Maryna Viazovska – Migoro Media

Maryna Viazovska: The Mathematician Who Solved the Sphere Packing Puzzle

A brilliant mind that unlocked an age-old problem and redefined geometry in higher dimensions

Maryna Sergiivna Viazovska (born in Kyiv, 1984) is a Ukrainian mathematician celebrated for one of the most elegant breakthroughs in modern mathematics. In 2016, she proved that the $E_{8}$ lattice gives the densest possible sphere packing in eight dimensions — a result that had puzzled mathematicians for centuries. Her work did not stop there: with collaborators, she extended the method to solve the problem in 24 dimensions, another long-standing challenge.

Today, Viazovska serves as Full Professor and Chair of Number Theory at the prestigious École Polytechnique Fédérale de Lausanne (EPFL) in Switzerland. In 2022, her groundbreaking achievements were recognized with the Fields Medal, often described as the “Nobel Prize of Mathematics.”

Her career exemplifies how deep theoretical insights can not only solve age-old problems but also reshape the way we think about symmetry, geometry, and the hidden structures of higher-dimensional spaces.

🌱 Early Life and Formative Influences

Birth & Family Background

Maryna Sergiivna Viazovska was born in Kyiv, Ukraine, on 2 December 1984 (some early biographies incorrectly state 2 November 1984; the authoritative date is used by EPFL and other official sources). She grew up as the eldest of three sisters in a family that valued both scientific curiosity and resilience.

Her father, a chemist, nurtured her analytical mindset.
Her mother, an engineer, encouraged precision and perseverance.

This blend of science and engineering at home shaped a household atmosphere where problem-solving and creativity were part of daily life.

Schooling and Early Aptitude

Maryna attended the Kyiv Natural Science Lyceum No. 145, a school renowned for cultivating mathematically gifted students. Admission was highly competitive, and the Lyceum provided intensive training in mathematics, physics, and the natural sciences.

One of her most influential teachers, Andrii Knyazyuk, recognized her exceptional abilities early on and gave her both encouragement and intellectual challenges. This mentorship proved crucial in shaping her confidence and mathematical maturity.

Competitions and Early Recognition

During her school years, Viazovska actively participated in national and international mathematics competitions. These contests not only sharpened her problem-solving skills but also introduced her to the global mathematical community.

Her consistent success in competitions:

Strengthened her ability to think creatively under pressure.
Exposed her to advanced topics outside the school curriculum.
Built a network of peers equally passionate about mathematics.

By the time she graduated, it was evident that she was destined for advanced study and research in mathematics.

🎓 University Education & Doctoral Work

🏛️ Undergraduate Studies in Kyiv

Maryna Viazovska began her formal higher education at the Taras Shevchenko National University of Kyiv, Ukraine’s leading university. She pursued a Bachelor of Science (BSc) in Mathematics, graduating around 2005.

At Kyiv, she deepened her interest in number theory and modular forms, areas that would later define her research.
The university’s rigorous curriculum combined classical mathematics with exposure to international literature, laying a strong foundation for graduate studies abroad.

🌍 Master’s in Germany

Following her undergraduate success, Viazovska enrolled at the University of Kaiserslautern in Germany, completing a Master of Science (MSc) in Mathematics in 2007.

Kaiserslautern was known for its strong emphasis on pure mathematics and applied research, offering Viazovska opportunities to broaden her technical background.
She gained experience working in a multicultural academic environment, preparing her for the highly international nature of modern mathematics.

📖 Candidate-Level Dissertation in Ukraine

Even as she pursued opportunities abroad, Viazovska maintained strong ties with Ukraine. In 2010, she defended a Candidate of Sciences dissertation (the Ukrainian equivalent of a PhD) at the Institute of Mathematics of the National Academy of Sciences of Ukraine.

This achievement reflected her commitment to advancing Ukrainian mathematical scholarship.
The candidate-level work sharpened her expertise in analytic number theory and modular functions, themes central to her later breakthroughs.

🧮 Doctorate (Dr. rer. nat.) at the University of Bonn

Viazovska’s most pivotal academic step came at the University of Bonn, one of Europe’s premier centers for mathematics.

In 2013, she earned her Doctor rerum naturalium (Dr. rer. nat.), the German equivalent of a doctoral degree in natural sciences.
Her dissertation, titled Modular Functions and Special Cycles, was supervised by Don Zagier, a world-leading authority in number theory, and Werner Müller, an expert in analysis and automorphic forms.
The work explored deep connections between modular forms, geometry, and arithmetic cycles, positioning her at the intersection of analytic number theory and geometric analysis.

The dissertation is publicly available through the University of Bonn’s online repository, serving as a resource for future researchers.

🧑‍🔬 Academic Positions & Career Timeline

🔎 Postdoctoral Research Years

Following her doctorate in Bonn, Maryna Viazovska embarked on a series of prestigious postdoctoral fellowships and visiting research appointments, which gave her both independence and exposure to leading international mathematical communities.

Berlin Mathematical School & Humboldt University of Berlin: She worked closely with specialists in number theory and analysis, expanding her collaborative network in Germany.
Institut des Hautes Études Scientifiques (IHÉS), France: This institute, long associated with breakthroughs in modern mathematics, provided Viazovska with a stimulating environment to deepen her ideas.
Princeton University (Minerva Distinguished Visitor): In the U.S., she interacted with leading figures in number theory and geometry, further shaping her trajectory toward higher-dimensional sphere packing problems.

These years were crucial: they allowed her to refine the analytic techniques and modular form insights that would culminate in her 2016 breakthrough.

🏫 Appointment at EPFL (École Polytechnique Fédérale de Lausanne)

Viazovska joined the École Polytechnique Fédérale de Lausanne (EPFL) in Switzerland in 2017 as a tenure-track Assistant Professor of Mathematics. Her rapid rise reflected both her extraordinary research achievements and her potential as a leader in pure mathematics.

In January 2018, she was promoted to Full Professor and Chair of Number Theory, a position of distinction and responsibility at one of Europe’s top technical universities.
At EPFL, she leads a research group on number theory and modular forms, supervising graduate students and postdocs.
Her group’s focus spans sphere packing, modular and automorphic forms, optimization in high dimensions, and connections to physics and coding theory.

🌍 Ongoing Academic Leadership

Beyond her position at EPFL, Viazovska is active in the global mathematical community:

She frequently lectures at international conferences and research institutes.
She serves as a mentor to young mathematicians, particularly encouraging students from Ukraine and other underrepresented regions in mathematics.
Her dual role as a research pioneer and educator ensures that her impact extends across both discovery and academic training.

🔑 The 2016 Breakthrough: Sphere Packing in Dimension 8

❓ The Sphere Packing Problem

The sphere packing problem asks: How densely can equal, non-overlapping spheres fill Euclidean space?

In low dimensions:
- 1D: Line of intervals — trivial.
- 2D: Hexagonal (triangular) packing — proven optimal by Gauss (1831).
- 3D: Kepler’s conjecture (face-centered cubic packing) — proven only in 1998–2014 by Thomas Hales using the Flyspeck project.
In higher dimensions:
The problem becomes far more challenging, with no classical geometric intuition. For decades, mathematicians speculated about certain exceptional lattices (notably $E_{8}$ in 8D and the Leech lattice in 24D) as candidates for optimal solutions.

🌟 Viazovska’s Result (2016)

In March 2016, Maryna Viazovska stunned the mathematical world with a concise, elegant paper proving that in $R8\mathbb{R}^{8}$ , the $E_{8}$ lattice achieves the maximum possible packing density.

Impact: Solved a conjecture that had resisted proof for decades.
Publication: First released as a preprint on arXiv, later published in the Annals of Mathematics.
Density: About 0.25367 (≈ 25.367%), meaning roughly a quarter of 8-dimensional space can be filled with non-overlapping spheres in the $E_{8}$ arrangement.

🧩 The Method: The “Magic Function”

What made her proof extraordinary was not only the result, but the unexpectedly simple structure:

She constructed an explicit radial Schwartz function (nicknamed the “magic function”).
This function perfectly matched the Cohn–Elkies linear programming bound, a method previously known to give only approximate answers.
The function exploited deep connections between:
- Modular and quasimodular forms
- Harmonic analysis
- Poisson summation

This combination allowed her to bridge abstract analytic number theory with concrete geometric packing.

✨ Why It Was Revolutionary

The proof was astonishingly short compared to the difficulty of the problem — about 23 pages.
It was conceptually elegant, avoiding heavy computational verification.
It showcased the power of modular form techniques in solving a longstanding geometric optimization problem.
Within weeks, her methods inspired extensions to 24 dimensions, confirming the Leech lattice as the densest packing in $R24\mathbb{R}^{24}$ .

Mathematicians hailed it as one of the most beautiful results of the 21st century, instantly placing Viazovska among the most influential researchers of her generation.

🌌 Extending the Idea: Dimension 24 and Universal Optimality

The 24-Dimensional Leech Lattice

In the wake of Maryna Viazovska’s 2016 proof in dimension 8, a natural question arose: Could the same techniques crack the long-standing conjecture in 24 dimensions?

The Leech lattice, a highly symmetric and exceptional structure in $R24\mathbb{R}^{24}$ , had long been suspected to give the densest possible packing.
Within weeks of Viazovska’s breakthrough, a team consisting of Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Viazovska extended her approach.
Their 2017 paper in the Annals of Mathematics constructed an analogue of the “magic” auxiliary function tailored to 24 dimensions.
The result: a rigorous proof that the Leech lattice is the densest sphere packing in $R24\mathbb{R}^{24}$ .

This achievement confirmed that both 8 and 24 dimensions — long considered “special” in mathematics due to their exceptional symmetries — indeed admit uniquely optimal packings.

Beyond Packing: Universal Optimality

The story did not end with densest packings. The same research group later pushed the methods much further, leading to a new landmark:

In 2019 (arXiv preprint, later Annals publication), Cohn, Kumar, Miller, Radchenko, and Viazovska proved that the $E_{8}$ lattice and the Leech lattice are universally optimal configurations.
Universal optimality means:
- These lattices minimize potential energy for an extremely broad class of radial interaction functions (not just packing density).
- In physical terms, if particles interact under almost any reasonable pair potential, they naturally “prefer” to arrange themselves in $E_{8}$ (8D) or the Leech lattice (24D).
The proofs again relied on Fourier interpolation formulas and modular form techniques, extending the analytic framework Viazovska pioneered.

Significance

Mathematical beauty: These results connected areas as diverse as geometry, optimization, harmonic analysis, and mathematical physics.
Physical relevance: Universal optimality links lattice theory to real-world models of matter and energy minimization.
Legacy: The extension from 8D to 24D, and then to universal optimality, showed how a single deep idea could resolve problems that had resisted proof for centuries.

🧩 Other Major Mathematical Contributions

🔵 Spherical Designs (2013)

In collaboration with Andriy Bondarenko and Danylo Radchenko, Viazovska solved a long-standing problem on spherical $t$ -designs.

Result: They proved optimal asymptotic bounds for spherical designs, confirming a conjecture by Korevaar and Meyers.
What it means:
- A spherical $t$ -design is a set of points on a sphere that allows exact integration of all polynomials up to degree $t$ .
- These designs are crucial in approximation theory, quadrature formulas, and discrete geometry.
Publication: Annals of Mathematics, 2013.
Impact: The result became a foundational theorem in the analysis of spherical approximation, influencing fields ranging from numerical analysis to coding theory.

🔧 Interpolation Techniques and Energy Minimization

Viazovska’s analytic innovations extend well beyond the 2016–2019 packing results:

She and collaborators developed a Fourier–interpolation framework that underpins both the 8D and 24D packing proofs.
These methods also yield:
- Energy-minimization theorems, showing optimality of special lattices for a wide variety of potential functions.
- Interpolation bases for radial Schwartz functions, technical but powerful tools in harmonic analysis and discrete optimization.
These contributions demonstrate her ability to transform abstract analytic techniques into concrete geometric insights.

📚 Selected Canonical Research Papers

Maryna Viazovska’s most influential works include:

Viazovska, M. S. — The sphere packing problem in dimension 8. Annals of Mathematics (2017).
Cohn, H.; Kumar, A.; Miller, S. D.; Radchenko, D.; Viazovska, M. — The sphere packing problem in dimension 24. Annals of Mathematics (2017).
Bondarenko, A.; Radchenko, D.; Viazovska, M. — Optimal asymptotic bounds for spherical designs. Annals of Mathematics (2013).
Cohn, H.; Kumar, A.; Miller, S. D.; Radchenko, D.; Viazovska, M. — Universal optimality of the $E_{8}$ and Leech lattices and interpolation formulas. (arXiv 2019; Annals publication).

✨ Together, these results place Viazovska at the center of modern discrete geometry and analytic number theory, showing a rare combination of technical power and conceptual clarity.

🏆 Awards, Honors & Recognition

Maryna Viazovska’s groundbreaking contributions have been acknowledged with some of the highest international distinctions in mathematics. Her awards highlight not only the depth of her research but also her role as a trailblazer in the global mathematical community.

📐 Early Recognition

2016 – Salem Prize: Awarded for outstanding contributions to analysis, particularly her proof of the sphere packing problem in 8 dimensions.

🔬 Major Breakthrough Prizes

2017 – Clay Research Award: Recognized for the creativity and elegance of her proof in dimension 8.
2017 – SASTRA Ramanujan Prize: Given to young mathematicians under 32, honoring her innovative use of modular forms.

🌟 Prestigious Young Researcher Distinctions

2018 – New Horizons in Mathematics Prize: Part of the Breakthrough Prizes, celebrating early-career achievements of exceptional promise.

📊 Broad Professional Recognition

2019 – Ruth Lyttle Satter Prize (AMS): For outstanding contributions to mathematics research by a woman.
2019 – Fermat Prize: Honoring contributions in fields inspired by Fermat, including number theory and geometry.
2020 – EMS Prize (European Mathematical Society): Awarded every four years to the best young European mathematicians.
2020 – National Latsis Prize (Switzerland): For excellence in scientific research, across disciplines.
2021 – Election to Academia Europaea: Recognition by one of Europe’s most distinguished academic bodies.

🧑‍🏫 Senior Leadership Roles

2022 – Senior Scholar, Clay Mathematics Institute: Supporting her continued high-level research.

🥇 Fields Medal (2022)

Awarded at the International Congress of Mathematicians, one of the highest honors in mathematics.
Viazovska became:
- The second woman in history to receive the Fields Medal (after Maryam Mirzakhani in 2014).
- The first laureate with a degree from a Ukrainian university, a point of pride for Ukraine’s mathematical tradition.
The citation praised her proof of the sphere packing problem in 8 dimensions and her development of new analytic methods with far-reaching consequences.

✨ Collectively, these honors place Maryna Viazovska among the most decorated mathematicians of her generation, with recognition spanning analysis, geometry, and number theory.

🌍 Personal Life, Outreach & Impact Beyond Mathematics

👨‍👩‍👧‍👦 Personal Details

Maryna Viazovska met her husband Daniil Evtushinsky during a high school physics group in Kyiv.
Evtushinsky is now a researcher at EPFL in condensed matter physics.
Together, they have two children, balancing family life with internationally acclaimed research careers.

📢 Public Engagement & Context

Viazovska’s achievements brought unprecedented global visibility to Ukrainian mathematics, especially significant during the geopolitical challenges of the 2020s.
She has been featured in interviews, EPFL media coverage, and international science press, becoming a public voice for both mathematics and Ukrainian science.
Her story resonates with students worldwide:
- The sphere-packing problem looks simple (“How can you stack spheres most efficiently?”).
- Yet its solution demanded some of the deepest tools of modern mathematics (modular forms, Fourier analysis, harmonic methods).
This duality makes her work a powerful example of how elementary curiosity can lead to profound discoveries.

🔗 Legacy & Broader Applications

While Viazovska’s 8D and 24D sphere-packing proofs are pure mathematics, the lattices she studied have deep real-world connections:

Coding Theory: The $E_{8}$ and Leech lattices underpin some of the most efficient error-correcting codes, essential in data transmission and digital communications.
Theoretical Physics: The same lattices appear in string theory and conformal field theory, where their exceptional symmetries play a central role.
Conceptual Advances: Her proof techniques explain why these lattices are exceptional — by making the Cohn–Elkies linear programming bound sharp through explicit auxiliary functions.

Her legacy lies not just in solving a long-standing problem, but in opening new methods and perspectives for geometry, analysis, and mathematical physics.

📚 Sources & Further Reading

📝 Selected Primary Sources (Viazovska’s Papers)

Viazovska, M. S. The sphere packing problem in dimension 8. Annals of Mathematics, 185(3), 991–1015 (2017).
Cohn, H.; Kumar, A.; Miller, S. D.; Radchenko, D.; Viazovska, M. The sphere packing problem in dimension 24. Annals of Mathematics, 185(3), 1017–1033 (2017).
Bondarenko, A.; Radchenko, D.; Viazovska, M. Optimal asymptotic bounds for spherical designs. Annals of Mathematics, 178(2), 443–452 (2013).
Cohn, H.; Kumar, A.; Miller, S. D.; Radchenko, D.; Viazovska, M. Universal optimality of the $E_{8}$ and Leech lattices and interpolation formulas. Annals of Mathematics (2022).

📖 Secondary & Authoritative References

International Mathematical Union (IMU): Fields Medal 2022 citation and official profile of Maryna Viazovska.
EPFL (École Polytechnique Fédérale de Lausanne): Faculty page, interviews, and institutional coverage of her research and awards.
Clay Mathematics Institute: Research profiles and press releases connected to the Clay Research Award and her role as Senior Scholar.
Academia Europaea: Membership records and biographical entry.
Wikipedia (curated and cited): “Maryna Viazovska” — for consolidated biographical details and award listings, with references to original sources.
Feature articles in science media:
- Quanta Magazine: Explainers on sphere packing and universal optimality.
- WIRED: Profiles connecting Viazovska’s work to applications in coding theory and physics.
- Nature News and Science: Coverage of the Fields Medal 2022.

✨ These references provide a complete learning pathway:

Primary papers for technical readers.
IMU, EPFL, Clay Institute for authoritative recognition.
Quanta, WIRED, Nature for accessible science communication.

📚 Publications & Writings

📖 Foundational Works

Éléments de Géométrie Algébrique (EGA): Co-authored with Jean Dieudonné, EGA systematically rebuilt algebraic geometry using the scheme framework.
Séminaire de Géométrie Algébrique (SGA): Collected seminar notes from IHÉS, spanning thousands of pages, introducing étale cohomology, topoi, and more.

✍️ Independent Works

Tôhoku Paper (1957): Laid the groundwork for abelian categories and modern homological algebra.
Esquisse d’un Programme (1984): Written for a research fellowship application, it sketched new directions like anabelian geometry and dessins d’enfants.
Pursuing Stacks (1983–84): Thousands of handwritten pages exploring higher category theory.
Récoltes et Semailles (1983–86): A monumental autobiographical and philosophical manuscript—part memoir, part critique of the mathematical establishment.

🏅 Honors & Recognition

Fields Medal (1966): Awarded for his transformation of algebraic geometry; he declined to travel to Moscow in protest of Soviet policies.
Crafoord Prize (1988): Jointly awarded with Pierre Deligne. Grothendieck declined, criticizing the pursuit of honors in science.
Posthumous Influence: His name lives on in countless concepts—Grothendieck groups, topologies, universes, duality, and the Grothendieck–Riemann–Roch theorem.

👉 His writings remain a living library for mathematicians, a source of both technical tools and philosophical reflection.

❓ Frequently Asked Questions (FAQs)

🔹 Who is Maryna Viazovska?

Maryna Viazovska is a Ukrainian mathematician, born in 1984 in Kyiv. She is best known for solving the sphere packing problem in 8 dimensions and for her contributions to discrete geometry, number theory, and harmonic analysis. Since 2018, she has been Full Professor and Chair of Number Theory at EPFL (Switzerland).

🔹 What is the sphere packing problem, and what did she solve?

The sphere packing problem asks: How can spheres be packed together most densely without overlaps?

In 2D (circles) and 3D (ordinary spheres), the densest packings were known.
For higher dimensions, the problem was open.
In 2016, Viazovska proved that in 8 dimensions, the $E_{8}$ lattice gives the optimal packing.
In 2017, together with collaborators, she extended this to 24 dimensions, proving that the Leech lattice is optimal.

🔹 Why was her proof considered groundbreaking?

Her proof was short, elegant, and conceptually deep compared to decades of failed attempts.

She used modular forms, Fourier analysis, and interpolation methods to construct a “magic function” that made the density bound sharp.
Mathematicians praised it as one of the most beautiful results of the 21st century.

🔹 What are the $E_{8}$ and Leech lattices, and why are they important?

$E_{8}$ lattice: A highly symmetric structure in 8D space, already famous in mathematics and physics.
Leech lattice: A 24D lattice with extraordinary symmetries, connected to error-correcting codes and string theory.
Both are examples of exceptional structures that appear in multiple areas of mathematics, coding, and physics.

🔹 What awards has Maryna Viazovska received?

She has received many top prizes, including:

Salem Prize (2016)
Clay Research Award (2017)
SASTRA Ramanujan Prize (2017)
New Horizons in Mathematics Prize (2018)
Fermat Prize and Ruth Lyttle Satter Prize (2019)
EMS Prize and National Latsis Prize (2020)
Fields Medal (2022) — the most prestigious award in mathematics.
She is the second woman in history to receive a Fields Medal (after Maryam Mirzakhani in 2014).

🔹 What is her connection to Ukraine?

Viazovska was born and educated in Kyiv, where she completed her first degree.
She is the first Fields Medalist with a degree from a Ukrainian university.
During the 2020s, her recognition brought worldwide attention to Ukrainian mathematics, especially amidst the country’s difficult geopolitical situation.

🔹 How does her work affect science outside mathematics?

While her proofs are pure mathematics, the lattices she studied appear in:

Digital communications — through error-correcting codes.
Theoretical physics — especially in string theory and conformal field theory.
Her work deepens our understanding of why these lattices are optimal and exceptional.

🔹 Does Maryna Viazovska have a family?

Yes. She met her husband, Daniil Evtushinsky, in a school physics group; he is now a researcher at EPFL. They have two children.

🔹 Where can students learn more about her work?

Primary sources: Viazovska’s Annals of Mathematics papers on sphere packing and spherical designs.
Accessible introductions: Articles in Quanta Magazine, Wired, and Nature.
Talks & lectures: EPFL hosts recorded lectures and interviews with her.
IMU & Fields Medal citations: Authoritative summaries of her contributions.