In the realm of abstract mathematics, hidden structures often reveal themselves in unexpected ways, much like the elusive images in Magic Eye books from the 1990s. Carnegie Mellon University mathematician Florian Frick delves into this intriguing world where fairness and geometry converge, offering a fresh perspective on complex problems.
Florian Frick, an associate professor in the Mellon College of Science at Carnegie Mellon University, specializes in the interplay of geometry, topology, and combinatorics. Geometry studies shapes, topology examines properties of shapes that remain constant despite deformation, and combinatorics focuses on discrete structures and patterns. By merging these fields, Frick transforms seemingly non-geometric problems into questions about spaces and symmetry.
“Mathematics is a kind of mapmaking for abstract worlds,” Frick explained. “I enjoy the process of taking an abstract problem and translating it into something that I can reason about visually.”
The concept of a “political landscape,” which describes a complex arrangement of views and alliances, finds a parallel in Frick’s approach to mathematics. He builds geometric spaces for various problems, revealing insights that aren’t apparent from any single perspective.
The Shape of Fairness
Imagine the challenge of dividing rent among roommates with different-sized bedrooms and varying preferences. Mathematically, this problem can be visualized as a geometric shape. For example, with three roommates, the possible rent divisions form a triangle. With four, it’s a tetrahedron, and with more, an even higher-dimensional simplex emerges.
Rather than examining each potential rent division individually, mathematicians assess the entire space of possibilities. “A single proposed rent split may fail. Many proposed splits may fail. But the overall shape of the space of possible splits may force a fair solution to exist somewhere,” Frick noted.
Frick and his team tackled a variation of this problem, considering scenarios where one roommate’s preferences are unknown. Their research demonstrated that a fair division is still achievable, providing a method for computing an envy-free rent division. This work was highlighted in an episode of PBS’s Infinite Series titled Splitting Rent with Triangles.
Geometry Beyond the Visible
Frick’s research embodies a broader theme: using geometry to uncover solutions to complex problems. “When I look at a problem, I parameterize the space of all possible solutions. I use the word space because it is a geometric object,” Frick stated. “And on this geometric object I use geometric and topological tools to find an actual solution. This is particularly powerful because geometry keeps track of global phenomena.”
From data science to economics, many mathematical problems involve global phenomena. Frick applies topological and geometric methods to questions of shape division, high-dimensional object intersection, and discrete structure partitioning. His work bridges fair division, combinatorics, convexity, and theoretical computer science, revealing hidden constraints within abstract problems through a geometric lens.
Geometry, in Frick’s hands, acts as a tool to shift focus, transforming abstract problems into geometric forms and unveiling the hidden structures that govern them.
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