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AMR

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Association for Mathematical Research

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Jul 1, 2026
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AMR – Association for Mathematical Research
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  1. From Ramsey’s Theorem to the Erdös–Hajnal Conjecture
    Jul 1, 2026 · original
    Ordnung muss sein! (Order is inevitable.) This saying , attributed to Theodore Motzkin, aptly summarizes Ramsey theory , which aims to solve problems of the form: “From what size on can a structure no longer avoid having a certain property ?"
  2. Build Boy’s Surface
    Apr 5, 2026 · original
    ARTICLE BY RICHARD EVAN SCHWARZ These notes give a fairly conceptual description of Boy’s surface that does not draw too much on three dimensional visualization. I will explain how to build Boy’s surface out of simple pieces.
  3. Charting the Worlds
    Dec 10, 2025 · original
    Etienne Ghys: Cartography has accompanied mathematics since its very beginnings and continually renews its set of problems. I would like to present a few selected pieces of this interaction between cartography and mathematics. I hope to show through these examples how the two disciplines mutually enrich one another.
  4. Triangles after Euclid, Gauss and Gromov
    Dec 10, 2025 · original
    Etienne Ghys: For centuries, geometry was Euclid’s geometry — the one we learn in school, with its right, isosceles and equilateral triangles and its theorems of Pythagoras and Thales; the geometry of “the world in which we live.” Euclid established its foundations in the third century BCE in the book — a landmark for mathematicians, titled “The Elements”. For more than twenty centuries, this book stood at the heart of mathematics, so definitive did it seem.
  5. A Bit of Geometric Group Theory
    Nov 2, 2025 · original
    Gilbert Levitt: Discrete groups appear in every area of mathematics — and even in Escher’s art. Even if the y are defined algebraically, we often understand them better by their action on geometric objects. More and more often, they are viewed as geometric entities in their own right. Their properties are especially striking when the curvature is negative.
  6. Dimers
    Nov 2, 2025 · original
    Adrien Kassel: This is an introductory text on the dimer model and its links to combinatorics, statistical physics, and geometry. This article is a translation of the French original published in 2016. In the meantime, the study of the dimer model has seen several interesting developments, reflected in the final section, which was added at the time of translation in October 2025.
  7. Not all subRiemannian geodesics are smooth
    Jun 8, 2025 · original
    AUTHOR: R. Montgomery. Are all subRiemannian geodesics smooth? The question was answered recently with a decisive “no” by Yacine Chitour, Fr´ederic Jean, Roberto Monti, Ludovic Rifford, Ludovic Sacchelli, Mario Sigalotti, and Alessandro Socionovo.
  8. SubRiemannian Geometry: Two Open Problems and One Falling Cat
    Aug 17, 2024 · original
    AUTHOR: R. Montgomery. This open problem review examines two open and closely related problems in subRiemannian geometry. These are introduced through the problem of the falling cat.
  9. Identities for sin x
    Jul 17, 2024 · original
    AUTHOR: P. Kuchment and S. Lvin. This open problem review examines some mysterious identities for sin(x) and asks for a connection to known theories.
  10. The Optimal Paper Moebius Band
    Oct 9, 2023 · original
    AUTHOR: Sergei Tabachnikov. Everyone knows how to make a Moebius band out of a paper rectangle: give it a $180^{\circ}$ twist and attach the ends to each other. It is easy to do if the rectangle is long and narrow, but it is impossible if the ratio of the length to width is sufficiently small (say, equal to 1). Thus there exists a number $\lambda$ such that if this ratio is greater than $\lambda$, a paper Moebius band can be made, and if it is smaller than $\lambda$, then a paper Moebius band does not exist.

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