Calculation of $K$-groups for the Eilenberg–McLane complexes. The sufficient and nesessary conditions for existence of elements of infinite filtration in complex $K$-theory was found in the terms of the Chern character. Calculation of the bordism groups of almost-complex manifolds with action of the group $Z_p$ in the terms of the local properties of stationary points. Construction of symmetric signature for nonsimply-connected manifolds as an element of the Wall group of the fundamental group of the manifold. The invariance of the symmetric signature both under the homotopy equivalence and bordisms was established. The formula for the obstruction for existence of a surgery of normal map to a homotopy equivalence was established modulo torsion. Theory of Fredholm representations of discrete groups and the proof of the Novikov conjecture for non-positivly curved manifolds. The index theory of elliptic operators over $C^*$-algebras and the Hirzebruch formula for non-simply connected manifolds. Proof of integrability of dynamic systems on the Lie groups and symmetric spaces. Proof the theorem that existence of a finite dimensional Lie algebra $F$ of the first integrals for a Hamiltonian dynamical system on a manifold $M$ such that $\dim M=\dim F+\index F$ implies the completely integrability of the dynamical system. Construction of homotopical invariants of the Lagrangian manifolds which are responsible for existence of the complex Maslov operator on the Lagrangian manifolds. Analitical solution of the linear differential game of pursuit. Estimating of number of stationary solutions of a differential stochastic equation. The theory analytical torsion over $C^*$-algebras was developed. Relations between asymptotic and Fredholm representations were established. Local combinatorial version of the Hirzebruch formula was developed.

Graduated from Departmet of Mechanics and Mathematics of M. V. Lomonosov Moscow State University (MSU) in 1965 (Chief of Higher Geometry and Topology). Ph.D. thesis was defended in 1968. D.Sci thesis was defended in 1973. A list of my papers contains more than 100 titles. Since 1968 I and my pupils have led the research seminars at MSU on geometry and topology.

In 1971 I was awarded the prize of the Moscow Mathematical Society for the paper on $K$-Theory of infinite dimensional $CW$-complexes (jointly with V. M. Buhshtaber). In 1996 I was awarded the State Prize of Russian federation in the area of sciences and technique for a cicle of papers "Investigations of invariants of smooth manifolds and Hamiltonian dynamic systems" (jointly with A. T. Fomenko). In 2001 I was awarded the M. V. Lomonosov prize of Moscow State University for pedagogical activity.

1. A. S. Mischenko, I. M. Gelfand, “Kvadratichnye formy nad kommutativnymi gruppovymi koltsami i $K$-teoriya”, Funkts. analiz i ego prilozheniya, 3:4 (1969), 28–33      
2. A. S. Mischenko, “Gomotopicheskie invarianity neodnosvyaznykh mnogoobrazii. 1. Ratsionalnye invarianty”, Izvestiya AN SSSR. Ser. matem., 34:3 (1970), 501–514      
3. A. S. Mischenko, “Beskonechnomernye predstavleniya diskretnykh grupp i vysshie signatury”, Izvestiya AN SSSR. Ser. matem., 38:1 (1974), 81–106      
4. A. S. Mischenko, A. T. Fomenko, “Obobschennyi metod Liuvillya integrirovaniya gamiltonovykh sistem”, Funkts. analiz i ego prilozheniya, 12:2 (1978), 46–56      
5. A. S. Mischenko, “Lokalnaya kombinatornaya formula Khirtsebrukha”, Trudy MIAN, 224, 1999, 249–263    

• M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
• Steklov Mathematical Institute, Russian Academy of Sciences