1964 - 1974 Secondary Scholl nr. 15 in Tallinn.
I studied at this school for 10 years, that is, it had ten classes. Starting from the seventh grade, I successfully participated in math olympiads. In the ninth and tenth grades, I studied in a special class with intensive study of mathematics, the so-called class with a mathematical bias. Gifted children from all over the city of Tallinn were selected for this class. A few words about mathematical Olympiads. At that time, mathematics olympiads had several levels. The first level is the school, the second level is the city Olympiad, then the Republican and finally the All-Union. Successful performance at the All-Union Olympiad made it possible to enter the USSR national team for the international Olympiad. My best result is the Republican Olympiad, I have not risen higher.1974 - 1976 Tallinn Polytechnical Institute.
After graduating from secondary school, I went to enter the Moscow University at the Faculty of Applied Mathematics and Cybernetics. However, the competition turned out to be very high and I did not score enough points to be enrolled as a student. Then I returned to Tallinn and entered the evening department of the Tallinn Polytechnic Institute with a speciality in computer technology. The evening department meant that during the day I worked at the "Dvigatel" plant, and in the evening I listened to lectures at the institute. The level of teaching mathematics at the Tallinn Polytechnic Institute did not satisfy me and I decided to enter the Faculty of Mathematics at the University of Tartu.
1976 - 1981 Faculty of Mathematics, University of Tartu.
I entered the Faculty of Mathematics at the University of Tartu in a group with Estonian as the language of teaching. It should be noted that there was also a small group with Russian as the language of teaching. Thus, I studied mathematics, as well as the Estonian language at the same time. The first year was not easy, but then I mastered the language and it became easier to study. It is worth to mention that I studied Estonian at a school in Tallinn, starting from the second grade, but the teaching was rather formal and, as a rule, after finishing ten years of high school, we did not know how to speak Estonian. The Faculty of Mathematics had several departments, but I was particularly attracted to differential geometry, so I chose the Department of Algebra and Geometry. In my third year, I was supervised by Associate Professor Aivo Parring, and under his supervision I wrote an undergraduate thesis on the Petrov's classification of Einstein spaces. Then Professor Ülo Lumiste became my supervisor and he suggested that I study the theory of connections from the point of view of applications of this theory in gauge theories.
| Scientific Degree | Candidate of Science in Physics and Mathematics (equivalent to Ph. D.) |
| Title of Thesis | Superspace of Connections and Geometric Structures of a Quantum Gauge Field Theory |
| Institution | A degree was conferred by the Council for the Defense of Doctoral Dissertations at the Byelorussian State University (Minsk, 1986) and approved by the Higher Attestation Commission (USSR) in 1987 |
| Supervisor | Ülo Lumiste, professor of the University of Tartu and academician of the Academy of Science of Estonia |
| Reviewers at defense | A. A. Burdun (Byelorussian State University), L. E. Evtushik (Moscow State University) |
| Summary of the Thesis | In the 70s of the last century, the quantum theory of Yang-Mills fields, proposed by Faddeev, Slavnov, Popov and others, was actively studied. Faddeev and Popov introduced the so-called ghost fields, which later became known as the Faddeev-Popov fields. The effective Lagrangian of the theory, which, in addition to the Yang-Mills fields, also contains the Faddeev-Popov fields, was invariant under the transformations that came to be known as the BRST-transformations. At that time, it was already known that the potentials of ? Yang-Mills field could be interpreted as the coefficients of a connection in a principal fiber bundle, and the field strength, as the curvature of this connection. Naturally, there was posed a question about a geometric interpretation of the Faddev-Popov fields within the framework of the theory of connections on fiber bundles, and this interpretation had to be such that it could be possible to derive the BRST transformations from a geometric nature of Faddeev-Popov fields. Since the Faddeev-Popov fields are fermion fields, it was proposed to identify them with differential forms on the bundle. In my dissertation, I pointed out that this interpretation has flaws. In quantizing the Yang-Mills theory, Faddeev and Popov used the infinite-dimensional Grassmann algebra described by Berezin in his monograph on second quantization. The generators of this algebra are the Faddeev-Popov fields. This algebra is functional (one should use distributions) and the generators depend on a point of a manifold, which serves as a continuous index of these generators. Generators given at different points of a manifold can be multiplied and the product is anticommutative (Grassmann algebra). On the other hand a multiplication of differential forms (wedge product) is defined pointwise, the multiplication of differential forms at different points of a manifold does not make sense. In my thesis, I extended Berezin's construction from a manifold to bundles by constructing an infinite-dimensional Grassmann algebra on a fiber bundle. Then I constructed a Banach supermanifold. The even sector of this supermanifold is the Banach manifold of connections on a fiber bundle, and the odd sector is the infinite-dimensional Grassmann algebra on a fiber bundle. In this approach, BRST transformations can be considered as an odd vector field on the above supermanifold, and the effective Lagrangian is a functional on this supermanifold. A group of gauge transformations acts on a Banach manifold of connections. I have shown that the vector field of BRST transformations is an extension of infinitesimal gauge transformations to the supermanifold of connections along the orbits of the action of the gauge transformation group. |
| 1984 - 1998 | Junior Researcher, Researcher, Senior Researcher at the Institute of Pure Mathematics, University of Tartu |
| 1998 - 2011 | Associate Professor at the Institute of Mathematics and Statistics, University of Tartu |
| 2011 - | Full Professor of Geometry and Topology at the Institute of Mathematics and Statistics, University of Tartu |
| 1993 | Research grant "Sonderforschungsbereiches 288, Differentialgeometrie und Quantenphysik", May - June, University of Humboldt, Berlin |
| 1994 | TEMPUS individual mobility grant nr. IMG-93-EE-2026, teaching stuff retraining and research, April - August, University Paris VI, Paris |
| 1997 | Senior Researcher (Chercheur Associé) at the Labboratory of Relativistic Cosmology and Gravitation, University Paris VI, position financed by CNRS, August - December, Paris |
| 2012 | PARROT (program of scientific cooperation between Estonia and France) travel grant, joint research, Laboratoire de Physique de la Matiere Condensee, University Paris VI, October, Paris |
| 2012 | ERASMUS teaching stuff retraining and research grant, Department of Mathematics, University of Aveiro, December, Portugal |
| 2015 | PARROT travel grant, joint research, Laboratoire de Physique de la Matiere Condensee, University Paris VI, January - February, Paris |
| 2015 | Visiting professor, Department of Mathematics, Federal University of Parana, September, Curitiba, Brazil |
| 2018 | Visiting Professor, Department of Mathematics, CIRM, University of Haute Alsace, 30.09.2018 - 28.10.2018, Mulhouse, France |
| 2019 | Visiting Professor, Department of Mathematics, University of Mälardalen, May 6 - May 31, Sweden |
| 2021 | PARROT travel grant, joint research, Department of Mathematics, Universite de Haute Alsace, Mulhouse and Laboratoire de Physique de la Matiere Condensee, Sorbonne University, November - December, Paris |
| 2022 | Simons Grant, visiting Professor, Department of Mathematics, University of South Florida, 23.04.2022 - 09.05.2022, Tampa, USA |
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