Lucía Martín-Merchán, a post-doctoral researcher in Pure Mathematics, has disproved a conjecture that mathematicians have been investigating for almost thirty years.

In her upcoming article, “Compact holonomy G_{2} manifolds need not be formal,” Martín-Merchán makes an important intervention at the intersection of the pure mathematics fields of differential geometry and algebraic topology.

Broadly speaking, she explains, differential geometry is about “measuring lengths and angles on a shape.” In algebraic topology, meanwhile, "we imagine that shapes are made of plasticine, and we study the properties that don’t change when we bend, stretch or distort them. One of these properties is the number of holes.” In topology, famously, a donut is equivalent to a coffee mug, because they both consist of surfaces interrupted by a single hole.

Most people only learn about shapes in school up to three dimensions. A line (one dimension) is squared to make a square (two dimensions) and cubed to make a cube (three dimensions).

Pure mathematicians, however, work in many dimensions beyond the three dimensions we can see, continuing to square shapes again and again. These higher-dimensional shapes, however, cannot be easily visualized, and must be explored mathematically instead.

Martín-Merchán’s paper looks at a specific group of shapes called “G_{2} manifolds,” in the seventh dimension. Since 1996, researchers have theorized that these shapes must have a specific algebraic-topological property, called being “formal”, but Martín-Merchán has proved that this is not the case.

“This is something people have been interested in for a long time,” Martín-Merchán says. “I’ve shown that in this case there is no connection between these two concepts, and so further inquiry won’t be productive. I’ve disproved a long-standing conjecture.”

Martín-Merchán is particularly grateful to have been able to do this research while part of a dedicated department of Pure Mathematics, a rarity in academic spaces. “It’s great to have an adviser who works in the same area and field I work in, and to have other people in the research group who think about topics related to mine,” she says. “This always pushes my research a little farther, because if I was alone, I couldn’t confront or test my ideas.”

While research into higher dimensions may seem obscure, Martín-Merchán emphasizes the importance of pursuing fundamental research in mathematics. “If we only do research based on what we already know is useful, we will limit our discoveries,” she says. “You never know what in the future you will need.”

Still, she acknowledges, a large part of what motivates her inquiry is pure curiosity. “I like visualizing things and doing experiments with these theoretical objects,” she says. “It’s kind of like you’re exploring other planets, but through abstract thinking and mathematics.”