IAS Researcher Debuts Counterexample to Viterbo’s Conjecture

Pazit Haim-Kislev has published a counterexample to Viterbo’s conjecture, demonstrating that some convex domains can achieve larger symplectic capacities than previously thought possible. With coauthor Yaron Ostrover, Haim-Kislev has discovered surprising features to the relationship between convex geometry and symplectic capacity. Their findings signal rich potential in further exploration of the unexpected ways these mathematical invariants behave. 

Symplectic geometry is a branch of theoretical mathematics that intersects with physics. Its name, “symplectic,” means “interwoven” and refers to the fundamentally emmeshed nature of its central object of investigation: phase space. Phase space knits together two variables – position and momentum – into one unified structure. Derived from Hamiltonian mechanics, phase space is a map of all possible states of a physical system over time. 

Whether we’re talking about a planet in orbit, an engine’s piston, or ballroom dancers, phase space describes every possible position-momentum pairing through which these systems travel. Each point in phase space will tell you where something is and how fast it’s moving in a certain direction. 

As a planet orbits the sun, for example, its position and momentum change. With phase space, we can map those changes to describe the totality of this system in motion. What’s important to keep in mind, however, is that as these coordinates change, the relationship between them does not. Phase space implies a certain relational structure that governs these systems – and this is where symplectic geometry comes in.

Symplectic geometry uses mathematics to abstract and study phase space. It employs a mathematical tool, called a symplectic manifold, to represent the combinatory points of phase space and assign them a symplectic structure that preserves the governing relationship between them. For ease of calculation or reasons of curiosity, mathematicians can transform these manifolds using symplectomorphisms, distortions that alter the shape of the manifold while preserving its underlying structure.

A relatively new field, symplectic geometry began in earnest in the 1980s when Mikhael Gromov published what became to be known as his “non-squeezing theorem,” which pushed a mysterious mathematical invariant into the spotlight.

This invariant is symplectic capacity. Gromov demonstrated that symplectic manifolds have an additional global property – what later came to be known as a symplectic capacity – that fundamentally defines them. Gromov’s non-squeezing theorem revealed that embedding a symplectic ball into a smaller symplectic cylinder – or “squeezing it” – would always change its capacity and therefore not preserve its essential character. 

Constant through symplectomorphisms, symplectic capacity measures the size of a domain. It describes the amount a phase space can be reduced without disrupting the foundational geometry of that space. Gromov therefore exposed fascinating limitations inherent in the finely-grained geometries offered by symplectic approaches. 

Often referred to as the “uncertainty principle of classical mechanics,” Gromov’s theorem kicked off decades of inquiry surrounding the complex geometries inherent in the relationship between movement, location, and change in physical systems. After Gromov’s discovery, Ekeland and Hofer would formalize the term “symplectic capacity” and offer additional ways to calculate the size of a symplectic manifold after discovering more examples. And since then, many capacities have been introduced, along with different tools and techniques to calculate these sizes. 

Today, symplectic capacity is a fundamental concept that highlights the special rigidity existing in symplectic geometry: what can and cannot be done using symplectomorphisms.

It's surprisingly possible to learn which domain of phase space – a ball, for example – is possible to embed into another domain – a cylinder, for instance – by investigating the special Hamiltonian dynamics associated with each of these domains. It’s also surprisingly difficult. 

“The space of symplectic mapping is huge,” Haim-Kislev says. “The question of embedding one domain into another is a very hard question, and even today, there are complete unknowns. We still don’t have the answer for ellipsoids in dimension six, for example.”

In previous work, Haim-Kislev developed a specialized tool to help. She created a formula to calculate symplectic capacities based on polytopes that significantly speeds up the process.

“In the past, you could calculate capacity for some of these cases, but it took a tremendous amount of time and analysis,” Haim-Kislev says. “This formula allows you to work more quickly and explore more examples. It’s not a universal key, but it is a useful way to refine and improve intuition.”

This refined intuition has paid off in her recent work, where the simplicity of her counterexample may appear to be counterintuitive. 

“Even to me, the result was unexpected,” Haim-Kislev says. “The case I explore is remarkably simple. And, in fact, when I first outlined this case, I was trying to prove Viterbo’s conjecture, not disprove it.”

Disprove it she did, with a paper that uses the relatively straightforward geometry of a pentagon to explore the special relationship between convex geometry and symplectic capacity that Viterbo’s conjecture gracefully figures. 

Convexity describes a shape in which you can draw a line segment from any point without crossing the boundaries of that shape; it applies to domains within symplectic manifolds – balls and cylinders for example.

This relationship has become the object of increasing interest since researchers proved that Viterbo’s conjecture – from symplectic geometry, the mapping of phase space – implies Mahler’s conjecture – from convex geometry, the study of Euclidean space. 

This discovery of an innate bridge between these two disparate fields – thought to be as different as two unrelated languages – has sparked the curiosity of many mathematicians, Haim-Kislev included.

“I focus on these unique intersections between symplectic and convex geometries,” Haim-Kislev says. “We’ve known for some time that convexity plays a role in symplectic geometry, which is surprising because convexity isn’t a symplectic property. Symplectomorphism doesn’t ‘see’ convexity. Convexity isn’t preserved under transformation. And yet, we know that domains behave differently when they are convex.”

Viterbo’s conjecture represents one of these special convex behaviors. It states in its 'weak' version that the ball has the largest capacity among all convex domains of the same volume. In its 'strong' version, the conjecture states that all capacities coincide when restricted to the class of convex domains. If the domain is convex, one would always get the same answer. 

Haim-Kislev’s counterexample disproves the weak version, and, by extension, the strong one. 

Her paper uses the product of two domains to create a dynamical system that behaves like a billiards table governed by a modified reflection law, one in which the angle of reflection might be different than the angle of incidence. Within this system, she set out to find an algorithm that would calculate the optimal geometry – the largest symplectic capacity possible. 

What started out as an intended proof for Viterbo’s conjecture, quickly became a counterexample, as she found evidence for a domain that would exceed the expected limits of what a domain’s capacity could be. 

“Viterbo’s conjecture is fascinating and impressive because it elegantly foregrounds the relationship between convexity and symplectic geometry. Our work builds on that achievement and gives us new avenues to explore,” Haim-Kislev says.