A team of mathematicians at the Institute for Advanced Study has released a breakthrough proof that sheds new light on one of number theory’s oldest and most established inquiries.

Their achievement highlights the power of a surprising partnership: number theory and dynamical systems.

At first glance, these fields couldn’t be more different. Number theory studies the properties and relationships of integers. The study of dynamical systems, on the other hand, originated in a very practical branch of physics that aims to understand the ways objects and particles move through space. 

And yet the fields do fruitfully interact. Homogeneous dynamics, a key subfield of dynamical systems theory, has introduced new and promising ways to discover profound mathematical truths. In a landmark achievement, Grigory Margulis was among the first to successfully connect number theory and homogeneous dynamics when he proved the Oppenheim conjecture in 1987. Before Margulis’ proof, this powerful number theoretic problem had remained unresolved for almost sixty years.

In the wake of this milestone, mathematicians have been working to sharpen and expand the ways that the study of motion can reveal new information about numbers. At IAS, Manfred Einsiedler, Elon Lindenstrauss, Amir Mohammadi, and Andreas Wieser have made tremendous progress with their recent paper, "Effective equidistribution of semisimple adelic periods and representations of quadratic forms." 

The collaboration debuts a powerful new toolkit from dynamical systems that goes far beyond what can be obtained by number theory’s own methods for studying one of its most classical preoccupations: representations of quadratic forms.

In 2007, Ellenberg and Venkatesh delivered the first application of homogeneous dynamics to this foundational number theory problem.  This work demonstrated that one could use a dynamical systems approach to represent quadratic forms. Their qualitative approach sparked further investigation into the ways that homogeneous dynamics could continue to shed light on this central problem in number theory. 

The new IAS work brings this interaction to the finish line by providing explicit, quantitative information that significantly deepens mathematicians’ conceptual understanding of these structures.

"The beautiful thing about our result is that it’s not just the quantitative complement to existing qualitative information,” Wieser says. “It’s also deepened our qualitative knowledge of the problem.”

The IAS paper applies tools from homogeneous dynamics to a classical number theory problem about representations of quadratic forms, a problem that has been in play for centuries. 

“It asks, basically, whether the set of numbers produced by plugging whole numbers into one quadratic form can be found inside a set produced by a larger quadratic form,” Wieser says. “It’s a problem that seems simple but is actually very complex, challenging us to replace elementary computational ideas with sophisticated conceptual ones. Previous purely number-theoretic methods have struggled to make progress here, but our approach comes at the issue from a very different perspective.”

This issue of quadratic forms is fundamental for a reason. Number theorists have been trying to find their way through this problem for many years, and even the smallest victories on their journey to solutions are considered major advancements in the field.

This new paper from IAS provides robust, complete answers, showing that the path to these profound, universal truths need not necessarily come from within number theory itself.

For the first time, the IAS team’s work makes it possible to precisely count the representations of a quadratic form in question. Translating the problem into the language of orbits and flows native to homogeneous dynamics has yielded “effective” results, a term that designates an outcome that is both numerical and computable.

Intriguingly, the dynamical tools developed by the IAS team are flexible, demonstrating powerful potential for application to other number theoretic mysteries. 

“Our homogeneous dynamics tools – based in geometry and group actions – greatly transform how mathematicians approach this problem,” Wieser says. “But what’s most exciting to us is how others will use these tools in the future. We believe they can be impactful in many areas in number theory.”

A fully effective, quantitative, and adelic equidistribution theorem 

The broader mathematical context for this work traces back to the foundational qualitative work of mathematician Marina Ratner. She revealed that unipotent flows – a specific kind of dynamical system – possess an internal structure. The discovery of these hidden regularities allowed homogeneous dynamics to flourish as a mathematical field.

Over time, Ratner’s achievement has inspired an effort to match her qualitative descriptions with quantitative ones. The new IAS paper constitutes a major advance within that effort, providing effective results. The IAS team’s analysis is notable for its generality, overcoming the limitations of earlier proofs requiring restrictive assumptions to function.

For example, in 2009, Manfred Einsiedler, Grigory Margulis, and Akshay Venkatesh published a spectacular breakthrough regarding periodic orbits of semisimple groups. The proof relied on assumptions that ruled out situations in which periodic orbits appear in continuous families. Those assumptions made the dynamics easier to control. But they also limited the range of arithmetic applications.

In 2024, Wieser showed that this obstacle could be overcome, proving that it is possible to obtain quantitative results even when these continuous families are present. Wieser’s approach provides tools to understand and quantify these masses of complicated flows, tracking their patterns in order to discern at exactly what point and how fast they spread evenly in homogeneous spaces.

Wieser’s progress was motivating for the IAS group to seek a new approach that could function in a fully adelic setting. Adelic spaces are wide-ranging and general, including all completions of a number field at once. Using tools from a variety of different fields, the IAS team’s work addresses the problem of semisimple periodic orbits without any limiting assumptions, adding a substantial layer of difficulty to the inquiry. 

The adelic capability is what renders the IAS toolkit such a powerful engine of number theoretic results. Taking into account real numbers and p-adic numbers, the adelic setting gives mathematicians the necessary freedom to productively apply these tools in number theory. As much an advantage as a challenge, this freedom requires significant discipline and creativity to manage, a standout success of the new IAS publication.

This IAS publication is significant for making homogeneous dynamics more relevant than ever in number theory. It’s an innovative framework that is expected to create even more profound connections between the two fields.

The team’s quadratic forms application serves as a powerful demonstration of the theorem. But it offers much more. By introducing a new framework capable of transforming qualitative assurances into effective, quantitative results, the IAS collaboration opens up a much wider horizon of possibility for what can be accomplished by studying orbits in homogeneous spaces.

Is there any specific direction the team would like to see the toolkit take?

“More than anything,” Wieser says, “I hope to be surprised.”