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 Statistics Math-Net.Ru Total publications: 40 Scientific articles: 34 Presentations: 1

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Professor
Doctor of physico-mathematical sciences (1988)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 27.02.1939
E-mail: ,
Keywords: theory of group representations; symmetric spaces; harmonic analysis on homogeneous spaces; quantization; canonical representations; boundary representations.

Subject:

In a series of my works (the 60 80 ies) the construction of harmonic analysis on semisimple symmetric spaces $G/H$ (non-Riemannian) of rank one was begun and completed. A description of the corresponding principal non-unitary series of representations was given. Principal notions of the theory were introduced ($H$-invariants, Fourier transform, Poisson transform, spherical functions) and corresponding methods were worked out. Plancherel formula was obtained explicitly ( in different variants, one of them is expansion of the delta function in terms of spherical functions). The Berezin quantization was transferred from Hermitian symmetric spaces to symplectic semisimple symmetric spaces. In particular, an important case of quantizations was described — the so–called polynomial quantization. A new form of the deformation decomposition (the decomposition of the Berezin transform) was offered using "generalized powers" (generalized Pochhammer symbols) instead of usual powers of a parameter. This form makes the decomposition natural and apparent and allows to compute it explicitly. Canonical representations on these symplectic spaces were studied - in connection with the construction of quantizations (decompositions into irreducible constitutients — right up to explicit formulae for one rank spaces). The canonical representations (sometimes called the Berezin representations) on Hermitian symmetric spaces were introduced by Berezin and Vershik–Gelfand–Graev. They are unitary representations. We consider the canonical representations in a much wider sense: we give up the condition of unitarity, they act on sufficiently extensive function spaces, in paricular, on spaces of distributions. Also boundary representations generated by canonical representations were studied. In particular, appearance of Jordan blocks in the decomposition of these representations was discovered. It is found that the decomposition of boundary representations is intimately connected with the meromorphic structure of Poisson and Fourier transforms associated with the canonical representations. These results (quantizations, canonical and boundary representations) can be transferred to a certain extent to some semisimple symmetric spaces which are not symplectic, for example, to hyperboloids of arbitrary signature. This work (quantizations, canonical and boundary representations etc.) is a part of what I call a non-unitary version of harmonic analysis, a new and promising field of research. For hyperboloids of Hermitian type, the holomorphic discrete series was investigated, Cauchy-Szego kernels were computed, projection operators on analytic and antianalytic series of irreducible unitary reresentations were explicitly found, an analogue of the Hilbert transform was introduced and computed. One of results — separation of series — was carried over to hyperboloids of arbitrary signature. For finite reflection groups, Poincare polynomials and series were explicitly computed.

Biography

Graduated with a first–class honours degree from Faculty of Mathematics and Mechanics of Lomonosov Moscow State University in 1962 (chair of theory of functions and functional analysis). Candidate dissertation (Ph.D. thesis) was defended in 1967. Doctor dissertation was defended in 1987. A list of my works contains about 100 titles. I have led the research seminar at Derzhavin Tambov State University on functional analysis.

Member of Moscow Mathematical Society Corresponding member of RANS (Russian Academy of Natural Science).

Main publications:
• Molchanov V. F. Quantization on para-Hermitian symmetric spaces // Amer. Math. Soc. Transl. Ser. 2, vol. 175 (Adv. Math. Sci., 31), 1996, 81–96.
• Dijk. G. van, Molchanov V. F. The Berezin form for rank one para-Hermitian symmetric spaces // J. Math. Pures Appl., 1998, 77, no. 8, 747–799.
• Dijk. G. van, Molchanov V. F. Tensor products of maximal degenerate series representations of the group $SL(n,\Bbb R)$ // J. Math. Pures Appl., 1999, 78, no. 1, 99–119.

http://www.mathnet.ru/eng/person17888
List of publications on Google Scholar
List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/195585

Publications in Math-Net.Ru
 2019 1. V. F. Molchanov, “Radon problems for hyperboloids”, Russian Universities Reports. Mathematics, 24:128 (2019),  432–449 2. V. F. Molchanov, E. S. Yuryeva, “Integer triangles, Pell's equation and Chebyshev polynomials”, Russian Universities Reports. Mathematics, 24:126 (2019),  179–186 2018 3. V. F. Molchanov, E. E. Kryukova, “Placements without neighbours”, Tambov University Reports. Series: Natural and Technical Sciences, 23:124 (2018),  655–665 4. V. F. Molchanov, “Polynomial quantiztion and overalgebra for hyperboloid of one sheet”, Tambov University Reports. Series: Natural and Technical Sciences, 23:123 (2018),  353–360 2017 5. V. F. Molchanov, “Berezin quantization as a part of the representation theory”, Tambov University Reports. Series: Natural and Technical Sciences, 22:6 (2017),  1235–1246 2015 6. V. F. Molchanov, “Poisson and Fourier Transforms for Tensor Products”, Funktsional. Anal. i Prilozhen., 49:4 (2015),  50–60    ; Funct. Anal. Appl., 49:4 (2015), 279–288 2012 7. V. F. Molchanov, “Radon transform on a space over a residue class ring”, Mat. Sb., 203:5 (2012),  119–134        ; Sb. Math., 203:5 (2012), 727–742 2006 8. V. F. Molchanov, “Canonical representations on two-sheeted hyperboloids”, Zap. Nauchn. Sem. POMI, 331 (2006),  91–124        ; J. Math. Sci. (N. Y.), 141:4 (2007), 1432–1451 2005 9. V. F. Molchanov, “Canonical Representations and Overgroups for Hyperboloids”, Funktsional. Anal. i Prilozhen., 39:4 (2005),  48–61        ; Funct. Anal. Appl., 39:4 (2005), 284–295 1999 10. V. F. Molchanov, “Representations of pseudo-unitary groups associated with a cone”, Lobachevskii J. Math., 3 (1999),  221–241 1997 11. V. F. Molchanov, “Separation of Series for Hyperboloids”, Funktsional. Anal. i Prilozhen., 31:3 (1997),  35–43        ; Funct. Anal. Appl., 31:3 (1997), 176–182 1992 12. V. F. Molchanov, “On the Poincaré series of representations of finite reflection groups”, Funktsional. Anal. i Prilozhen., 26:2 (1992),  82–85      ; Funct. Anal. Appl., 26:2 (1992), 143–145 1990 13. V. F. Molchanov, “Harmonic analysis on homogeneous spaces”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 59 (1990),  5–144 1986 14. V. F. Molchanov, “The Plancherel formula for pseudo-Riemannian symmetric spaces of rank $1$”, Dokl. Akad. Nauk SSSR, 290:3 (1986),  545–549 15. V. F. Molchanov, “Spherical functions on pseudo-Riemannian symmetric spaces of rank $1$”, Dokl. Akad. Nauk SSSR, 287:5 (1986),  1054–1058 1984 16. V. F. Molchanov, “Plancherel's formula for pseudo-Riemannian symmetric spaces of the universal covering group of $SL(2,\mathbf{R})$”, Sibirsk. Mat. Zh., 25:6 (1984),  89–105      ; Siberian Math. J., 25:6 (1984), 903–917 1983 17. V. F. Molchanov, “Orbits of a stationary subgroup on a pseudo-Riemannian symmetric space of rank one”, Uspekhi Mat. Nauk, 38:5(233) (1983),  203–204      ; Russian Math. Surveys, 38:5 (1983), 158–159 1982 18. V. F. Molchanov, “Poincaré polynomials of representations of finite groups generated by reflections”, Mat. Zametki, 31:6 (1982),  837–845      ; Math. Notes, 31:6 (1982), 423–427 19. V. F. Molchanov, “Harmonic analysis on pseudo-Riemannian symmetric spaces of the group $SL(2,\mathbf R)$”, Mat. Sb. (N.S.), 118(160):4(8) (1982),  490–503      ; Math. USSR-Sb., 46:4 (1983), 493–506 20. V. F. Molchanov, “Plancherel's formula for the pseudo-Riemannian space $SL(3,\mathbf{R})/GL(2,\mathbf{R})$”, Sibirsk. Mat. Zh., 23:5 (1982),  142–151      ; Siberian Math. J., 23:5 (1982), 703–711 1981 21. V. F. Molchanov, “The Plancherel formula for the tangent bundle of a projective space”, Dokl. Akad. Nauk SSSR, 260:5 (1981),  1067–1070 1980 22. V. F. Molchanov, “Quantization on the imaginary Lobachevskii plane”, Funktsional. Anal. i Prilozhen., 14:2 (1980),  73–74      ; Funct. Anal. Appl., 14:2 (1980), 142–144 23. V. F. Molchanov, “Plancherel's formula for hyperboloids”, Trudy Mat. Inst. Steklov., 147 (1980),  65–85      ; Proc. Steklov Inst. Math., 147 (1981), 63–83 1979 24. V. F. Molchanov, “Tensor products of unitary representations of the three-dimensional Lorentz group”, Izv. Akad. Nauk SSSR Ser. Mat., 43:4 (1979),  860–891      ; Math. USSR-Izv., 15:1 (1980), 113–143 1978 25. V. F. Molchanov, “Elementary representations of the Laguerre group”, Mat. Zametki, 23:1 (1978),  31–40      ; Math. Notes, 23:1 (1978), 19–23 26. V. F. Molchanov, “Reduction of representations of the complementary series of the $2+3$ de Sitter group with respect to the Lorentz group”, TMF, 37:2 (1978),  274–280      ; Theoret. and Math. Phys., 37:2 (1978), 1017–1022 1977 27. V. F. Molchanov, “The restriction of a representation of the complementary series of a pseudo-orthogonal group to a pseudo-orthogonal group of lower dimension”, Dokl. Akad. Nauk SSSR, 237:4 (1977),  782–785 28. V. F. Molchanov, “The decomposition of the tensor square of a representation of the complementary series of the unimodular group of real matrices of order two”, Sibirsk. Mat. Zh., 18:1 (1977),  174–188      ; Siberian Math. J., 18:1 (1977), 128–138 1976 29. V. F. Molchanov, “Spherical functions on hyperboloids”, Mat. Sb. (N.S.), 99(141):2 (1976),  139–161      ; Math. USSR-Sb., 28:2 (1976), 119–139 1975 30. V. F. Molchanov, “Decomposition of the tensor square representation of the complementary series of a group”, Funktsional. Anal. i Prilozhen., 9:4 (1975),  79–80      ; Funct. Anal. Appl., 9:4 (1975), 344–345 1971 31. V. F. Molchanov, “On the caluculation of weight multiplicity”, TMF, 8:2 (1971),  251–254      ; Theoret. and Math. Phys., 8:2 (1971), 810–812 1970 32. V. F. Molchanov, “Representations of pseudo-orthogonal groups associated with a cone”, Mat. Sb. (N.S.), 81(123):3 (1970),  358–375      ; Math. USSR-Sb., 10:3 (1970), 333–347 1968 33. V. F. Molchanov, “An analog of Plancherel's formula for hyperboloids”, Dokl. Akad. Nauk SSSR, 183:2 (1968),  288–291 1966 34. V. F. Molchanov, “Harmonic analysis on a hyperboloid of one sheet”, Dokl. Akad. Nauk SSSR, 171:4 (1966),  794–797 2019 35. A. M. Borodin, Aleksandr I. Bufetov, Aleksei I. Bufetov, A. M. Vershik, V. E. Gorin, A. I. Molev, V. F. Molchanov, R. S. Ismagilov, A. A. Kirillov, M. L. Nazarov, Yu. A. Neretin, N. I. Nessonov, A. Yu. Okounkov, L. A. Petrov, S. M. Khoroshkin, “Grigori Iosifovich Olshanski (on his 70th birthday)”, Uspekhi Mat. Nauk, 74:3(447) (2019),  193–213    ; Russian Math. Surveys, 74:3 (2019), 555–577 2013 36. A. M. Vershik, A. A. Kirillov, V. F. Molchanov, Yu. A. Neretin, G. I. Olshanski, V. V. Ryzhikov, V. M. Tikhomirov, A. A. Shkalikov, “Rais Sal'manovich Ismagilov (on his 75th birthday)”, Uspekhi Mat. Nauk, 68:4(412) (2013),  185–190      ; Russian Math. Surveys, 68:4 (2013), 783–788 2008 37. A. M. Vershik, I. M. Gel'fand, S. G. Gindikin, A. A. Kirillov, G. L. Litvinov, V. F. Molchanov, Yu. A. Neretin, V. S. Retakh, “Mark Iosifovich Graev (to his 85th brithday)”, Uspekhi Mat. Nauk, 63:1(379) (2008),  169–182        ; Russian Math. Surveys, 63:1 (2008), 173–188 1997 38. V. F. Molchanov, “Tambov School-Seminar on Harmonic Analysis”, Uspekhi Mat. Nauk, 52:6(318) (1997),  216 1989 39. A. A. Kirillov, V. I. Man'ko, V. F. Molchanov, I. I. Shitikov, “School-Seminar “Group Presentations in Physics””, Uspekhi Mat. Nauk, 44:6(270) (1989),  171–172 1988 40. S. G. Gindikin, V. F. Molchanov, Yu. G. Reshetnyak, I. I. Shitikov, “XII School on Operator Theory in Functional Spaces”, Uspekhi Mat. Nauk, 43:1(259) (1988),  223–224

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