Can numbers reveal the shape? CMU mathematics graduate student explores the hidden geometry in mathematical data

What if you could hide the shape of an object and reveal only a collection of numbers connected to it? How much could those numbers tell you about the shape you can no longer see?

That question is at the heart of research being conducted by Central Michigan University mathematics graduate student Brandon Hart with Dr. Debraj Chakrabarti in the Department of Mathematics.

Hart’s research explores the relationship between geometry and numerical information generated by mathematical objects associated with regions inside the unit disk. At its core, the project asks whether numbers can reveal important characteristics of the shape that produced them.

“The main idea of the project is to understand how the shape of a region can be connected to certain numbers that come from it,” Hart said. “If I hide the shape from you but give you enough numerical information connected to it, how much could you figure out about the shape?”

The question leads to a series of mathematical problems. Hart and Chakrabarti are studying how these numerical measurements change as the shape of a region changes. For example, if several regions have the same hyperbolic area, is there a particular shape that produces the largest possible value for one of the measurements? And if there is, can mathematicians prove why?

One result has already provided an important piece of the puzzle: for one of the measurements Hart and Chakrabarti are studying, a disk produces the largest possible value among regions with the same hyperbolic area. The result is even more significant because equality occurs only for a disk.

That means the measurement is capturing more than the size of a region.

“It is also detecting something about its shape,” Hart said.

The research sits at the intersection of geometry and analysis. Geometry provides the shapes, areas and distances, while analysis provides the functions and numerical measurements associated with those shapes. Hart said the surprising part is discovering that information generated through analysis can retain information about the original geometry.

That connection became clearer after one of the project's early turning points.

Hart and Chakrabarti spent about a month working through a lengthy calculation that rewrote one of the quantities they were studying as a complicated series involving special functions. The calculation was correct, but it was not providing the geometric insight they were looking for.

So Hart suggested looking at the expression from a different perspective.

He initially expected the relationship to involve the hyperbolic cosine function. Instead, the correct expression involved its reciprocal, known as the hyperbolic secant.

That discovery changed the direction of the project by establishing a much more direct connection between the analytic formula and the geometry of the disk.

Hart then recognized that the formula appeared to contain even more information. One part of the expression reflected distances between points, while another captured information about the hyperbolic area enclosed by those points.

“That idea that what we were interested in encoded distance and hyperbolic area led us to many of the ideas that we are currently researching,” Hart said.

The work has also demonstrated that mathematical research rarely follows a straight line.

In a classroom, students typically receive problems that are known to have solutions. Research is different. The question itself may be new, and researchers may not know whether a particular statement is true or whether a proposed approach will work.

For Hart, that has meant learning when to pursue an idea, when to change direction and how to turn an informal insight into a proof that another mathematician can carefully verify.

“You can spend a long time on an approach and then realize it does not work,” he said.

Another challenge has been bringing together ideas from different areas of mathematics. A problem that begins with functions and operators may ultimately require geometric arguments.

Hart and Chakrabarti have also had to adapt existing mathematical results to their particular setting. Some theorems they use were developed in more general or different geometric contexts, requiring the researchers to examine the assumptions, definitions and constants carefully and, in some cases, rework proofs so the results apply consistently to their problem.

The process can be painstaking, but Hart said it is also one of the most valuable parts of the experience.

“The biggest difference is that in a classroom, the problems are usually chosen because someone already knows they can be solved,” he said. “Research is not like that.”

Hart became interested in working with Chakrabarti because of his interest in complex analysis and Chakrabarti's work in several complex variables. After passing his qualifying exams, Hart reached out to Chakrabarti, and the two began developing the project together.

Chakrabarti has helped guide the overall direction of the research and identify problems to investigate, while Hart has taken on much of the detailed mathematical work, including calculations, proofs, examples, reading related research and writing the results.

He has also developed ideas that have contributed to the project's progress.

The research is ongoing and is being developed into a manuscript. Hart is also preparing a 20-minute presentation, “Bergman-Toeplitz operators with Indicator Symbols,” for a Joint Mathematics Meetings special session focused on the interplay of geometry and analysis in several complex variables.

The presentation will highlight results from the project and explore the connection between the numerical information Hart and Chakrabarti are studying and the geometry of regions in the disk.

The researchers also hope to determine how far their findings can be extended, including whether similar relationships hold for more complicated measurements and in higher-dimensional spaces.

For Hart, the appeal of the work comes from the same reason he chose mathematics in the first place: a desire to understand why something works rather than simply learning how to produce an answer.

“I have always been more interested in understanding why something works than simply memorizing how to do it,” he said.

That curiosity has taken him from Flint, Michigan, to his sixth year in CMU's mathematics Ph.D. program, where he is exploring questions that ask mathematicians to look at familiar objects in an unfamiliar way.

Instead of starting with a shape and calculating what numbers it produces, Hart's research asks the question in reverse: What can the numbers tell us about the shape?

After completing his Ph.D., Hart hopes to continue working in mathematics at the college level, combining teaching with continued research.