Peter M. Akhmet'ev was preparing his Ph.D. in 1989 under Prof. M. M. Postnikov in the Moscow Steklov Mathematical Institute. His thesis was dedicated to geometrical problems in Algebraic L-theory, concerning splitting obstructions. After he start to work in Differential and Geometric Topology and Mathematical Physics. His main results are following. 1) Application of Stable Homotopy in Geometric Topology. For example, Hopf invariant one problem: an arbitrary map $S^n\to S^n$, $n\ne 2$ can be approximated by an embedding in $R^{2n}$ if $n\ne 1,3$, or $7$. 2) A number of results concerning applications of Immersion Theory in Pseudo-isotopy Theory [1]. 3) New geometrical proofs of classical results in Stable Homotopy groups of Spheres (join with P. J. Eccles and A. Szucs) [2], [3]. 4) A number of results in Mathematical Physics, concerning 3D analytical models of Electromagnetic Field and application of Link Theory in Magnetic Hydrodynamic.
Biography
1980–1985 Diploma in Mathematics (BS & MS), Moscow State University.
1985–1989 PhD, Steklov Institute.
2000 Dr. in Sc. (Doktorskaja dissertazija), Steklov Institute.
1994–1998 Teacher of Mathematics, Moscow International University, Leningradski, 17.
1989–2000 Scientific Researcher, Institute of Terrestrial Magnetizm, Ionosphere and Radio Wave Propagation, Russian Academy of Science.
Main publications:
P. M. Akhmetev, “Vlozheniya kompaktov, stabilnye gomotopicheskie gruppy sfer i teoriya osobennostei”, UMN, 55:3(333) (2000), 3–62; Russian Math. Surveys, 55:3 (2000), 405–462
P. Akhmet'ev and A. Szűcs, “Geometric proof of the easy part of the Hopf invariant one theorem”, Math. Slovaca, 49:1 (1999), 71–74
P. M. Akhmet'ev and P. J. Eccles, “A geometric proof of Browder's result on the vanishing of the Kervaire invariant”, The report at the Conference on 60-th Birthday of S. P. Novikov, Proc. Steklov Math. Inst., 225 (1999), 40–44
M. S. Dvornikov, P. M. Akhmet'ev, “Evolution of the magnetic field in spatially inhomogeneous axion structures”, TMF, 218:3 (2024), 601–618; Theoret. and Math. Phys., 218:3 (2024), 515–529
P. M. Akhmet'ev, Yu. V. Muranov, “Arf invariants of codimension one in a Wall group of the dihedral group”, Mat. Sb., 214:5 (2023), 3–17; Sb. Math., 214:5 (2023), 613–626
2021
3.
P. M. Akhmet'ev, “Topological meaning of the slope of the Kolmogorov spectrum of magnetic turbulence”, TMF, 209:2 (2021), 351–366; Theoret. and Math. Phys., 209:2 (2021), 1620–1632
P. M. Akhmet'ev, “Magnetic helicity flux for mean magnetic field equations”, TMF, 204:1 (2020), 130–141; Theoret. and Math. Phys., 204:1 (2020), 947–956
P. M. Akhmetiev, S. A. Melikhov, “Projected and near-projected embeddings”, Zap. Nauchn. Sem. POMI, 498 (2020), 75–104
2016
8.
P. M. Akhmet'ev, F. Yu. Popelenskii, “Local coefficients and the Herbert formula”, Fundam. Prikl. Mat., 21:6 (2016), 79–91; J. Math. Sci., 248:6 (2020), 719–727
9.
P. M. Akhmet'ev, O. D. Frolkina, “On properties of skew-framed immersions cobordism groups”, Fundam. Prikl. Mat., 21:5 (2016), 19–46; J. Math. Sci., 248:5 (2020), 524–543
P. M. Akhmet'ev, “Quadratic helicities and the energy of magnetic fields”, Trudy Mat. Inst. Steklova, 278 (2012), 16–28; Proc. Steklov Inst. Math., 278 (2012), 10–21
P. M. Akhmet'ev, V. B. Semikoz, D. D. Sokolov, “Hypermagnetic helicity flux in the nuclei of a new phase in the electroweak phase transition”, Pis'ma v Zh. Èksper. Teoret. Fiz., 91:5 (2010), 233–236; JETP Letters, 91:5 (2010), 215–218
P. M. Akhmet'ev, O. V. Kunakovskaya, “Integral Formula for a Generalized Sato–Levine Invariant in Magnetic Hydrodynamics”, Mat. Zametki, 85:4 (2009), 524–537; Math. Notes, 85:4 (2009), 503–514
P. M. Akhmet'ev, O. V. Kunakovskaya, V. A. Kutvitskii, “Remark on the dissipation of the magnetic helicity integral”, TMF, 158:1 (2009), 150–160; Theoret. and Math. Phys., 158:1 (2009), 125–134
P. M. Akhmet'ev, “Geometric approach to stable homotopy groups of spheres. Kervaire invariants. II”, Fundam. Prikl. Mat., 13:8 (2007), 17–41; J. Math. Sci., 159:6 (2009), 761–776
15.
P. M. Akhmet'ev, “Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants”, Fundam. Prikl. Mat., 13:8 (2007), 3–15; J. Math. Sci., 159:6 (2009), 753–760
P. M. Akhmet'ev, D. Repovš, M. Cencelj, “Some Algebraic Properties of Cerf Diagrams of One-Parameter Function Families”, Funktsional. Anal. i Prilozhen., 39:3 (2005), 1–13; Funct. Anal. Appl., 39:3 (2005), 165–174
P. M. Akhmet'ev, A. V. Khokhlov, “Classification of Harmonic Functions in the Exterior of the Unit Ball”, Mat. Zametki, 75:2 (2004), 182–191; Math. Notes, 75:2 (2004), 166–174
P. M. Akhmet'ev, “A Remark on the Realization of Mappings of the 3-Dimensional Sphere into Itself”, Trudy Mat. Inst. Steklova, 247 (2004), 10–14; Proc. Steklov Inst. Math., 247 (2004), 4–8
P. M. Akhmet'ev, “$K_2$ for the simplest integral group rings and topological applications”, Mat. Sb., 194:1 (2003), 23–30; Sb. Math., 194:1 (2003), 21–29
P. M. Akhmet'ev, I. Maleshich, D. Repovš, “On the Euler characteristic of multiple selfintersection points of immersed manifolds”, Sibirsk. Mat. Zh., 44:2 (2003), 256–262; Siberian Math. J., 44:2 (2003), 208–212
2002
21.
P. M. Akhmet'ev, D. Repovš, I. Maleshich, “On Milnor's Invariants of 4-Component Links”, Mat. Zametki, 71:4 (2002), 496–507; Math. Notes, 71:4 (2002), 455–463
P. M. Akhmet'ev, I. Maleshich, D. Repovš, “A formula for the generalized Sato–Levine invariant”, Mat. Sb., 192:1 (2001), 3–12; Sb. Math., 192:1 (2001), 1–10
P. M. Akhmet'ev, “A higher-order analog of the helicity number for a pair of divergent-free vector fields”, Zap. Nauchn. Sem. POMI, 279 (2001), 15–23; J. Math. Sci. (N. Y.), 119:1 (2004), 5–9
P. M. Akhmet'ev, “Embedding of compacta, stable homotopy groups of spheres, and singularity theory”, Uspekhi Mat. Nauk, 55:3(333) (2000), 3–62; Russian Math. Surveys, 55:3 (2000), 405–462
P. M. Akhmet'ev, S. A. Melikhov, “On isotopic realizability of continuous mappings”, Zap. Nauchn. Sem. POMI, 267 (2000), 53–87; J. Math. Sci. (N. Y.), 113:6 (2003), 759–776
P. M. Akhmet'ev, “Generalized Daverman's problem”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 68 (1999), 5–15; J. Math. Sci. (New York), 105:2 (2001), 1813–1818
P. M. Akhmet'ev, P. J. Eccles, “A Geometrical Proof of Browder's Result on the Vanishing of the Kervaire Invariant”, Trudy Mat. Inst. Steklova, 225 (1999), 46–51; Proc. Steklov Inst. Math., 225 (1999), 40–44
P. M. Akhmet'ev, D. Repovš, “A Generalization of the Sato–Levine Invariant”, Trudy Mat. Inst. Steklova, 221 (1998), 69–80; Proc. Steklov Inst. Math., 221 (1998), 60–70
P. M. Akhmet'ev, Yu. V. Muranov, “Obstructions to splitting manifolds with infinite fundamental group”, Mat. Zametki, 60:2 (1996), 163–175; Math. Notes, 60:2 (1996), 121–129
P. M. Akhmet'ev, “$\operatorname{Prem}$-mappings, triple self-intersection points of oriented surfaces, and the Rokhlin signature theorem”, Mat. Zametki, 59:6 (1996), 803–810; Math. Notes, 59:6 (1996), 581–585
P. M. Akhmet'ev, “On isotopic and discrete realizations of maps of an $n$-dimensional sphere in Euclidean space”, Mat. Sb., 187:7 (1996), 3–34; Sb. Math., 187:7 (1996), 951–980
P. M. Akhmet'ev, “Solution of the problem of realizing a mapping of an $n$-sphere in a Euclidean $2n$-space”, Trudy Mat. Inst. Steklova, 212 (1996), 37–45; Proc. Steklov Inst. Math., 212 (1996), 32–39
P. M. Akhmet'ev, “An elementary proof of the Freedman immersion theorem”, Algebra i Analiz, 7:5 (1995), 93–100; St. Petersburg Math. J., 7:5 (1996), 749–754
P. M. Akhmet'ev, “Smooth immersion of manifolds of small dimension. II. Cobordism group of critical points of multiparameter families of functions”, Mat. Sb., 186:12 (1995), 37–62; Sb. Math., 186:12 (1995), 1727–1751
P. M. Akhmet'ev, “Splitting homotopy equivalences along a one-sided submanifold of codimension one”, Izv. Akad. Nauk SSSR Ser. Mat., 51:2 (1987), 211–241; Math. USSR-Izv., 30:2 (1988), 185–215
P. M. Akhmet'ev, Yu. B. Rudyak, “Behavior of Bockstein homomorphisms of infinite cellular spaces”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1985, no. 5, 31–33
2021
38.
O. V. Anashkin, P. M. Akhmet'ev, D. V. Balandin, M. K. Barinova, I. V. Boykov, A. N. Bezdenezhnyh, V. N. Belykh, P. A. Vel'misov, I. Yu. Vlasenko, O. E. Galkin, S. Yu. Galkina, V. K. Gorbunov, S. D. Glyzin, S. V. Gonchenko, A. S. Gorodetski, E. V. Gubina, E. Ya. Gurevich, A. A. Davydov, L. S. Efremova, R. V. Zhalnin, A. Yu. Zhirov, E. V. Zhuzhoma, N. I. Zhukova, S. Kh. Zinina, Yu. S. Ilyashenko, N. V. Isaenkova, A. O. Kazakov, A. V. Klimenko, S. A. Komech, Yu. A. Kordyukov, V. E. Kruglov, E. V. Kruglov, E. B. Kuznetsov, S. K. Lando, Yu. A. Levchenko, L. M. Lerman, S. I. Maksimenko, M. I. Malkin, D. S. Malyshev, V. K. Mamaev, T. Ph. Mamedova, V. S. Medvedev, T. V. Medvedev, D. I. Mints, T. M. Mitryakova, A. D. Morozov, A. I. Morozov, E. V. Nozdrinova, E. N. Pelinovsky, Ya. B. Pesin, A. S. Pikovsky, S. Yu. Pilyugin, G. M. Polotovsky, O. V. Pochinka, I. D. Remizov, P. E. Ryabov, A. S. Skripchenko, A. V. Slunyaev, S. V. Sokolov, L. A. Sukharev, E. A. Talanova, V. A. Timorin, S. B. Tikhomirov, V. F. Tishkin, D. V. Treschev, D. V. Turaev, N. G. Chebochko, E. E. Chilina, P. A. Shamanaev, D. D. Shubin, E. I. Yakovlev, “To the 75th anniversary of Vyacheslav Zigmundovich Grines”, Zhurnal SVMO, 23:4 (2021), 472–476
2009
39.
P. M. Akhmet'ev, “Corrections to the paper “Geometric Approach to Stable Homotopy Groups of Spheres. The Adams–Hopf Invariants” (Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, 3–15 (2007))”, Fundam. Prikl. Mat., 15:5 (2009), 211; J. Math. Sci., 172:5 (2011), 759
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
Рациональный гомотопический тип для представляющего пространства стабильнооснащенных кобордизмов П. М. Ахметьев International Scientific Conference KOLMOGOROV READINGS – IX. General Control Problems and their Applications (GCP–2020), dedicated to the 70-th birth anniversary of Alexander Ivanovich Bulgakov and to the 90-th anniversary of the Institute of Mathematics, Physics and Information
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