Maryna Viazovska: The Magic Function That Packed Eight Dimensions
Within hours of Maryna Viazovska posting a short paper online in March 2016, the mathematician Akshay Venkatesh emailed a colleague with a one-word subject line: "Wow!" The paper solved, in a proof running to a handful of pages, a problem about stacking spheres in eight-dimensional space that had resisted the field for decades — and it did so, according to Princeton's Peter Sarnak, in a way so elegant that his paper was "one of those papers you pick up, and you don't put down before you've read it." Six years later Viazovska became only the second woman in history to win the Fields Medal.
A Chemist's Daughter in Post-Soviet Kyiv
Maryna Sergiivna Viazovska was born December 2, 1984, in Kyiv, the eldest of three sisters, to a father who worked as a chemist at the Antonov aircraft factory and a mother who was an engineer. She grew up amid the economic turmoil of post-Soviet Ukraine and attended Kyiv Natural Science Lyceum No. 145, a specialized school for scientifically gifted students, where a former research mathematician turned teacher, Andrii Knyazyuk, shaped her early mathematical development. She competed seriously in olympiads — placing 13th nationally in a competition selecting twelve students for International Mathematical Olympiad training, narrowly missing the cut — and as an undergraduate at Taras Shevchenko National University of Kyiv she won the International Mathematics Competition for University Students outright in both 2002 and 2005, co-authoring her first research paper that same year.
From Kyiv to Bonn
Her formal training crossed several countries: a master's degree from the University of Kaiserslautern in Germany (2007), a PhD from the Institute of Mathematics of the National Academy of Sciences of Ukraine (2010), and a doctorate (Dr. rer. nat.) from the University of Bonn in 2013, where she studied under Don Zagier and Werner Müller on modular functions and special cycles in analytic number theory. Before her signature breakthrough, she had already established a research reputation: a 2011 paper with Andriy Bondarenko and Danylo Radchenko, proving a conjecture of Korevaar and Meyers about small spherical designs in arbitrary dimensions, was published in the Annals of Mathematics — one of the field's most selective journals — years before she became widely known outside number theory circles.
The Sphere-Packing Problem
The question Viazovska set out to answer has an old and famous history: how can identical spheres be packed into space to fill the largest possible fraction of volume? In three dimensions, this is the Kepler conjecture, finally proven only in the late 1990s and 2000s through an exhaustive, computer-assisted argument. In certain higher dimensions the problem takes on unexpected structure because of special, highly symmetric lattices — most notably the E₈ lattice in eight dimensions and the Leech lattice in twenty-four. Mathematicians had long suspected these lattices gave the densest possible sphere packings in their respective dimensions, but no one had a way to prove it.
The "Magic Function"
In 2016, working largely alone after postdoctoral positions at the Berlin Mathematical School, Humboldt University, and a Minerva fellowship at Princeton, Viazovska constructed what became known as a "magic function" for dimension 8 — built not from the modular forms most number theorists expected such a proof to require, but from a quasimodular form with deliberately imperfect symmetries. The function let her prove, in a remarkably short and clean argument by the standards of the field, that the E₈ lattice does indeed give the densest possible sphere packing in eight dimensions. Within a week, working with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko, she extended the same technique to solve the 24-dimensional case using the Leech lattice — a burst of collaborative work Radchenko later called "probably the craziest week of my life." The two results, released within days of each other, settled sphere packing in the only two dimensions beyond three where an exact answer was then known.
Beyond Packing: Universal Optimality
Viazovska and her collaborators pushed further in subsequent years, proving in 2019 that the E₈ and Leech lattices are "universally optimal" — meaning they minimize potential energy not just for one physical model but across an entire broad class of energy functions, with implications reaching from electron repulsion to polymer configurations. The mathematician Sylvia Serfaty compared the result to "great breakthroughs of the 19th century," calling it "a great advancement of science." The work also advanced Fourier analysis more broadly, through Viazovska's approach to reconstructing functions from partial spectral information — a technique with applications well beyond the packing problem that inspired it.
The Fields Medal and Recognition
In July 2022 Viazovska received the Fields Medal, becoming only the second woman to win it after Maryam Mirzakhani, the second mathematician born in the Ukrainian SSR after Vladimir Drinfeld, and the first Fields Medalist to hold a degree from a Ukrainian university. The honor followed a run of major prizes: the Salem Prize (2016), the Clay Research Award (2017), the SASTRA Ramanujan Prize (2017), the European Prize in Combinatorics (2017), the New Horizons Prize (2018), the Ruth Lyttle Satter Prize (2019), the Fermat Prize (2019), the EMS Prize (2020), and the National Latsis Prize (2020). She was named to the BBC's 100 Women list later in 2022. Since January 2018 she has held the Chair of Number Theory at EPFL in Lausanne, Switzerland, where she lives with her husband, physicist Daniil Evtushinsky — whom she met through an after-school physics group as a teenager — and their two children.
Why Maryna Viazovska Is Called a Genius
The specific mathematical quality behind the "genius" label is Viazovska's construction of an exact, closed-form proof where the field expected — and had previously needed, in the three-dimensional Kepler case — an exhausting computer-assisted argument spanning hundreds of pages. Finding the precise quasimodular form that made the eight-dimensional proof work required an unusual and hard-to-teach combination of deep technical command of modular forms and an almost aesthetic instinct for which function would satisfy the necessary interpolation conditions; that instinct is exactly what mathematicians who reviewed the paper, from Sarnak to Venkatesh, reacted to with open astonishment rather than routine professional approval. The honest limit on the framing is that the sphere-packing problem in eight and twenty-four dimensions was tractable precisely because those two dimensions possess exceptional lattice structures (E₈ and the Leech lattice) not available in general dimensions — the achievement does not generalize to arbitrary dimensions, and the field's broader sphere-packing problem remains open almost everywhere else. The 24-dimensional extension, moreover, was explicitly a rapid group collaboration with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko rather than solo work, and even the celebrated 8-dimensional proof drew on an existing linear-programming approach to sphere-packing bounds that other researchers, Cohn among them, had already developed before Viazovska found the specific function needed to complete it.
Legacy
Viazovska's methods have already become a standard reference point in the study of packing, energy minimization, and Fourier interpolation problems, and her Fields Medal — awarded amid the first months of Russia's full-scale invasion of Ukraine — carried particular symbolic weight for Ukrainian science, a country whose mathematical institutions she has continued to publicly champion since. At EPFL she continues to train the next generation of number theorists working at the boundary of lattice geometry and analytic number theory that her own career helped define.
