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How do mathematicians reveal patterns in structures too complex to see directly? That is at the heart of the work of 2026 Wolf Prize in Mathematics Laureates David Kazhdan and Joseph Bernstein. Their discoveries helped build what is now known as geometric representation theory — a field that connects...

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Wolf Prize

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88,284 просмотров • 4 дней назад •via X (Twitter)

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dan with glasses3 дней назад

@HebrewU @HebrewU_heb @TelAvivUni @WolfPrize_ @grok "connects algebra, geometry and symmetry" is broad. Specify one Kazhdan–Lusztig or D-module construction where structure actually transfers: all exact relations lift SDG5 to @leanprover: connection →PiπΠ lift →theorem #LeanProver 🔢 🧠

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For 300 years mathematicians solved polynomials of degree 1, 2, 3, and 4 with increasingly complex formulas. Then they hit degree 5 and the formula simply doesn't exist - not undiscovered, provably impossible. The reason is group theory. The 5 solutions of a quintic can be scrambled in ways that no finite chain of nested roots can replicate - the same symmetry mathematics that underlies modern cryptography and the attention mechanism in transformers. 28 minutes. Bookmark & watch today. The proof is more beautiful than the formula would have been.

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I think one of the roles we play as mathematicians in society is to help people become acquainted with the underlying secret patterns. I have been working for several years on projects in crystallography, where we study crystal structures. But few people know that such periodic patterns come with severe constraints on their symmetry. In the plane, there are 17 different symmetry types, a fact well known even to the designers of the mesmerizing patterns of the Alhambra. Here you can find and experiment with such tilings in the plane, gaining insight into the intrinsic beauty of the so-called wallpaper groups, the crystal symmetries in dimension 2. The app interactively helps you design symmetric patterns with colors and shows how changes in the structure of the unit cell propagate via symmetry. In dimension 3, if you look into International Tables for Crystallography, Vol. 1, you will find a theorem due to Schoenflies and Fedorov stating that there are 230 such symmetry types, a cornerstone of modern chemistry. Beyond that, in dimensions 4 and higher, a count can be made, but it requires a proof of the general theorem due to Frobenius and Bieberbach. This was an answer to the first part of Hilbert’s famous eighteenth problem. One of the fun consequences of such a classification is that in dimensions 2 and 3, 5-fold symmetry is forbidden in regular periodic arrangements. Intrinsically, this fact is related to the existence of matrices with a fifth-root-of-unity eigenvalue. For integral matrices, this is possible only in dimensions 4 and higher. If you generalize the square and cube tilings to dimensions 4 and 5, obtaining hypercubic tilings, the 5-fold symmetry pattern emerges. Skew projections of the 5D hypercubic tiling onto a 2-dimensional plane give rise to a quasicrystalline tiling known as the Penrose tiling. You can find such patterns in front of the Andrew Wiles Building at the Oxford Mathematical Institute. In later posts this summer, I will take a deep dive into group homology, a modern tool for studying the geometry of crystals. There are still many open questions, for example, how many symmetry types exist exactly in dimensions beyond 6. This is still largely unknown; at present, we only have asymptotic lower bounds.
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