Aaron Lauda of USC Dornsife has been awarded an exceptional $8 million grant from the Simons Foundation to mentor researchers from 11 prestigious institutions on a project that could turn science fiction dreams into reality.

the main points:

- USC Dornsife manages an 11-institutional project to push the boundaries of knowledge about how the universe and our reality work.
- Discoveries from the Initiative may lead to developments that are now considered science fiction.
- The four-year project, funded by an $8 million Simons Foundation grant, brings together researchers to tackle some of the most pressing questions in mathematics and theoretical physics.
- A key differentiator of this type of funding: It aims to expand the frontiers of knowledge by bringing together scholars, scholars, and scholars who would not normally interact.

This story involves two types of mathematics: simple and highly complex.

Simple mathematics adds an extraordinary amount of money – $8 million, in fact – to study other, more complex mathematics over the next four years. That’s $2 million a year, and a few numbers more than what’s usually found in grants awarded for math research.

Most science-funded grants go to experimental equipment, postdoctoral scientists, graduate students, and other expensive resources.

“In math, all we usually need is a paper and a pencil, so grants tend to be much lower,” says Aron Lauda, a professor of mathematics, physics and astronomy in the UCSD College of Letters, Arts and Sciences. Lauda led the assignment for the prestigious grant, which establishes the Simons Collaboration on New Structures in Low Dimensional Topology.

The grant is unique in that it is large enough to fund a large team working in a similar direction and will support extended visits by team members and other experts who will enhance USC’s research and community activities, says Lauda, who directs the collaboration.

Funding comes from the Simmons Foundation. Lauda as principal, along with co-principal investigator Chris Negron, associate professor of mathematics, brings approximately 14% of the total — about $1.1 million — to USC. It is one of the largest awards ever received for the UCSD Department of Mathematics.

The remaining $7.9 million is divided among 10 other institutions: Caltech, MIT, UCLA, University of North Carolina at Chapel Hill, Columbia, Stanford, Princeton, and Harvard universities, as well as the Center for Quantum Mathematics of the South. . University of Denmark and University of Zurich.

## The Simons collaboration aims to solve complex math problems

The complex mathematics that Lauda and other researchers are exploring focuses on topology. In greatly simplified language, topology refers to properties of a geometry that don’t change no matter how much you use it.

Lauda points to nodes as an example.

A length of rope is twisted and intertwined, then tied at its end to form one contiguous knot, which has its own topology. No matter how the node is placed, it is still the same.

“If I shake it, as long as I don’t cut or break it, it’s basically the same,” says Lauda.

If two people arbitrarily twist and tangle two different but identical ropes of the same size and seal the ends together, even the most discerning eye may find it difficult to determine whether the knot is identical or different.

As Lauda explains, we can try to animate one node to look the same as another, but if we don’t succeed, we can’t necessarily infer that the node is different. Maybe we didn’t try hard enough. This is where the math of node constants can help.

A static node is a way to assign a number or polynomial to a node that remains the same no matter how the node is shaken or deformed. If our constant assigns different numbers, or polynomials, to two different nodes, then we can be sure that the node is different.

## Mathematicians are pushing for progress on par with Einstein

As fun as knots can be, Lauda and his collaborators on the project have set their sights on more big issues in topology, with the goal of building on existing theory and tools to answer fundamental questions about the universe.

“Knot theory is interesting because it is the intersection point where ideas from theoretical physics, such as string theory, intersect with some of our more complex mathematical tools,” says Lauda. “But what Collaboration is really trying to do goes beyond knots, to understand the ways in which even our reality can have these kinds of topological properties.”

Lauda refers to knowledge on par with that revealed by Einstein and his general theory of relativity, namely, that the three dimensions of space are closely related to the fourth dimension, which is time. And mass — planets, stars, even nodes — warps space-time, something we experience as gravity.

In revealing the gravitational curvature of space-time, Einstein revolutionized perceptions of the universe, and this in turn made technologies such as GPS navigation possible.

When it comes to topology, Lauda says, calculating the fourth dimension has led to important advances in mathematics and physics, and he and his collaborators aim to push the boundaries of knowledge.

“Mathematically, it’s important to understand and classify 4D topological objects in the same way we do nodes,” he says. “That’s the kind of big frontier we’re trying to achieve with this collaboration.”

## Math collaborators work to progress in four dimensions

Mathematicians attempt to describe space-time by defining four dimensions: three spatial dimensions (up and down, back and forth, right and left) and one time dimension.

For mathematicians and physicists, working on this fourth dimension is the most difficult, a fact that Lauda finds interesting. “It is a curious fact of nature that the fourth dimension is unique in that our understanding is largely limited to precisely this dimension which happens to coincide with the four dimensions that make up our universe.”

The Simons Collaboration aims to tackle this confounding dimension head-on, developing the next generation of tools for studying 4D topology, unlocking new techniques emerging from theoretical physics and solving long-standing questions that have plagued mathematicians for decades, says Lauda.

For example, the project could shed important new light on theories of quantum gravity, or a theory that combines general relativity and quantum physics.

“General relativity works fantastically well when you’re talking about planets and stars and the like,” Lauda explains. Quantum mechanics works when you talk about very small things like subatomic particles. But once you start to combine those little things with big things, like one might encounter near a black hole, or near the Big Bang, theories start to fall apart.”

He says the project’s efforts could pave the way for finally bridging the gap, giving deeper insight into how the universe works on all scales.

Moreover, any breakthroughs in understanding could lead to scientific and technological advances in the same way that Einstein’s discoveries led to the Global Positioning System, he says.

“What we expect to learn from this project can serve as the basis for future technologies and developments that seem like pure science fiction now.”

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