Kristin Elizabeth Gabe

Project title

Classification and Dynamics: Beyond the Limit

What is your project about?

I study operator algebras, the abstract mathematical objects underpinning quantum technology. While the most basic examples of operator algebras come from finite-dimensional matrices (grids of numbers), capturing real-world phenomena often requires infinitely many dimensions or parameters. But how does one study infinite dimensions? Just as we examine an architectural marvel by looking at its bricks, we can understand an infinite mathematical object through its finite building blocks. My work exposes this underlying architecture for operator algebras, showing that they are built up from finite-dimensional matrices. This project focuses on these building blocks and how they are put together in order to uncover unique mathematical fingerprints that identify these intangible objects, as well as hidden symmetries underlying the complex physical and information systems they model.

How did you become interested in your particular field of research?

I have always had a deep appreciation for logical thinking and for finding patterns underlying the world I experienced, which is why I was originally drawn to mathematics. What captivated me about operator algebras, or more generally functional analysis, was the elegance. I saw professors take questions that demanded cluttered computations, distill the essence of what was really being asked, and put that essence into an abstract framework that could deliver the solution effortlessly. By embracing abstraction, it felt like transitioning from walking to flying.

What are the scientific challenges and perspectives in your project?

Just as focusing on individual bricks can make one lose sight of the overall structure of a building, focusing on the finite-dimensional building blocks underlying so many operator algebras can obscure their large-scale features and symmetries. Yet in our setting the whole structure leaves an imprint on its building blocks. The challenge lies in decoding this subtle signature. My project aims to detect traces of their distinct global features and large-scale symmetries at this small scale. Pivoting to the phenomena that these operator algebras model, this amounts to picking up on global patterns and pathologies of a large complex system from only rough approximations.

What is your estimate of the impact, which your project may have to society in the long term?

As quantum technology rapidly advances, understanding the capabilities and limitations predicted by its fundamental mathematical models is more critical than ever. This project delivers tools to decode and manipulate the mathematics of quantum information, such as the long-term behavior of communication channels. But even beyond quantum, theoretical mathematics has a rich history of sparking unanticipated breakthroughs. Non-Euclidean geometry made Einstein’s relativity and GPS possible; counting holes in high dimensional topological shapes became a powerful tool in big data analysis; and pure number theory became the foundation of global cryptography. The abstract nature of pure mathematics gives it unparalleled versatility, and thus we always keep one eye on our motivation and the other on the unknown.

Which impact do you expect the Sapere Aude programme will have on your career as a researcher?

Receiving a Sapere Aude grant is a great honor and opportunity, enabling me to build my own research group here at the University of Southern Denmark. It also allows me to bring international experts to Odense to better discuss and exchange ideas on the project and where it fits into the global research framework. This in-person interaction is a crucial facet of mathematics that gives us the time and space to communicate highly complex ideas. Moreover, it also has the benefit of drawing international attention to the work we do here. This grant is the next step in establishing myself as a global research leader and makes me highly competitive for major international grants such as the ones from the European Research Council (ERC).