Riemann–Roch type theorems,
oriented cohomology theories,
linear algebraic groups,
a conjecture of Grothendieck,
Characteristic classes and numbers,
principal homogeneous spaces.
Algebraic K-groups of twisted forms of flag varieties and principal homogeneous spaces over semi-simple algebraic groups.Index reduction formulas for skew-fields (with A. Merkurjev, A. Wadsworth) are proved. Gersten conjecture is proved (equi-characteristic case). Purity problem for quadratic forms is solved (with M. Ojanguren). A concept of oriented cohomology theory on algebraic varieties is introduced, Riemann–Roch theorem is proved for a ring morphism of oriented theories (with A. Smirnov). An explicite formulae for the Todd genus of such a ring morphism is found. Suslin's rigidity theorem is extended (S. Yagunov) to any oriented theory.
Graded in 1981 by Department of Mathematic and Mechanic Leningrad State University. Ph.D. Theses got in 1984 from Leningrad Branch of Steklov Mathematical Institute. Defended Habilitation in 1996. Head of a laboratory in Steklov Mathematical Institute (St. Petersburg) since 1999.
Since 1995 till 1999 I was applying the technic of V. Voevodsky's article "Homology of schemes. II" to a conjecture of Grothendieck (with Suslin and Ojanguren) and to purity problems (with Ojanguren). Since 1999 I began to transplant (with A. Smirnov) the concept of oriented cohomology theory (to algebra from topology). The notion has been fixed, for a ring morphism of oriented theories an explicite formulae for Todd genus is given and a Riemann–Roch theorem is proved. With S. Yagunov the rigidity theorem of A. Suslin was extended to an oriented theory. Now a variant of Poincare duality is working out.
Panin I. A. On the algebraic $K$-theory of twisted flag varieties // K-Theory 8 (1994), no. 6, 541–585.
Merkurjev A. S., Panin I. A., Wadsworth A. R. Index reduction formulas for twisted flag varieties. I // K-Theory 10 (1996), no. 6, 517–596.
Panin I. A. Splitting principle and K-theory of simply connected semisimple algebraic groups. (Russian) Algebra i Analiz 10 (1998), no. 1, 88–131; translation in St. Petersburg Math. J. 10 (1999), no. 1, 69–101.
Ojanguren M., Panin I. A purity theorem for the Witt group // Ann. Sci. Ecole Norm. Sup. (4) 32 (1999), no. 1, 71–86.
Panin I., Smirnov A. Push-forwards in oriented cohomology theories of algebraic varieties. Preprint http://www.math.uiuc.edu/K-theory/0459/index.html.
To the anniversary of Sergei Vladimirovich Vostokov D. G. Benua, M. V. Bondarko, N. A. Vavilov, M. A. Vsemirnov, A. I. Generalov, N. L. Gordeev, I. B. Zhukov, G. A. Leonov, B. B. Lur'e, I. A. Panin, A. L. Smirnov, I. B. Fesenko, A. V. Yakovlev Algebra i Analiz, 27:6 (2015), 3–5
Anatolii Vladimirovich Yakovlev N. A. Vavilov, S. V. Vostokov, A. I. Generalov, N. L. Gordeev, B. B. Lur'e, A. S. Merkur'ev, V. V. Ishkhanov, I. A. Panin, A. I. Skopin, A. A. Suslin Zap. Nauchn. Sem. POMI, 272 (2000), 5–13