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Friday, July 31, 2020

How Physics Found a Geometric Structure for Math to Play With | Mathematics - Quanta Magazine

Symplectic geometry is a relatively new field with implications for much of modern mathematics. Here’s what it’s all about, explains Kevin Hartnett, senior writer at Quanta Magazine covering mathematics and computer science.

Photo: Olena Shmahalo/Quanta Magazine
In the early 1800s, William Rowan Hamilton discovered a new kind of geometric space with nearly magical properties. It encoded motion and mathematics into a single, glinting geometric object.

This phenomenon birthed a field called symplectic geometry. Over the last few decades it has grown from a small collection of insights into a dynamic area of research with deep connections to more areas of math and physics than Hamilton ever could have imagined.

Symplectic geometry is ultimately the study of geometric spaces with a symplectic structure. But exactly what it means for a space to have a structure — let alone this particular structure — takes a little explaining.

Geometric spaces can be floppy like a tarp or rigid like a tent. “The tarp is very malleable but then you get, whatever, a bunch of sticks or scaffolding to shape it,” said Emmy Murphy of Northwestern University. “It makes it a more concrete thing.”...

Vladimir Arnold made several foundational conjectures in the study of symplectic geometry.

Beginning in the 1960s, Vladimir Arnold made several influential conjectures that captured the specific ways in which symplectic spaces are more rigid than ordinary topological ones (like the floppy sphere). One of them, known as the Arnold conjecture, predicts that Hamiltonian diffeomorphisms have a surprisingly large number of “fixed” points, which don’t move during a transformation. By studying them, you can put your hands on just what it is that makes a symplectic space different from other kinds of geometric spaces.

Source: Quanta Magazine