying structure is based on the circle group. The smallest of the Gowers norms, the $U^2$-norm, is closely related to this classical theory. However, the $U^d$ norms of higher order (i.e. for d > 2) cannot be analyzed in terms of classical Fourier characters, requiring instead more subtle underlying objects, such as nilmanifolds. This motivated the above-mentioned generalization known as higher-order Fourier analysis. This theory includes as a central result the so-called Inverse Theorem for the Gowers norms, proved first in the integer setting by Green, Tao and Ziegler in 2010 [4]. This theorem asserts that if a function $f$ of modulus at most 1 on a large cyclic group $\mathbb{Z}_N$ has non-trivially large Gowers $U^{d+1}$-norm, then $f$ has large inner product with a $d$-step nilsequence of bounded complexity, a function which plays the role of a Fourier character relative to this norm, and which is defined using a $d$-step nilmanifold (instead of the circle group for classical characters). The initial proofs of this inverse theorem did not provide effective bounds for the complexity of the nilsequence or the magnitude of the inner product in the conclusion for general order d (good bounds were known in special cases, notably for the $U^3$-norm due to Green and Tao). It was an outstanding achievement of Manners to provide the first reasonable bounds for these quantities for general order $d$.

The second highlight is the groundbreaking proof, given in 2023 by Manners jointly with Gowers, Green, and Tao, of Marton’s conjecture (also known as Polynomial Freiman–Ruzsa conjecture) in additive combinatorics, in the case of vector spaces over finite fields [2]. This conjecture, well-known in this area at least since the 1990s, can be stated as follows: suppose that a subset $A$ of $\mathbb{F}_2^n$ has a small sumset in the sense that $\vert A + A\vert \leq K\vert A\vert$ for some constant $K>1$. Then $A$ can be covered by at most $2K^C$ cosets of some subgroup $H\leq \mathbb{F}_2^n$ of cardinality at most $\lvert A\rvert$ (for some absolute constant $C$). The conjecture was widely known to be of central importance in the field, with many equivalent reformulations, and strong connections with other principal results in this area. Notably, an equivalence had been established by Green and Tao in 2010 [3] between this conjecture and an effective version of the inverse theorem for the $U^3$ norm with polynomial bounds. The proof of Marton’s conjecture given by Manners and his coauthors is outstanding in many aspects, especially for the use it makes of entropy tools.

These examples are only a few of the important mathematical contributions of Manners to the development of the highly conceptual and technical fields of arithmetic combinatorics and higher-order Fourier analysis.

References

1. W.T. Gowers, A new proof of Szeméredi’s theorem, GAFA 11 (2001), 465–588.
2. W. T. Gowers, B. Green, F. Manners, T. C. Tao, On a conjecture of Marton, arXiv:2311.05762
3. B. J. Green, T. C. Tao, An equivalence between inverse sumset theorems and inverse conjectures for the $U^3$ norm, Math. Proc. Cambridge Philos. Soc. 149 (2010), no.1, 1–19.
4. B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers $U^{s+1}[N]$-norm, Ann. of Math. (2) 176 (2012), no. 2, 1231–1372.
5. F. Manners, Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups, arXiv:1811.00718.

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Pablo Candela is a CSIC researcher at the Institute of Mathematical Sciences (ICMAT), Madrid, Spain.