# Exploring the Shape of Our Universe–and the Multiverse

A professor of mathematics considers popular culture’s current obsession with multiverse theory and explains how algebraic topology and differential geometry can shed light on the subject

The multiverse: it’s a hot topic these days, dominating the plotlines of films like Everything Everywhere All at Once, countless productions from Marvel Comics, television shows like Russian Doll and Shining Girls, and a slew of other cultural artifacts.

Tufts Now asked Professor Loring Tu, an expert in algebraic topology and differential and algebraic geometry at the School of Arts and Sciences, for insights about the universe (or universes) in which we live, as well as advice about the questions we really should be asking.

#### Tufts Now: How do you explain algebraic topology to, say, a first-year high schooler?

Loring Tu: It’s important to understand that mathematicians specialize—they become algebraic topologists or differential geometers, but those are just particular approaches. It makes more sense to talk about topology or geometry in a general way. And I’m purposely mentioning geometry, because when we’re discussing multiverse theory, geometry is as relevant as topology. So, a more useful explanation for the student might be one focused on the difference between topology and geometry.

#### What is the difference?

Topology is the study of shapes up through continuous deformation—that is, deforming a shape without cutting or tearing it in any way. For example, a basketball is perfectly round and a football is oval-shaped. But in topology, they’re considered to be the same; imagine a basketball made of malleable material that you can squash into a football. Topologists study the properties of a shape that don’t change even when the shape is deformed. You might think of it as the study of a shape’s essence.

Geometry, on the other hand, considers how a shape looks. In geometry, you can’t deform shapes, because that changes properties that geometry measures—for example, distance, volume, or curvature. Concepts such as those don’t make sense in topology.

So, to your original question, algebraic topology is just the topological study of shapes using the methods of algebra.

#### It sounds like, whether we use methods of geometry or methods of topology, the possible existence of a multiverse is related to the shape of the universe?

Popular-culture references to the multiverse don’t mean what we mean from a theoretical perspective. Movies, TV shows, fiction—these seem to center around the idea of parallel universes existing simultaneously: whole universes that are separate from one another. But in science, it’s a question of what’s in the observable universe. The observable universe is the totality of all the events and objects that we can see, measure, or make contact with.

The minute someone crosses over from one parallel universe to another, as they do in movies and the like, they are making contact with what’s observable—and therefore, they’re in the same universe. So, it’s a paradox. If it’s a parallel universe, it should be unobservable, but if it’s unobservable, we can never make contact with it or know for sure that it exists. If we do make contact with it, it’s by definition a part of the observable universe—meaning our universe, the same one we’re in.

The shape of the universe that we’re in is relevant because the multiverse can be interpreted as different universes existing simultaneously or it can be interpreted as different models of the universe we live in. Topology and geometry can help us create those models. These fields study the possible shapes of any universe.

And when you’re considering shape, you’re considering fundamental characteristics, like whether something is finite or infinite; whether it’s curved or flat; whether it has a boundary, like an eggshell, or no boundary; how many dimensions it has (some physicists say the universe has ten!).

The fact is, we don’t really know for sure what kind of universe we live in. Are we inside a ball? A donut? Maybe the universe is pretzel-shaped!

#### Can topology and geometry reveal definitively what shape our universe is or whether parallel universes exist (and, if so, how many)?

No. Topology and geometry can only provide possible models for our universe and for parallel universes. It is possible that parallel universes exist. But mathematics alone cannot prove what the universe is nor whether parallel universes exist.

#### What else is needed?

We require physical data. In astronomy, there is an observed phenomenon called "red shift," which means that light from distant galaxies is shifted to a longer wavelength. You could also think of this as "becoming redder.” The standard Big Bang cosmology is based on an interpretation of the observed increase of red shift with an increase in distance as being due to expansion of the universe; in other words, more distant galaxies have larger red shifts because they are moving away from us faster. I recently learned from Krzysztof Sliwa, a colleague in physics, that by analyzing the number of galaxies as a function of their red shift, one can distinguish among different models of the universe. If we can make accurate enough measurements, then we can tell which model fits the data best.

#### Does Sliwa’s point—that it is possible to exclude some models of the universe—change anything for mathematicians?

Yes, it means that mathematicians need to construct models of the universe compatible with the physical constraints.

#### Why should we care about the shape of the universe?

I believe, as humans, we have a natural curiosity to understand the universe we live in. Thousands of years ago, people wanted to know whether we live in a flat world or a round world, or whether Earth is the center of the universe, or it revolves around the sun. It used to be a life-or-death question: think of Galileo recanting because otherwise the Catholic church would have punished him for his findings.

On a practical note, if we want to do interplanetary travel someday, it’s crucial for us to determine the shape of the universe. We need to have some idea of what will happen when we set out. If we keep traveling, will we travel forever? Or will we end up back where we started, as would happen on Earth?

#### To return to the question of multiverse theory as it’s represented—or misrepresented—in popular culture, why do you think it’s such a popular concept right now?

Since antiquity, people have been fascinated by the idea of alternative worlds. But I think there are at least two factors for the recent explosion of interest. One is the polarization of contemporary America. With regard to Trump, abortion, race, gender identity, and a host of other issues, it almost seems like people are living in parallel universes with alternative realities, alternative sets of facts.

Another factor is what’s been happening around us these past few years: the pandemic, the social justice movement, war, economic inequality. I think people yearn for another world, a better one, in which good triumphs over evil and justice always prevails.

#### As someone who has spent a career exploring big ideas like the math behind multiverse theory, how do you feel about this current fascination?

It is disappointing to me that in the current slew of movies and shows about it, people are more interested in superheroes than in the shape of the universe. What we really need to know is: What is this universe we live in? That’s the question to be asking.