About

Frederick Manners received his D.Phil. from the University of Oxford in 2016. He then spent three years as a Szego Assistant Professor of Mathematics at Stanford University. His main area of research is in Additive Combinatorics, but he has broad interests across a number of areas of pure mathematics including number theory, combinatorics, analysis, and ergodic theory. His particular focus is in notions of arithmetic pseudorandomness: testing whether a particular arithmetic object 'looks random', or if not, finding some precise kind of structure that helps analyze it. Recently, this has taken the form of an in-depth study of Gowers norms and their inverse theory. He is also interested in the application of these ideas to combinatorics, number theory, and other areas as they arise.

Research topics

• Combinatorics
• Pure mathematics
• Mathematics

Selected publications

• Marton’s conjecture in abelian groups with bounded torsion

  Annales de la faculté des sciences de Toulouse Mathématiques · 2026-01-13

  articleOpen access

  We prove a Freiman–Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> be an abelian group of torsion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> (meaning <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> ) and suppose that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is a non-empty subset of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>K</mml:mi> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> . Then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> can be covered by at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> translates of a subgroup <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> of cardinality at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> . The argument is a variant of that used in the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi mathvariant="bold">F</mml:mi> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> in a recent paper of the authors.

  PublisherDOI

• The motivic class of the space of genus $0$ maps to the flag variety

  arXiv (Cornell University) · 2026-01-12

  preprintOpen access

  Let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}$ and let $Ω^{2}_β(\operatorname{Fl}_{n+1})$ be the space of based maps $f:\mathbb{P}^{1}\to \operatorname{Fl}_{n+1}$ in the class $f_{*}[\mathbb{P}^{1}]=β$. We show that under a mild positivity condition on $β$, the class of $Ω^{2}_β(\operatorname{Fl}_{n+1})$ in $K_{0}(\operatorname{Var})$, the Grothendieck group of varieties, is given by \[ [Ω^{2}_β(\operatorname{Fl}_{n+1})] = [\operatorname{GL}_{n}\times \mathbb{A}^{a}]. \] The proof of this result was obtained in conjunction with Google Gemini and related tools. We briefly discuss this research interaction, which may be of independent interest. However, the treatment in this paper is entirely human-authored (aside from excerpts in an appendix which are clearly marked as such).

  PublisherDOI

• The motivic class of the space of genus $0$ maps to the flag variety

  ArXiv.org · 2026-01-12

  articleOpen access

  Let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}$ and let $Ω^{2}_β(\operatorname{Fl}_{n+1})$ be the space of based maps $f:\mathbb{P}^{1}\to \operatorname{Fl}_{n+1}$ in the class $f_{*}[\mathbb{P}^{1}]=β$. We show that under a mild positivity condition on $β$, the class of $Ω^{2}_β(\operatorname{Fl}_{n+1})$ in $K_{0}(\operatorname{Var})$, the Grothendieck group of varieties, is given by \[ [Ω^{2}_β(\operatorname{Fl}_{n+1})] = [\operatorname{GL}_{n}\times \mathbb{A}^{a}]. \] The proof of this result was obtained in conjunction with Google Gemini and related tools. We briefly discuss this research interaction, which may be of independent interest. However, the treatment in this paper is entirely human-authored (aside from excerpts in an appendix which are clearly marked as such).

  PublisherOA PDF

• On a conjecture of Marton

  Annals of Mathematics · 2025-03-01 · 10 citations

  articleOpen access

  We prove a conjecture of K. Marton, widely known as the polynomial Freiman--Ruzsa conjecture, in characteristic 2. The argument extends to odd characteristic, with details to follow in a subsequent paper.

  PublisherOA PDFDOI

• Sumsets and entropy revisited

  Random Structures and Algorithms · 2024-07-31 · 9 citations

  articleOpen access

  Abstract The entropic doubling of a random variable taking values in an abelian group is a variant of the notion of the doubling constant of a finite subset of , but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the “skew dimension” of subsets of with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over implies the (weak) Polynomial Freiman–Ruzsa conjecture over .

  PublisherOA PDFDOI

• Transversals in quasirandom latin squares

  Proceedings of the London Mathematical Society · 2023-05-31 · 4 citations

  articleOpen access

  Abstract A transversal in an latin square is a collection of entries not repeating any row, column, or symbol. Kwan showed that almost every latin square has transversals as . Using a loose variant of the circle method we sharpen this to . Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.

  PublisherOA PDFDOI

• Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

  arXiv (Cornell University) · 2023-11-09

  preprintOpen access

  Affine extractors give some of the best-known lower bounds for various computational models, such as AC⁰ circuits, parity decision trees, and general Boolean circuits. However, they are not known to give strong lower bounds for read-once branching programs (ROBPs). In a recent work, Gryaznov, Pudlák, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of ROBPs where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs (SROLBPs). Their main result gives explicit constructions of directional affine extractors for entropy k > 2n/3, which implies average-case complexity 2^{n/3-o(n)} against SROLBPs with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (CCC' 23) improves the hardness to 2^{n-o(n)} at the price of increasing the correlation to polynomially large, via a new connection to sumset extractors introduced by Chattopadhyay and Li (STOC' 16) and explicit constructions of such extractors by Chattopadhyay and Liao (STOC' 22). Both works left open the questions of better constructions of directional affine extractors and improved average-case complexity against SROLBPs in the regime of small correlation. This paper provides a much more in-depth study of directional affine extractors, SROLBPs, and ROBPs. Our main results include: - An explicit construction of directional affine extractors with k = o(n) and exponentially small error, which gives average-case complexity 2^{n-o(n)} against SROLBPs with exponentially small correlation, thus answering the two open questions raised in previous works. - An explicit function in AC⁰ that gives average-case complexity 2^{(1-δ)n} against ROBPs with negligible correlation, for any constant δ > 0. Previously, no such average-case hardness is known, and the best size lower bound for any function in AC⁰ against ROBPs is 2^Ω(n). One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in 𝖥₂ⁿ, recently proved by Gowers, Green, Manners, and Tao (arXiv' 23).

  PublisherOA PDFDOI

• Sumsets and entropy revisited

  arXiv (Cornell University) · 2023-06-23

  preprintOpen access

  The entropic doubling $σ_{\operatorname{ent}}[X]$ of a random variable $X$ taking values in an abelian group $G$ is a variant of the notion of the doubling constant $σ[A]$ of a finite subset $A$ of $G$, but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the ``skew dimension'' of subsets of $\mathbf{Z}^D$ with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of $\mathbf{Z}^D$ with small doubling; (3) A proof that the Polynomial Freiman--Ruzsa conjecture over $\mathbf{F}_2$ implies the (weak) Polynomial Freiman--Ruzsa conjecture over $\mathbf{Z}$.

  PublisherOA PDFDOI

• The Apparent Structure of Dense Sidon Sets

  The Electronic Journal of Combinatorics · 2023-02-24 · 7 citations

  articleOpen accessSenior author

  The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essentially no new examples. There are many further examples arising from nondesarguesian planes. We conjecture that all dense Sidon sets arise from finite projective planes in this way. If true, this implies that all abelian groups of most orders do not have dense Sidon subsets. In particular if $\sigma_n$ denotes the size of the largest Sidon subset of $\mathbb{Z}/n\mathbb{Z}$, this implies $\liminf_{n \to \infty} \sigma_n / n^{1/2} &lt; 1$. We also give a brief bestiary of somewhat smaller Sidon sets with a variety of algebraic origins, and for some of them provide an overarching pattern.

  PublisherOA PDFDOI

• Transversals in quasirandom latin squares

  arXiv (Cornell University) · 2022-09-06

  preprintOpen access

  A transversal in an $n \times n$ latin square is a collection of $n$ entries not repeating any row, column, or symbol. Kwan showed that almost every $n \times n$ latin square has $\bigl((1 + o(1)) n / e^2\bigr)^n$ transversals as $n \to \infty$. Using a loose variant of the circle method we sharpen this to $(e^{-1/2} + o(1)) n!^2 / n^n$. Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.

  PublisherOA PDFDOI

Frequent coauthors

• Yonatan Gutman

  Polish Academy of Sciences

  15 shared

• Sean Eberhard

  11 shared

• Péter P. Varjú

  9 shared

• Rudi Mrazović

  University of Zagreb

  7 shared

• Ben Green

  5 shared

• Terence Tao

  University of California, Los Angeles

  4 shared

• Jacob Fox

  Stanford University

  3 shared

• W. T. Gowers

  Collège de France

  2 shared