Toward a Deeper Understanding of the Manifolds Behind Motion

Up close, the surface of a beach ball will not look round, but flat. So will the surface of an inner tube — even though these objects look nothing alike. Mathematically speaking, these two summertime accessories can be classified as two-dimensional manifolds: Locally they look the same, but when you zoom out and look at them in their entirety, they’re obviously different. Understanding the properties that determine these differences, both for these and for much more complicated examples, is a central theme of modern geometry.

However, no geometer studies manifolds in a vacuum, says Simons Junior Fellow Tim Large; instead, they focus on related components that lend additional understanding. In Large’s case, this means a focus on symplectic manifolds, which deals directly with Newton’s theory of motion and attempts to make sense of the movement of particles in curved space. While Newtonian mechanics is over 300 years old, symplectic geometry has only come into its own within the last few decades, through new theoretical developments and applications to quantum physics and other areas of pure and applied mathematics.

Large earned his undergraduate degree at the University of Cambridge and a Ph.D. from the Massachusetts Institute of Technology. He is now completing a postdoctoral fellowship in mathematics at Columbia University, under the mentorship of Mohammed Abouzaid.

Large and I recently discussed his work and its implications. Our conversation has been edited for clarity.

What are symplectic manifolds?

Symplectic manifolds have a technical definition that can look bizarre even to mathematicians, but, they are everywhere — from basic physics all the way to complex string theory.

To explain, let’s turn to physics 101: the motion of a pendulum. If we know the pendulum’s angular position and momentum, we can predict its movements. All the possible different angular positions and momenta of the pendulum taken together form a two-dimensional manifold, which we call the phase space.

Up close, the phase space looks like a flat plane, but there is a cylindrical aspect to its shape: If we rotate the pendulum 360 degrees, we’re back to where we started. This is one of the most basic examples of a symplectic manifold. More complex examples include the motion of planets around their stars. Symplectic manifolds are critical for helping to make sense of Newton’s equations as they relate to gravity.

Tell me about more complex examples of symplectic manifolds.

Symplectic manifolds turn up all over modern math and physics. For example, one of the central postulates of string theory is that the universe comes with extra dimensions (usually six) too small for even atomic microscopes to see. These extra dimensions are wrapped together into a special type of symplectic manifold called a Calabi-Yau manifold, which can in turn reveal new information about the observable particles in the known universe.

The same symplectic manifolds appearing in string theory also crop up in algebraic geometry, a very different area of pure mathematics that focuses on the spaces and shapes defined by polynomial equations. Some algebraic spaces have a hidden ‘mirror’ symplectic manifold. In these cases, the string theory of the mirror manifold encodes a huge wealth of information about the algebra of the original space.

Mathematicians call this phenomenon mirror symmetry because it enables us to take an algebraic problem, look at it in a different way, and see some surprising symplectic geometry. This concept has been an influential part of geometry and theoretical physics for the last 30 years, and I hope that symplectic geometry has a greater part to play over the next century as we develop a better experimental understanding of the fundamental fabric of our universe.

What draws you to the study of symplectic manifolds and symplectic geometry?

What really gets me excited about symplectic manifolds is that for all the different places that they exist, we know fundamentally very little about them. It’s a very young subject, compared to other manifolds such as algebraic manifolds, which have a well-established collection of tools that mathematicians have long used in their studies.

The basic toolkit for symplectic geometry is very much still in development, although there has already been marked success in proving new theorems. One of the things I enjoy most about working in the subject is that to build these new tools, I need to draw upon knowledge from a wide range of mathematical fields — from partial differential equations to homotopy theory. Symplectic manifolds are at a crossroads of so many parts of contemporary math.

Can you elaborate on those tools and theorems?

One of my favorite examples goes back to the pendulum. Pendulums move periodically, meaning that they come back to where they started with the same momentum. Is this just a happy accident of the explicit solution to Newton’s equations, or is it in part a manifestation of the cylindrical topology of the pendulum’s phase space? In the 1960s, Russian mathematician Vladimir Arnold asked a version of this question in great generality: When is there a periodic solution to Newton’s equations in the symplectic phase space of any classical mechanical system? He conjectured that the existence of periodic solutions must depend on the global topology of the symplectic manifold.

The modern viewpoint of symplectic geometry really began with German mathematician Andreas Floer in the 1980s, some 20 years after Arnold’s conjecture. Instead of thinking about the dynamics in terms of a point tracing out a line on a symplectic manifold, Floer envisaged loops or strings sweeping out surfaces within the manifold, much as string theorists do. Whereas the particle motion is given by Newton’s equations — which are ordinary differential equations — these surfaces must satisfy a much more complicated partial differential equation.

At first sight, this makes the problem far more complicated since partial differential equations are typically much harder to solve explicitly. But Floer ingeniously showed that more indirect questions about these equations — such as the existence and nonexistence of solutions or what sort of spaces families of nearby solutions form — encode much of the topology of the symplectic manifold. He defined an algebraic invariant of certain types of symplectic manifolds, which we now call Floer homology, and used it to prove several of Arnold’s conjectures.

Since its introduction, Floer homology has been at the center of symplectic geometry. Much work has gone into extending its definition to apply to different varieties of symplectic manifold, to find out what it means in the context of mirror symmetry, and to enhance its capabilities to see more of the topology of a symplectic manifold.

How does your work build on Floer homology?

Homology itself is almost 100 years old; it is probably the most basic metric of how many ‘holes’ a manifold has. It will tell you that the beach ball has no holes, while the inner tube has one, for example. But this turns out to be a coarse way to look at a manifold, and there are many finer and more powerful ways you could answer that question.  This is the subject of homotopy theory.

I work on building and applying additional structures within Floer homology that mimic structures in homotopy theory. Floer homology and homotopy theory appear very analogous, so I want to know: To what extent can you use the machinery of homotopy theory to directly answer questions in symplectic geometry? I am building a dictionary that connects Floer homology and homotopy theory.

Can you describe that work in a bit more detail?

Modern homotopy theorists have a powerful notion called a stable homotopy type, which is an abstraction of a manifold that loses some of its geometric features but retains many of the most important topological ones. Many of the more complex tools used by homotopy theorists are in fact for manipulating stable homotopy types. In my work, I want to use Floer homology to create stable homotopy types associated with symplectic geometry — to essentially use the homotopy theorist’s toolkit to prove new results about symplectic geometry. At MIT, I constructed a stable homotopy version of Floer homology for a specific variety of symplectic manifolds called Liouville domains and put my construction into a broader structure called a Fukaya category. Fukaya categories are basically a way to organize the algebra of Floer homology. They can be very hard to compute, but if you manage to, you can get a strong handle on a symplectic manifold.

My focus now at Columbia is on getting a computational grip on these homotopy-theoretic versions of Floer homology and seeing what new things they can tell us about both symplectic geometry and mirror symmetry. Many of the earliest examples of mirror symmetry arise from symplectic manifolds that are related in various ways to another famous family of manifolds called the complex projective spaces. Complex projective spaces have a lot of extra symmetries, which can make Floer homology and computations more amenable. I’m currently working on constructing homotopy-theoretic versions of Floer homology and Fukaya categories for these spaces and using the extra symmetries to understand them concretely. My hope is that this can give some insight into a deeper, homotopy-theoretic interpretation of mirror symmetry.

Finally, what are your thoughts on the Simons Junior Fellowship?

I interviewed for the Junior Fellowship during the depths of the pandemic, and it was the first time I’d spoken to someone who wasn’t a mathematician in maybe six months. When things went remote all of my own department meetings kept going, but basically it became impossible to meet up and connect with someone in another field. To finally be able to talk to a biologist during my interview was such a breath of fresh air! And I really enjoy learning and speaking about the wide variety of scientific fields now that I’m a Junior Fellow.

And even within math, I’m learning a lot from the other Junior Fellows. For example, I never appreciated the importance of random matrices until I spoke to my Junior Fellow colleague Jiaoyang Huang. The Simons Junior Fellowship is transformative both within and beyond my field, and I really appreciate the intellectual community within New York City that it fosters.