Elizabeth Nelson
Elizabeth Nelson

Shapes of Mathematical Elegance

by J. Edward Anthony

Like a language, math evokes ideas that exist independently of any particular set of symbols. In algebra, for example, a polynomial usually appears as numbers and letters strung together by plus and minus signs, such as x2 + xy + y2. Algebraic expressions are such a familiar way of denoting polynomials that it’s easy to lose the distinction between the concept and its representation, as with a dollar bill or a cartoon cat.

But polynomials are not a tangle of numbers, variables, and plus or minus signs any more than a tiger is the five-letter word. A poem, a child’s picture book, and a Netflix series each conjures a tiger in its own way. So it is with mathematicians. “Different people see polynomials in different lights,” says Karola Mészáros, Mathematics. “I explore the properties of important polynomials by thinking of them geometrically, as polytopes.”

If the term polytope calls to mind a polygon (a flat shape with straight edges, such as triangles and octagons), there’s good reason: Polygons are a subset of polytopes that have two dimensions. Polytopes, as the more inclusive concept, may have any number of dimensions. A line segment is a one-dimensional polytope, a pyramid is a three-dimensional polytope. But a polytope can also have four dimensions, or four hundred.

That may sound out of this world, but it simply means that polytopes have the capacity to embody, in geometric terms, any polynomial—including polynomials that have a huge number of variables. From genomics to climate science, that level of complexity is all around us.

Visualizing Polynomials

Every polynomial can be visualized geometrically. “I see a polynomial as points in space with chips stacked on them,” Mészáros says. “Having that view, I can sometimes understand properties of these polynomials that hadn’t yet been established. Or I recast established properties of polynomials in a different kind of framework. Sometimes, the geometric view may help me discover properties of these polynomials that have not been considered before.”

How can a polynomial and a polytope describe the same mathematical tiger? Mészáros explains how to plot x2 + xy + y2 as three points on a flat, two-dimensional plane. Algebra isn’t part of everyone’s daily banter, so a demonstration of key terms might be useful here. The polynomial x2 + xy + y2 is composed of three monomials: x2, xy, and y2. The variables are x and y, and the superscript 2s are exponents. Coefficients appear as a multiplier preceding the variable, such as the 5 in 5x2. Each monomial in x2 + xy + y2 has an implied coefficient of 1.

“Because any number raised to the zero power equals one, you can think of x2 as x2y0,” Mészáros says. “You can plot that as the point (2,0) on a grid. Each variable in xy has an exponent of one, so xy is really x1y1. This you can plot as (1,1). And y2, or x0y2, represents the point (0,2).” The result is a line segment—a one-dimensional polytope—that lives in a two-dimensional space.

“In mathematics, somehow the truth often appears beautiful.”

But maybe it’s not so simple. “When I think of a polynomial like x2 + xy + y2, I don’t think of it as just three points on a Euclidian plane,” Mészáros says. “I think of it as all the points on a Euclidian plane. Three of the points have ones on them. The rest appear with a coefficient of zero.” The 1s on those three points are the implied coefficients: 1x2 + 1xy +1y2. For Mészáros, however, zeros are important too, and x3y or (3,1) is part of the polynomial even if we can’t see it. It just has a coefficient of zero: 0x3y. In whatever multidimensional space Mészáros is working, every integer point gets a value that corresponds to the coefficient, which can be represented as chips stacked on the point. Sometimes she renders polynomials by color-coding the number of chips on each point to produce a sort of topographic map.

Unexpected Beauty

This example, which Mészáros provides in concrete algebraic terms, leaves little to be imagined. She herself is consumed with a few distinguished genera of polynomials, with names like Schur, Schubert, and Grothendieck, that are defined in ways far more abstract than a simple algebraic expression. “When I started working with Schubert polynomials, it was amazing to me that if I asked, does x11y99 have a nonzero coefficient, that wasn’t an easily answered question,” Mészáros says.

Her research has cast aside some of the mystery that surrounds Schur and Schubert polynomials. With collaborators Alex Fink (Queen Mary University of London) and Avery St. Dizier, PhD ’20—now a postdoctoral fellow at the University of Illinois, Urbana-Champaign—Mészáros proved that every Schubert polynomial hits exactly the integer points of a particular kind of polytope, called a generalized permutahedron. The revelation has made these classic polynomials even more fascinating.

Generalized permutahedra have shown up in recent work in quantum field theory. It’s an intriguing connection, and a glimmer of math’s ability to describe the natural world. Mészáros says that her primary motivation, however, is math’s inherent beauty: “When I think about math, I think about it because I love it. I think there’s an obsessive curiosity in all research mathematics. A search for truth. In mathematics, somehow the truth often appears beautiful. It’s nice to have something where beauty and the truth usually coincide.”

One such coincidence of beauty and truth occurs in the topography of Schubert polynomials’ coefficients—all those chips stacked on the polytope’s various integer points. Mészáros, in collaboration with St. Dizier, June Huh (Stanford University), and Jacob P. Matherne (now a postdoctoral fellow at the University of Oregon), proved that for every Schubert polynomial (of which there are infinitely many), the coefficients form a hill-like structure that rises and then falls. It never goes up and down and back up again. In mathematical terms, the coefficients are log-concave.

“Schubert polynomials haven’t been around such a long time—just since 1982,” Mészáros says. “But they include Schur polynomials as special cases, and Schurs were introduced in 1815. This hill-like structure of the coefficients, it’s a beautiful result. You know, these Schur polynomials have been around more than 200 years. For a mathematician, it's a remarkable thing to be able to formally say something about that structure.”

Math as a language can sometimes be authoritative, sometimes instrumental. It describes the world around us and gives us tools with which to glimpse the future. For a pure mathematician like Mészáros, math is also evocative and poetic. It’s a storyteller’s language, opening our eyes to truths that we couldn’t see without it. “Algebra is a fundamental part of mathematics, and polynomials are a building block of algebra,” she says. “They’re always going to be here. To understand them differently, and understand new properties of them—that’s interesting in and of itself.”