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Home , Pontryagin duality

Sergei Akbarov

E-mail address: [email protected] Skype: Sergei Akbarov Languages: Russian, English, French Web presence: Mathnet.ru, ORCID, SCOPUS, Zentralblatt, Mathscinet, Arxiv, Google Scholar, MathOverflow, ResearchGate, SPIN РИНЦ Research interests: functional analysis, , , differential geometry, complex geometry, quantum groups, theory

Academic degrees 2010 Doctor of Physical and Mathematical Sciences (Doctor Habilitatus), Moscow Lomonosov State University, Faculty of Mechanics and Mathematics. Thesis title: Stereotype and for Stein groups. 1989 PhD degree, Moscow Institute of Electronic Engineering (currently Moscow State Institute of Electronics and Mathematics, MIEM). Thesis title: Hamiltonian mechanics and quantization on locally compact groups. Supervisor: Professor V.P. Maslov, Member of Russian Academy of Sciences. 1986 Master of Sciences in Applied Mathematics, Moscow Institute of Electronic Engineering (MIEM).

Academic rank

1994 Docent in Mathematical Analysis.

Academic experience 2018-present Full Professor, Higher School of Economics (National Research University), Moscow. 2017-present Leading researcher, Department of Mathematics, All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences (VINITI), Moscow. 2016-2017 Full Professor, Higher School of Economics (National Research University), Moscow. 2013-2016 Full Professor, Moscow Aviation Institute (National Research University), Moscow. 2002-2012 Group leader, Department of Mathematics, All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences (VINITI), Moscow. 1999-2001 Associate Professor (Docent), Department of Mathematical Analysis, Moscow Institute of Electronics and Mathematics (MIEM), Moscow. 1996-1999 Associate Professor (Docent), Department of Higher Mathematics, Bauman Moscow State Technical University (MSTU), Moscow. 1993-1995 Associate Professor (Docent), Department of Mathematical Analysis, Moscow Institute of Electronics and Mathematics (MIEM), Moscow. 1989-1993 Assistant Professor, Department of and Analysis, Moscow Institute of Electronics and Mathematics (MIEM), Moscow. 1978-1996 High school mathematics teacher, Moscow.

List of Publications

1. The Stereotype Approximation Property for the Stereotype Group Algebra of Measures, Матем. заметки, 104:3 (2018), 465–468; Math. Notes, 104:3 (2018), 465–468. 2. Continuous and smooth envelopes of topological algebras. Part I, Itogi Nauki i Techniki, Modern Mathematics and its Applications, Thematical Surveys. V. 129: 3 -132, 2017; Journal of Mathematical Sciences, 227(5):531-668, 2017, arxiv. 3. Continuous and smooth envelopes of topological algebras. Part II, Itogi Nauki i Techniki, Modern Mathematics and its Applications, Thematical Surveys. V. 130:1 -110, 2017; Journal of Mathematical Sciences, 227(6):669-789, 2017, arxiv. 4. Algebra of continuous functions as a continuous envelope of its subalgebras, Functional Analysis and Applications, 50(2): 75-77, 2016. 5. Envelopes and refinements in categories, with applications to functional analysis, Dissertaciones mathematicae, 513(1): 1-188, 2016; arxiv. 6. C∞(M) as a smooth envelope of its subalgebras, Mathematical Notes, 97(4): 489-492, 2015. 7. The Gelfand transform as a C*-envelope. Mathematical Notes. 94(5):777-779, 2013. 8. Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity, Fundamentalnaya i prikladnaya matematika 14(1): 3-178, 2008; Journal of Mathematical Sciences, 162(4): 459-586, 2009; arxiv. 9. The structure of modules over the stereotype algebra L(X) of operators, Funkts. Anal. Prilozh., 40(2): 1-12, 2006. 10. Pontryagin duality and topological algebras, in: "Topological Algebras, their Applications and Related Topics eds. K. Jarosz and A. Soltysiak, Banach Center Publications, 67: 55 - 71, 2005. 11. (with E.T.Shavgulidze) On two classes of spaces reflexive in the sense of Pontryagin, Mat. Sbornik, 194(10):3-26, 2003. 12. Pontryagin duality in the theory of topological vector spaces and in Topological Algebra, Journal of Mathematical Sciences, 113(2):179-349, 2003. 13. Stereotype spaces, algebras, homologies: an outline, In: "Topological homology", Editor: A.Ya.Helemskii, Nova Science Publishers, 1-29, 2000. 14. Stereotype locally convex spaces, Izv. Ross. Akad. Nauk, 64(4):4-46, 2000. 15. Absolute homology theory of stereotype algebras, Funkts. Anal. Prilozh., 34(1):76-79, 2000. 16. Stereotype algebras with reflection and the double commutant theorem, Mathematical Notes, 66(5):789-792, 1999. 17. Stereotype group algebras, Mathematical Notes, 66(6): 777–780, 1999. 18. Stereotype approximation property and the uniqueness problem for the trace, Funkts. Anal. Prilozh., 33(2): 137–140, 1999. 19. The Pontryagin duality in the theory of topological modules, Funkts. Anal. Prilozh., 29(4): 68-72, 1995. 20. The Pontryagin duality in the theory of topological vector spaces, Mathematical Notes, 57(3): 463-466, 1995. 21. The structure of the cotangent bundle of a locally , Izv. Ross. Akad. Nauk, 59(3): 3-30, 1995. 22. Differential geometry and quantization on a , Izv. Ross. Akad. Nauk, 59(2): 47-62, 1995. 23. Smooth structure and differential operators on a locally compact group, Izv. Ross. Akad. Nauk, 59(1): 3-48, 1995. 24. A criterion of the completeness of an analytical vector field on a locally compact , Mat. Sbornik, 185(11): 3-22, 1994. 25. The of invariant differential operators on a locally compact Abelian group, Mathematical Notes, 56(2): 132-136, 1994. 26. The structure of differential operators on a locally compact Abelian group, Mathematical Notes, 52(4):3-14, 1992. 27. Derivations of the algebra of smooth functions on a locally compact Abelian group, Mathematical Notes, 50(2):3-13, 1991. 28. Vector fields on locally compact Abelian groups, Mathematical Notes, 45(1):132-134, 1989. 29. On power series with coefficients in the spaces without norm, Izv. Akad. Nauk Arm. SSR, 23(2):172- 181, 1989.

Books in preparation

30. Mathematical analysis (draft available at arxiv, in Russian). 31. Stereotype spaces, algebras, geometries.

Statement of Research Interest I am the author of the theory of stereotype spaces. This is the class of topological vector spaces defined by the identity

X** = X, where X* denotes the space of continuous linear functionals on X, endowed with the topology of uniform convergence on totally bounded sets, and the identity is understood as an isomorphism of topological vector spaces. The class Ste of stereotype spaces is very wide and forms a category with unexpectedly good properties. It provides the most natural setting, and an effective instrument for development and simplification of Functional analysis. The following facts support this assertion:  Ste includes all Frèchet spaces (and, thus, all Banach spaces), and admits standard constructions of new spaces, like projective and injective limits, tensor products, etc., see [12]. In practice this means that virtually each topological used in Analysis is stereotype. This resembles the situation in theory, where the attention of the researcher shifts from the class of all sets to the class of measurable sets, which is narrower, but because of this it doesn’t contain unexpected ugly counterexamples, and at the same time it is still sufficiently wide for all the necessary constructions.

 Ste is a closed monoidal category [12]. As a corollary, this category provides a convenient setting for a theory of topological algebras, called stereotype algebras. The theory of stereotype algebras generalizes the pure algebraic theory of associative algebras and, at the same time, the theory of Banach algebras (see the detailed discussion in [10], [12], [13]).

 Ste allows reduction in the number of counterexamples in Analysis. For instance, the approximation property is inherited in Ste by the tensor products and the spaces of operators [12]. As a corollary, the famous A. Szankowski counterexample (which states that the B(H) of operators on a Hilbert space H does not have the approximation property) loses its significance in the stereotype theory. It is a consequence of the fact, that for any stereotype space X with the approximation property, the space L(X) of operators on X also has the approximation property.

 If a stereotype space X has the approximation property, all modules over the algebra L(X) of operators on X allow a simple description (my paper [9] is devoted to this topic). This is completely unexpected, since up to now in the Banach theory no examples of (infinite- dimensional) spaces X were found with such a description. My current research is focused on the applications of the theory of stereotype spaces in the duality theories. I am studying different generalizations of Pontryagin duality to non-commutative and quantum groups. This is a meeting point of Analysis, Geometry and Algebra, and it is in vogue in Europe, since the investigations in this area give hope to find a proper definition of quantum groups.

In my monograph [8] I constructed a generalization of the Pontryagin duality from the class of compactly generated commutative complex Lie groups to the class of all (not necessarily commutative) compactly generated complex Lie groups whose connected component of identity is an affine algebraic group. This is a rather wide and important class of groups, since it contains all affine algebraic groups. Moreover, as it was shown in [8], the suggested construction of generalization is not limited to the class of Lie groups, but can be extended to certain quantum groups as well. An important inspiring detail of this picture is that the enveloping category, to which we embed the category of groups in such a generalization, consists of Hopf algebras (in the category of stereotype spaces). This is the first duality theory with this property, because in the other theories built by now, the objects of the enveloping category are not Hopf algebras.

From the results of [8] it follows, in fact, that there are many non-equivalent generalizations of the Pontryagin duality, and the scheme described in [8] can be applied to developing duality theories in all “big geometric disciplines” in mathematics, namely:  Topology,  Differential Geometry,  Complex Geometry. It is clear now that in each of these fields there exists its own, special duality theory (and even many of them, since there are many ways to understand what these disciplines actually are). In [8] I suggested such a construction for Complex Geometry, but if we replace one of the key in this scheme, the operation of taking the Arens-Michael envelope, with another similar , we come to another theory. The functors used in this scheme can be defined in purely categorical terms and are called envelopes [5]. In mathematics envelopes play a role similar to that of observation tools in physics: each envelope generates a visible picture of the objects under observation, and this leads to the construction of geometry as a categorical discipline. In [8] this scheme was applied to Complex Analysis, and in my recent monograph [1], [3] I showed how it works in Differential Geometry and in Topology. As a corollary, all the geometric disciplines mentioned above – Topology, Differential Geometry, Complex Analysis – turn out to be results of application of some envelope functors (continuous, smooth, holomorphic) to the category of stereotype algebras. Generally, each envelope in a category of stereotype algebras generates a geometry as a discipline, and this correspondence makes it possible to construct new geometries, to compare them, to find common properties, differences, to predict results, etc. This can be considered as a development of the famous Klein Erlangen program, and I see here a rare possibility to look at mathematics as a whole. The monograph [31] that I am writing now is devoted to this topic.

Teaching dossier A. Teaching experience I have been teaching mathematics at the university level since 1989, excluding the period of time I worked at VINITI. Before 1989, when I was a student, I taught at the mathematical schools in MEI and MIEM. Here is the list of courses that I conducted. • University Level (here HSE is the abbreviation of the Higher School of Economics, MAI is the Moscow Aviation Institute, MIEM is the Moscow Institute of Electronics and Mathematics, MSTU is the Bauman Moscow State Technical University, MPEI is the Moscow Power Engineering Institute)

Course and main topics University Dates

Mathematical analysis HSE Spring 2017 Mathematical analysis HSE Autumn 2016 Complex analysis, Equations of Mathematical Physics MAI Spring 2016 Probability theory, Complex analysis, Linear algebra MAI Autumn 2015 Mathematical analysis (differentiation of functions of several MAI Spring 2015 variables, integration of functions of several variables), Complex analysis, Computational methods Mathematical analysis (differentiation of functions of one variable, MAI Autumn integration of functions of one variable), Complex analysis 2014 Mathematical analysis (differentiation of functions of several MAI Spring 2014 variables, integration of functions of several variables), Complex analysis Mathematical analysis (differentiation of functions of one variable, MAI Autumn integration of functions of one variable), Complex analysis 2013 Probability theory, Computational methods MAI Spring 2013 Mathematical analysis (differentiation of functions of several MIEM Spring 2001 variables, integration of functions of one variable), Complex analysis Mathematical analysis (integration of functions of several MIEM Autumn variables), Functional analysis 2000 Mathematical analysis (differentiation of functions of several MIEM Spring 2000 variables, integration of functions of one variable), Functional analysis Mathematical analysis (differentiation of functions of one variable), MIEM Autumn Mathematical analysis (integration of functions of several variables) 1999 Complex analysis, Differential equations MSTU Spring 1999 Mathematical analysis (integration of functions of several variables) MSTU Autumn 1998 Complex analysis, Differential equations, Linear algebra and MSTU Spring 1998 analytical geometry Mathematical analysis (integration of functions of several MSTU Autumn variables), Linear algebra and analytical geometry 1997 Mathematical analysis (differentiation of functions of several MSTU Spring 1997 variables, integration of functions of one variable), Linear algebra and analytical geometry Mathematical analysis (differentiation of functions of one variable), MSTU Autumn Linear algebra and analytical geometry 1996 Mathematical analysis (integration of functions of several MIEM Autumn variables), Complex analysis, Differential equations 1994 Mathematical analysis (differentiation of functions of several MIEM Spring 1994 variables, integration of functions of one variable), Complex analysis Mathematical analysis (differentiation of functions of one variable), MIEM Autumn Mathematical analysis (integration of functions of several 1993 variables), Differential equations Mathematical analysis (differentiation of functions of several MIEM Spring 1993 variables, integration of functions of one variable), Complex analysis, Linear algebra and analytical geometry Mathematical analysis (differentiation of functions of one variable), MIEM Autumn Mathematical analysis (integration of functions of several 1992 variables), Differential equations, Linear algebra and analytical geometry Mathematical analysis (differentiation of functions of several MIEM Spring 1992 variables, integration of functions of one variable), Complex analysis, Linear algebra and analytical geometry Mathematical analysis (differentiation of functions of one variable), MIEM Autumn Mathematical analysis (integration of functions of several 1991 variables), Differential equations, Linear algebra and analytical geometry Mathematical analysis (differentiation of functions of several MIEM Spring 1991 variables, integration of functions of one variable), Complex analysis, Linear algebra and analytical geometry Mathematical analysis (integration of functions of MIEM Autumn several variables), Differential equations, Linear algebra and 1990 analytical geometry Mathematical analysis (differentiation of functions of several MIEM Spring 1990 variables, integration of functions of one variable), Complex analysis, Linear algebra and analytical geometry Mathematical analysis (differentiation of functions of one variable), MIEM Autumn Linear algebra and analytical geometry, Mathematical logics 1989

• Teaching at the Physical and mathematical science school Mathematics MIEM, Physical and mathematical school 1982-1989 Mathematics MPEI, Physical and mathematical school 1978-1982

• Student supervision: Student Area Graduation Year School Program Kirill Gusev Applied mathematics 1993 MIEM Master of Sciences Maria Filatova Applied mathematics 1995 MIEM Master of Sciences

B. Teaching goals I had the great chance and pleasure to encounter many outstanding professors when I was a student. Some of them, in particular Rais Salmanovich Ismagilov and Yuri Borisovich Orochko, became role models and serve as a constant source of inspiration in my own teaching career. They demonstrated to me early on that a teacher can always make a course interesting and easy to understand. One important lesson they taught me was that you should be in touch with the audience, and choose the sequence of themes according to the mathematical background of your students. Another idea I got from their classes was that students don’t usually learn by digesting generalities: one needs to arrange a system of examples and exercises in such a way that a transition from easy things to complicated ones creates a rhythm in which achieving the objectives becomes easy. These are the teaching methods that I use myself in order to show people the beauty of mathematics, to inculcate in them the understanding of its ideas and to accustom them to various uses of mathematical methods.

C. Teaching philosophy Most people treat mathematics as a science of computations, but for me it always has not been the most essential thing. As a teacher I have always tried to show students another component of mathematics, which I find to be much more important: the method of logical deduction, which allows one to construct chains of corollaries from elementary premises that often lead to complicated and non-obvious conclusions. The fact that the varied laws of the physical (and not only physical) world could be deduced from a small number of postulates which can easily be taken on trust – this, in my opinion, is the main point that mathematics teaches (and the main point that a mathematician should teach). Understanding when a proposition implies some other propositions is not inborn: it should be taught. The stark difference in manners of thinking and reasoning of people with different education demonstrates the importance of teaching these cognitive tools and the benefits of education rooted in reason and logic.

D. Textbook on Mathematical Analysis This is, in short, what can be called my philosophical views on teaching. How I understand the practice of my profession, I hope, is clearly demonstrated in my textbook on mathematical analysis [30] posted in arXiv. I am planning to publish this textbook after adding some supplementary material and correcting errors. In accordance with what I described as "philosophy", the goal of the author of a new textbook should be to simplify and clarify the logical connections, which are under-represented in existing textbooks and handbooks. For instance, the topic which is traditionally poorly connected to the axiomatic structure of mathematical analysis has always been the parts related to elementary functions. The very definition of elementary functions either is not given at all, or (in the only one case that I know, in the textbook by H. Grauert, I. Lieb and W. Fischer) it is given after Taylor series, at a time when elementary functions are supposed to be used for almost a whole semester. In my textbook elementary functions are introduced in the first chapters by two extra axioms, which are later proved in the text. Another important topic, which is traditionally unsatisfactorily explained in the existing textbooks on analysis is, in my opinion, the integration over curves and surfaces, which is usually presented as an application of the theory of integration over chains in manifolds with (smooth) boundary. But the class of such manifolds is too narrow to allow picking enough examples and exercises for working with students. And the attempts to widen it (for example, by including the manifolds with pseudo- boundary, as it is done in the textbook by L. Schwartz) in all the textbooks which I know leads to unjustified burdening of formulations. In my textbook this difficulty is resolved by introducing the class of manifolds, whose singularities in the parameterization have zero measure. Under such a definition the choice of the parameterization does not play an essential role in integration, and the Stokes theorem can be extended to a sufficiently wide class of manifolds with singularities.