It usually starts with her black notebook, where Maryna Viazovska records the results of her thoughts, a list of complex formulas. Maryna Viazovska is a mathematician and professor at the EPFL Chair of Number Theory. Her passion is the science of arithmetic, numbers and geometric figures. She successfully participated in school mathematics competitions in her Ukrainian homeland at the age of 12. Some 25 years later in 2022, the 37-year-old received the prestigious Fields Medal, known officially as the “International Medal for Outstanding Discoveries in Mathematics”. This award, which gets its name from its Canadian benefactor, John Charles Fields (1864–1932), has been presented every four years since 1936 and is considered the “Nobel Prize of Mathematics”. Among the 60 researchers to receive the award, Viazovska was the second woman to be honoured. For the “proof that the E8 lattice provides the densest packing of identical spheres in eight dimensions and other contributions to related extremal problems”, according to the statement of the International Mathematical Union, which confers the prize.
The work sounds complicated, and it is. The fundamental question is how many identical, non-overlapping spheres can be packed into a given volume. Researchers have concerned themselves with this problem for some time. For example, German mathematician Johannes Kepler postulated in 1611 that the best way to solve this problem in three-dimensional space was by arranging the spheres into a pyramid. However, this conjecture was unproven, and it took until 1998 for a proof to be found. This question was then settled, at least in three-dimensional space. However, in mathematics, countless dimensions exist in which the same question can be posed. “What we are talking about here is pure abstraction; while a three-dimensional space has three coordinates, a point in eight-dimensional space has eight coordinates,” Viazovska says.
This is the area in which she initially conducted her research. “The question was how to optimally fill the abstract space with spheres up to the upper limit,” the researcher explains. Dimension 8 was a good choice because preliminary work had already been done in this area. For example, Massachusetts Institute of Technology (MIT) mathematician Henry Cohn and Harvard scientist Noam Elkies demonstrated more than 10 years ago that it is possible to pack the spheres almost perfectly in this dimension – perfectly within a billionth of a percent. Viazovska developed this work further by combining various fields of mathematics and also by using “modular forms” – the primary topic of her dissertation.
After two years of intensive work, she developed the definitive function and provided 23 pages of proof for the perfect packing of the spheres in eight-dimensional space in March 2016. The mathematical experts were impressed, and MIT researcher Cohn congratulated and motivated her to extend her method to dimension 24, into the “Leech lattice”, which is used to construct sphere packings that are particularly efficient in 24-dimensional space. It was a week later when Viazovska, together with Cohn and two other scientists, posted a theorem on the open-source platform arXiv.org proving the perfection of the Leech lattice packing in dimension 24, thus also confirming the importance of the idea that Viazovska originally developed for dimension 8.
EPFL took notice of the gifted young researcher and invited her to serve as an associate professor on Lake Geneva. She later published her research results in the Annals of Mathematics, a leading mathematics journal, and has been a full professor of number theory at EPFL since 2018. It’s no coincidence that she chose the university. In addition to the EPFL’s reputation, the fact that the institution helped in the establishment of a dual career was crucial. With this assistance, it was also possible for her husband, who is likewise a natural scientist, to find employment there so the family of four would be able to live and do research in one location.
Mathematics is everywhere
“I am doing basic research,” Viazovska says, “but certainly with the goal and hope in mind that these interpolation formulas can be applied in order to solve problems in areas such as differential equations or signal processing.” After all, mathematics is everywhere in everyday life, usually without people noticing. “Without mathematics, there is no Skype,” she says, “and also no optimised timetable for the Swiss train system.” However, what counts for Viazovska is “the magic of 8 and 24”. This was also the title of her acceptance speech given at the Fields Medal ceremony.
Mara nyingi, yeye huanza kwa kutumia kitabu chake cha maandishi, ambapo Maryna Vyacheslavska huandika matokeo ya mawazo yake, orodha ya kanuni tata. Maryna Viazovska ni mwanahisabati na profesa katika Katibu wa EPFL wa Nadharia ya Nambari. Shauku yake ni sayansi ya hesabu, namba na takwimu za kijiometri. Alifanikiwa kushiriki katika mashindano ya hisabati ya shule katika nchi yake ya Ukraine akiwa na umri wa miaka 12. Mwaka 2022 alipewa tuzo ya Fields Medal, ambayo ni tuzo ya kimataifa kwa uvumbuzi bora katika hisabati. Tuzo hiyo, ambayo ilipewa jina kutokana na mfadhili wake wa Canada, John Charles Fields (1864-1932), imekuwa ikitolewa kila baada ya miaka minne tangu 1936 na inachukuliwa kuwa "Tuzo ya Nobel ya Hisabati". Miongoni mwa watafiti 60 waliopokea tuzo hiyo, Viazovska alikuwa mwanamke wa pili kuheshimiwa. "Kwa ""uthibitisho kwamba E8 lattice hutoa pakiti densest ya spheres sawa katika vipimo nane na michango mingine kwa matatizo yanayohusiana extreme"" (kwa mujibu wa taarifa ya Umoja wa Kimataifa Mathematical), ambayo inatoa tuzo." Kazi hiyo inaonekana kuwa ngumu, na ni ngumu. Swali la msingi ni ni ngapi sawa, non-overlapping spheres inaweza kuwa packed katika kiasi fulani. Watafiti wamekuwa wakijishughulisha na tatizo hilo kwa muda fulani. Mwanahisabati wa Ujerumani Johannes Kepler aliwahi kusema kwamba njia bora ya kutatua tatizo hili katika nafasi ya pande tatu ni kwa kupanga mviringo katika umbo la piramidi. Hata hivyo, dhana hiyo haikuthibitishwa, na ilichukua hadi mwaka wa 1998 ili kuthibitishwa. Swali hilo lilitatuliwa, angalau katika nafasi ya pande tatu. Hata hivyo, katika hisabati, kuna vipimo vingi ambavyo swali lilelile laweza kuulizwa. "Kile tunachozungumza hapa ni abstract safi; wakati nafasi ya pande tatu ina kuratibu tatu, hatua katika nafasi ya pande nane ina kuratibu nane", Viazovska anasema. Hii ni eneo ambalo mwanzoni alifanya utafiti wake. "Swali lilikuwa ni jinsi ya kujaza nafasi abstract na spheres hadi kikomo cha juu", mtafiti anaelezea. "Dimensions 8 ilikuwa chaguo nzuri kwa sababu kazi ya awali ilikuwa tayari imefanywa katika eneo hili. """ Kwa mfano, mwanahisabati Henry Cohn wa Taasisi ya Teknolojia ya Massachusetts (MIT) na mwanasayansi Noam Elkies wa Chuo Kikuu cha Harvard walionyesha zaidi ya miaka kumi iliyopita kwamba inawezekana kufunga mviringo karibu kikamilifu katika mwelekeo huu - kikamilifu ndani ya bilioni moja ya asilimia. Viazovska aliendeleza kazi hii zaidi kwa kuchanganya nyanja mbalimbali za hisabati na pia kwa kutumia "maumbo ya modular" mada ya msingi ya dissertation yake. Baada ya miaka miwili ya kazi ya kina, aliendeleza kazi ya mwisho na kutoa kurasa 23 za uthibitisho wa kufunga kamili ya mviringo katika nafasi ya nane-dimensional mnamo Machi 2016. "Mtaalamu wa MIT Cohn alimpongeza na kumtia moyo kupanua mbinu yake hadi mwelekeo wa 24, katika ""Lich Lattice,"" ambayo hutumiwa kujenga vifurushi vya mviringo ambavyo ni bora sana katika nafasi ya 24-dimensional." Vyacheslav Vyacheslav, pamoja na Cohn na wanasayansi wengine wawili, walichapisha theorem kwenye jukwaa la chanzo wazi la arXiv.org, ikithibitisha ukamilifu wa pakiti ya Leech lattice katika mwelekeo wa 24, na hivyo pia kuthibitisha umuhimu wa wazo ambalo Vyacheslav Vyacheslav awali aliendeleza kwa mwelekeo wa 8. EPFL iliona mtafiti huyo mchanga mwenye kipawa na kumwalika kutumikia kama profesa msaidizi kwenye Ziwa Geneva. Alitoa utafiti wake katika Annals of Mathematics, jarida la kuongoza la hisabati, na amekuwa profesa wa nadharia ya idadi katika EPFL tangu 2018. Si bahati mbaya kwamba alichagua chuo kikuu hicho. Mbali na sifa ya EPFL, ukweli kwamba taasisi alisaidia katika kuanzisha kazi mbili ilikuwa muhimu. Kwa msaada huo, mume wake, ambaye pia ni mwanasayansi wa asili, aliweza kupata kazi huko ili familia ya watu wanne iweze kuishi na kufanya utafiti katika eneo moja. "Viazovska anasema, ""Ninafanya utafiti wa msingi, lakini hakika na lengo na matumaini katika akili kwamba hizi interpolation formula zinaweza kutumika ili kutatua matatizo katika maeneo kama vile differential equations au signal processing." "Bila hisabati, hakuna Skype, "anasema, "na pia hakuna ratiba iliyoboreshwa kwa mfumo wa treni ya Uswisi". "Hata hivyo, kile kinachohesabiwa kwa Viazovska ni "chawi ya 8 na 24". Hii pia ilikuwa kichwa cha hotuba yake ya kukubali iliyotolewa katika sherehe ya medali ya Fields.