Young Stars of Mirror Symmetry Come Together in Moscow
Mirror symmetry is a relatively new field of mathematics which came into being in the 1990s. In 2017, HSE opened the International Laboratory for Mirror Symmetry and Automorphic Forms. The December conference, ‘Mirror Symmetry and Applications’, was a commemoration of its first year of operation.
What is mirror symmetry?
Mirror symmetry was discovered as a foundation for elementary particle theory models by physicists in the 1990s, in the form of a duality between superconformal field theories. The study of three-dimensional Calabi–Yau manifold geometry plays an important role in mirror symmetry. Calabi–Yau manifolds are split into dual (mirror) pairs. Characteristics of the first manifold correlate with completely different characteristics of its mirror partner, and vice versa.
Maxim Kontsevich, a renowned French mathematician hailing from Moscow, Professor at the Institut des Hautes Études Scientifiques in Paris and winner of the Fields Medal, reinvented the concept of modern theoretical physics as an incredibly deep and comprehensive mathematical duality. It is known today as homological mirror symmetry. He is responsible for radically changing the mathematical concept of space itself, and drew the attention of the entire mathematical community to this topic. This has enabled the synergy of various mathematical fields, such as symplectic geometry, algebraic geometry, and theory of categories. Today, homological mirror symmetry is the cornerstone of a vast field of mathematical research related to the use of constantly developing high-energy physics technologies. ‘Once supersymmetry can be proven, all these structures will have a physical meaning’, says Ludmil Katzarkov, American-Bulgarian mathematician and academic supervisor at the HSE Laboratory of Mirror Symmetry. ‘However, it hasn’t been proven yet. Today, mirror symmetry solves theoretical problems. I believe that, in 40 years, or maybe 70, we’ll know more about it. After all, it took almost a century before researchers discovered the gravitational waves described by Einstein’.
The Laboratory of Mirror Symmetry has existed as part of the HSE Faculty of Mathematics for almost a year. In 2016, Katzarkov was awarded a ‘megagrant’ from the Russian government for its creation. Thanks to his efforts, within just one year, the laboratory has become a leading international center for the study of mirror symmetry- a highly relevant field both in theoretical mathematics and theoretical physics. ‘Mathematicians began applying mirror symmetry as a tool to solve problems in enumerative geometry. This is a language’, explains Katzarkov. He adds, ‘Mirror symmetry offers three languages at once: symplectic, algebraic-geometrical, and numerical, which enables us to examine phenomena in algebraic geometry, for example, from different perspectives’.
‘When the Laboratory of Algebraic Geometry was created at HSE in 2010, we began organizing conferences and inviting interesting people’, says Fedor Bogomolov, Academic Supervisor at theLaboratory of Algebraic Geometry and its Applications and Professor at the Courant Institute of Mathematical Sciences in New York. ‘Ludmil Katzarkov took part in our seminars and conferences, and we wrote some joint papers with him. And when we were thinking about who could apply for the megagrant, it was obvious that he could. Ludmil is one of the world’s leading mathematicians in the field of mirror symmetry, and we are happy that this project has been a success. The conference has attracted researchers from all over the globe, many of them very young- and they are the very people who will define this science in future. We’ll be hearing about them soon. 2017 has been a successful year in my opinion, and we have great hopes for this new laboratory’.
Professor Katzarkovhas written his latest working paper with Maxim Kontsevich, a mathematician who has played a significant role in the development of mirror symmetry. A French researcher, originally from Moscow, winner of the Fields Medal and two Milner Prizes in mathematics and physics, Kontsevich is one of the world’s leading academics. Furthermore, he has agreed to become an associate member of the HSE Laboratory of Mirror Symmetry. Mirror symmetry is an essential tool, which, in addition to string theory, can be used to understand some aspects of quantum field theory and mathematical formalism. They are used by physicists to explain the distribution and interaction of elementary particles. The December conference focused on these very issues.
The aim of the December conference, ‘Mirror Symmetry and Applications’, was to summarize international research in mirror symmetry and outline future prospects for the field’s development. The conference covered a large geographical area, from Kyoto to Berkeley, from Harvard to London to Moscow. It included a mini-course run by Takuro Mochizuki, an outstanding Japanese mathematician, and 34 one-hour presentations by scholars from ten different countries. Almost all of the participants were very young, with an average age of 35.
‘Every conference gives us a push forward. It is mostly at conferences where I meet my colleagues from other countries, where I can talk to them and learn about what they do. We are not competitors, rather, we complement each other. Together, we can cover a bigger research field’, explained Tony Yue Yu, a young researcher from Paris-Sud University.
The HSE Laboratory of Mirror Symmetry is a symbiosis of mathematical schools located across the world, including Moscow, St. Petersburg and the US. ‘The Russian approach to research, with its specific problem-setting, is somewhat similar to the Japanese approach’, says Takuro Mochizuki, a leading Japanese mathematician who very rarely agrees to speak at conferences. Mochizuki was awarded a Japanese Academy Prize in 2011 for his research into D-modules in algebraic analysis and in 2014, he was a plenary speaker at the International Congress of Mathematicians. At the December conference, Mochizuki delivered a mini-course on ‘Asymptotic behavior of families of harmonic bundles’. Many young researchers travelled to Moscow specially to meet him.
Anthony Blanc, from the International School for Advanced Studies (SISSA) in Trieste, says: ‘Ludmil Katzarkov invited me to participate in the conference. He helped me a lot with my research in Vienna and I’m very glad I met him. The approach of the Russian school differs from ours, as we focus more on theory and theoretical conclusions. However, this greatly benefited our collaboration- our joint work is more exciting because of our different approaches.’
‘I’m happy to be here today’, says Alexey Bondal, one of the founders of derived algebraic geometry, research fellow at the HSE Laboratory for Mirror Symmetry, and researcher at theKavli Institute for the Physics and Mathematics of the Universe in Japan. ‘I see the Laboratory for Mirror Symmetry as a logical step in the development of the Moscow school, since Ludmil graduated in this city. And he remains very much a part of it, in spirit.’
Unlike physicists, chemists, and biologists, mathematicians don’t rely on complicated expensive equipment. All of our high-profile professional mathematicians are a kind of independent research laboratory. They might have an idea, a brainwave, at any moment, and so the main tools are always at hand: a pen, a sheet of paper, and a laptop. Many of them start working before they even begin their PhD, the tasks gradually increasing in complexity. The December conference attracted this very kind of researcher – those who, despite their youth, have been involved in research for quite a long time.
‘Ludmil Katzarkov invited me to the laboratory when he was only applying for the grant’, says Alexander Efimov, a talented young mathematician from Moscow. ‘We met rather a long time ago, in 2008, when I was a university student, and we have similar research interests. The laboratory’s conferences and the triweekly research seminars facilitate interaction between mathematicians. I’ve already picked up some interesting ideas at this conference. Research is a continuous process’. Katzarkov’s students also spoke at the Moscow conference. ‘I was lucky to meet Katzarkov when I was a student’, recalls George Dimitrov. ‘I completed my PhD in Vienna under his supervision. Working with such a high-profile scholar is a great opportunity and my presentation at the conference presents the results of our joint work. It focuses on how the count of non-commutative curves opens up ways to new categorical structures and connections to number theory and classical geometry.’
‘I’ve cooperated with the HSE Faculty of Mathematics for many years’, says Evgeny Shinder, associate member of the HSE Laboratory for Mirror Symmetry andlecturer at the University of Sheffield. ‘I’ve always been impressed by the level of the students, who know many things as well as I do. I now teach in England, but I always enjoy coming back to Russia. I studied in St. Petersburg, which, like Moscow, has a great mathematical school. Ideas and developments are shared openly. It goes without saying that this enriches and empowers the Russian mathematical community’. Another representative of the St. Petersburg mathematical school, Valery Gritsenko, head of the HSE Laboratory of Mirror Symmetry, Professor at the Lille University of Science and Technology, and member of Institut Universitaire de France, says: ‘The ‘Mirror Symmetry and Applications’ conference is the laboratory’s sixth and the most comprehensive conference yet. The meeting provided talented young people in mirror symmetry with an opportunity to present their results and initiate sustainable long-term cooperation’.
PhD and Master’s students at the HSE Faculty of Mathematics also numbered among the conference attendees. The laboratory organized the first research school, ‘Geometry 2017’, in St. Petersburg in July 2017, which attracted 120 students from numerous Russian cities. The participants attended five mini-courses run by leading international researchers, including Dmitry Orlov and Alexander Kuznetsov from the RAS Steklov Mathematical Institute, Sergey Ivanov from PDMI RAS, Misha Verbitsky, Professor at HSE and the Free University of Brussels, and Laurent Manivel, a renowned French mathematician. ‘At the end of January 2018, the laboratory is organizing a new research school, this time in physics, automorphic forms, and Kac-Moody algebras’, explains Valery Gritsenko. ‘It will be held at the Bogoliubov Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research in Dubna. The laboratory is also planning a research school in arithmetical issues of mirror symmetry’.
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‘Mirror Symmetry Was Discovered by Physicists, But Very Quickly Got the Attention of Mathematicians…’
The HSE International Laboratory for Mirror Symmetry and Automorphic Forms, which is among several international laboratories to recently open within the Higher School of Economics, was created in December 2016 as part of the Russian government’s mega-grants program. Below, the lab’s academic supervisor, Ludmil Katzarkov, along with deputy heads Valery Gritsenko and Viktor Przyjalkowski, explain why the laboratory is fully capable of becoming a unique multidisciplinary unit dedicated to the study of mirror symmetry, automorphic forms, and number theory.
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