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New math revives geometry's oldest problems

news.ycombinator.com Joseph Howlett 2025-09-28 09:57:57

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight. Such questions, which ask for the number of solutions that satisfy a set of geometric conditions, were a favorite of the ancient Greeks. And they’ve continued to entrance mathematicians for millennia. How many lines lie on a cubic surface? How many quadratic curves lie on a quintic surface? (Twen

Topics: enumerative geometry mathematicians questions theory
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New knot theory discovery overturns long-held mathematical assumption

news.ycombinator.com Max Springer 2025-11-04 14:35:22

Scanning the crowd at a fancy soiree may reveal a wide array of neckties, each fastened with a highly complex mathematical object masquerading as fashion. An entire field of mathematics is devoted to understanding mathematical knots, which one can obtain from any traditional knot by gluing the loose ends together. Mathematicians long believed that if you attach cut ends of two different knots to each other, the new knot will be just as complex as the sum of the individual knots’ complexity. But

Topics: knot knots mathematicians number unknotting
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New Knot Theory Discovery Overturns Long-Held Mathematical Assumption

news.ycombinator.com Max Springer 2025-11-05 08:35:22

Scanning the crowd at a fancy soiree may reveal a wide array of neckties, each fastened with a highly complex mathematical object masquerading as fashion. An entire field of mathematics is devoted to understanding mathematical knots, which one can obtain from any traditional knot by gluing the loose ends together. Mathematicians long believed that if you attach cut ends of two different knots to each other, the new knot will be just as complex as the sum of the individual knots’ complexity. But

Topics: knot knots mathematicians number unknotting
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This New Pyramid-Like Shape Always Lands With the Same Side Up

wired.com Elise Cutts 2025-12-24 23:00:00

The original version of this story appeared in Quanta Magazine. In 360 BC, Plato envisioned the cosmos as an arrangement of five geometric shapes: flat-sided solids called polyhedra. These immediately became important objects of mathematical study. So it might be surprising that, millennia later, mysteries still surround even the simplest shape in Plato’s polyhedral universe: the tetrahedron, which has just four triangular faces. One major open problem, for instance, asks how densely you can p

Topics: conway domokos faces mathematician tetrahedra
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Efforts to Ground Physics in Math Are Opening the Secrets of Time

wired.com Leila Sloman 2026-01-07 14:00:00

Now, three mathematicians have finally provided such a result. Their work not only represents a major advance in Hilbert’s program, but also taps into questions about the irreversible nature of time. “It’s a beautiful work,” said Gregory Falkovich, a physicist at the Weizmann Institute of Science. “A tour de force.” Under the Mesoscope Consider a gas whose particles are very spread out. There are many ways a physicist might model it. At a microscopic level, the gas is composed of individual

Topics: boltzmann gas mathematicians model particles
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A ‘Grand Unified Theory’ of Math Just Got a Little Bit Closer

wired.com Joseph Howlett 2026-01-21 04:00:00

“We mostly believe that all the conjectures are true, but it’s so exciting to see it actually realized,” said Ana Caraiani, a mathematician at Imperial College London. “And in a case that you really thought was going to be out of reach.” It’s just the beginning of a hunt that will take years—mathematicians ultimately want to show modularity for every abelian surface. But the result can already help answer many open questions, just as proving modularity for elliptic curves opened up all sorts of

Topics: abelian curves elliptic mathematicians modular
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After 20 years, math couple solves major group theory problem

news.ycombinator.com Leila Sloman 2025-12-10 08:09:56

After the conjecture was posed in the 1970s, dozens of mathematicians tried their hand at proving it. They made partial progress — and in the process they learned a great deal about groups, which are abstract objects that describe the various symmetries of a mathematical system. But a full proof seemed out of reach. Then Späth came along. Now, 20 years after she first learned about the problem and more than a decade after she met Cabanes, the two mathematicians have finally completed the proof.

Topics: conjecture group groups mathematicians mckay
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After 20 Years, Math Couple Solves Major Group Theory Problem

news.ycombinator.com Leila Sloman 2025-12-10 20:09:56

After the conjecture was posed in the 1970s, dozens of mathematicians tried their hand at proving it. They made partial progress — and in the process they learned a great deal about groups, which are abstract objects that describe the various symmetries of a mathematical system. But a full proof seemed out of reach. Then Späth came along. Now, 20 years after she first learned about the problem and more than a decade after she met Cabanes, the two mathematicians have finally completed the proof.

Topics: conjecture group groups mathematicians mckay
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