operads; homology and homotopy groups; spectral sequence.
Subject:
The field of scientific interests includes Operad Theory and its applications to determine the homotopy type of topological spaces, describing the Adams spectral sequence, calculation the homology groups of iterated loop spaces and the homotopy groups of spheres. The operad theory for the category of chain complexes is constructed. It is proved that the $E_\infty$-coalgebra structure on the singular chain complex of a 1-connected topological space determines the weak homotopy type of the spaces. The $E_2$- term of the Adams spectral sequence is described. The homology of iterated loop spaces over the real and complex projective spaces are calculated. The $E_\infty$- structure on the Adams spectral sequence is described. Some higher differentials of this spectral sequence are calculated.
Biography
Graduated from Faculty of Mathematics of the Moscow State Pedagogical University (MSPU) in 1967 Ph.D. thesis was defended in 1976. D.Sci. thesis was defended in 1986. A list of my works contains more than 60 titles.
Main publications:
Smirnov V. A., “A general algebraic approach to the problem of describing the second term of the Adams spectral sequence of stable homotopy groups of spheres”, Tensor and Vector Analisis, Gordon and Breach Science Publishers, Amsterdam, 1998, 213–250
Smirnov V. A., “Bioperady i bialgebry Khopfa v teorii kobordizmov”, Matematicheskie zametki, 65:2 (1999), 270–279
Smirnov V. A., “Algebra Daiera–Lashofa i algebra Stinroda dlya obobschennykh gomologii i kogomologii”, Matematicheskii sbornik, 190:12 (1999), 93–128
Smirnov V. A., “$A_\infty$-struktury i funktor $\mathscr D$”, Izvestiya RAN, ser. matem., 64:5 (2000), 148–162
Smirnov V. A., Simplicial and operad methods in algebraic topology, Transl. Math. Monogr., 198, American Mathematical Society, Providence, RI, 2001, 235 pp.
V. A. Smirnov, “Bott's Periodicity Theorem and Differentials of the Adams Spectral Sequence of Homotopy Groups of Spheres”, Mat. Zametki, 84:5 (2008), 763–771; Math. Notes, 84:5 (2008), 710–717
V. A. Smirnov, “The $A_\infty$-structures and differentials of the Adams spectral sequence”, Izv. RAN. Ser. Mat., 66:5 (2002), 193–224; Izv. Math., 66:5 (2002), 1057–1086
V. A. Smirnov, “The Dyer–Lashof algebra and the Steenrod algebra for generalized homology and cohomology”, Mat. Sb., 190:12 (1999), 93–128; Sb. Math., 190:12 (1999), 1807–1842
V. A. Smirnov, S. V. Kuznetsova, I. V. Mayorova, “Description of the cohomology of Banach algebras and locally convex algebras in the language of $A_\infty$-structures”, Izv. RAN. Ser. Mat., 62:4 (1998), 155–172; Izv. Math., 62:4 (1998), 789–805
V. A. Smirnov, “Lie algebras over operads and their applications in homotopy theory”, Izv. RAN. Ser. Mat., 62:3 (1998), 121–154; Izv. Math., 62:3 (1998), 549–580
V. A. Smirnov, “The $E_2$ term of the spectral sequence of unstable homotopy groups of topological spaces”, Mat. Zametki, 64:5 (1998), 792–796; Math. Notes, 64:5 (1998), 684–688
13.
V. A. Smirnov, “Homotopy type and $A_\infty$-group structure”, Mat. Sb., 189:10 (1998), 135–144; Sb. Math., 189:10 (1998), 1563–1572
V. A. Smirnov, “Description of the homology of groups and algebras”, Izv. RAN. Ser. Mat., 61:3 (1997), 203–212; Izv. Math., 61:3 (1997), 663–671
15.
V. A. Smirnov, “Calculation of the $E_\infty$-structure on the Milnor coalgebra”, Mat. Zametki, 62:3 (1997), 479–480; Math. Notes, 62:3 (1997), 399–401
16.
V. A. Smirnov, “$E_\infty$-structures on homotopy groups”, Mat. Zametki, 61:1 (1997), 152–156; Math. Notes, 61:1 (1997), 127–130
V. A. Smirnov, “Description of stable homotopy groups of spheres in the language of $A_\infty$-algebras”, Uspekhi Mat. Nauk, 51:1(307) (1996), 171–172; Russian Math. Surveys, 51:1 (1996), 171–172
1994
18.
V. A. Smirnov, “Homology of $B$-constructions and co-$B$-constructions”, Izv. RAN. Ser. Mat., 58:4 (1994), 80–96; Russian Acad. Sci. Izv. Math., 45:1 (1995), 79–95
V. A. Smirnov, “Secondary operations in the homology of the operad $E$”, Izv. RAN. Ser. Mat., 56:2 (1992), 449–468; Russian Acad. Sci. Izv. Math., 40:2 (1993), 425–442
V. A. Smirnov, V. A. Sheikhman, “Continuation of homogeneous functionals with a given convexity”, Mat. Zametki, 50:5 (1991), 90–96; Math. Notes, 50:5 (1991), 1157–1161
V. A. Smirnov, “Homology of symmetric products”, Mat. Zametki, 49:1 (1991), 104–113; Math. Notes, 49:1 (1991), 73–80
1990
24.
V. A. Smirnov, Chan Khuen, “On the funktor Ext in the category of linear topological spases”, Izv. Akad. Nauk SSSR Ser. Mat., 54:1 (1990), 188–200; Math. USSR-Izv., 36:1 (1991), 199–210
V. A. Smirnov, “On the chain complex of an iterated loop space”, Izv. Akad. Nauk SSSR Ser. Mat., 53:5 (1989), 1108–1119; Math. USSR-Izv., 35:2 (1990), 445–455
V. A. Smirnov, “Twisted tensor products and strong homotopy”, Dokl. Akad. Nauk SSSR, 222:2 (1975), 288–290
1974
37.
V. A. Smirnov, “Twisted tensor products and Hirsch's theory”, Dokl. Akad. Nauk SSSR, 219:2 (1974), 294–296
2015
38.
I. Smirnova, V. Smirnov, “Каскады из правильных многогранников”, Kvant, 2015, no. 3, 32–33
2006
39.
I. Smirnova, V. Smirnov, “Вписанные и описанные многоугольники”, Kvant, 2006, no. 4, 31
1998
40.
D. V. Anosov, P. M. Akhmet'ev, A. A. Bolibrukh, V. M. Buchstaber, V. A. Kolosov, A. A. Mal'tsev, S. P. Novikov, V. A. Smirnov, A. V. Chernavskii, “Mikhail Mikhailovich Postnikov (on his 70th birthday)”, Uspekhi Mat. Nauk, 53:2(320) (1998), 183–184; Russian Math. Surveys, 53:2 (1998), 431–433