Akhmet'ev, Petr Mikhailovich

Statistics Math-Net.Ru
Total publications: 36
Scientific articles: 34
Presentations: 33

Number of views:
This page:3567
Abstract pages:8990
Full texts:3543
Doctor of physico-mathematical sciences (2000)
Speciality: 01.01.04 (Geometry and topology)
Birth date: 2.02.1963
E-mail: ,
Keywords: stable homotopy groups of spheres, singularities theory, algebraic K-theory, induction equation.
UDC: 515.164.6, 516.16, 515.164, 515.1, 514, 515.142, 515.163.6, 515.126.2, 515.162.8, 515.168.3, 517.958, 515.12.124.552, 515.146.23
MSC: 19-XX, 57-XX, 76-XX


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.


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:
  1. P. M. Akhmetev, “Vlozheniya kompaktov, stabilnye gomotopicheskie gruppy sfer i teoriya osobennostei”, UMN, 55:3(333) (2000), 3–62  mathnet  mathscinet  zmath  adsnasa; Russian Math. Surveys, 55:3 (2000), 405–462  crossref  mathscinet  adsnasa  isi
  2. 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  mathscinet
  3. 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  mathscinet
List of publications on Google Scholar
List of publications on ZentralBlatt

Publications in Math-Net.Ru
1. P. M. Akhmet'ev, “On the properties of the cobordism group of stably-framed immersions in codimension $k$”, Chebyshevskii Sb., 21:2 (2020),  26–36  mathnet
2. P. M. Akhmet'ev, “Knot Invariants in Geodesic Flows”, Trudy MIAN, 308 (2020),  50–64  mathnet  mathscinet  elib; Proc. Steklov Inst. Math., 308 (2020), 42–55  isi  scopus
3. P. M. Akhmet'ev, “Magnetic helicity flux for mean magnetic field equations”, TMF, 204:1 (2020),  130–141  mathnet  mathscinet  elib; Theoret. and Math. Phys., 204:1 (2020), 947–956  isi  scopus
4. P. M. Akhmetiev, S. A. Melikhov, “Projected and near-projected embeddings”, Zap. Nauchn. Sem. POMI, 498 (2020),  75–104  mathnet
5. P. M. Akhmet'ev, F. Yu. Popelenskii, “Local coefficients and the Herbert formula”, Fundam. Prikl. Mat., 21:6 (2016),  79–91  mathnet
6. P. M. Akhmet'ev, O. D. Frolkina, “On properties of skew-framed immersions cobordism groups”, Fundam. Prikl. Mat., 21:5 (2016),  19–46  mathnet
7. P. M. Akhmet'ev, “Quadratic helicities and the energy of magnetic fields”, Trudy MIAN, 278 (2012),  16–28  mathnet  mathscinet  elib; Proc. Steklov Inst. Math., 278 (2012), 10–21  isi  elib  scopus
8. 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  mathnet; JETP Letters, 91:5 (2010), 215–218  isi  scopus
9. 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  mathnet  mathscinet  zmath; Math. Notes, 85:4 (2009), 503–514  isi  scopus
10. 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  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 158:1 (2009), 125–134  isi  scopus
11. P. M. Akhmet'ev, “Geometric approach to stable homotopy groups of spheres. Kervaire invariants. II”, Fundam. Prikl. Mat., 13:8 (2007),  17–41  mathnet  mathscinet  zmath; J. Math. Sci., 159:6 (2009), 761–776  scopus
12. P. M. Akhmet'ev, “Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants”, Fundam. Prikl. Mat., 13:8 (2007),  3–15  mathnet  mathscinet  zmath; J. Math. Sci., 159:6 (2009), 753–760  scopus
13. 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  mathnet  mathscinet  zmath  elib; Funct. Anal. Appl., 39:3 (2005), 165–174  isi  scopus
14. 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  mathnet  mathscinet  zmath  elib; Math. Notes, 75:2 (2004), 166–174  isi
15. P. M. Akhmet'ev, “A Remark on the Realization of Mappings of the 3-Dimensional Sphere into Itself”, Trudy MIAN, 247 (2004),  10–14  mathnet  mathscinet  zmath; Proc. Steklov Inst. Math., 247 (2004), 4–8
16. P. M. Akhmet'ev, “$K_2$ for the simplest integral group rings and topological applications”, Mat. Sb., 194:1 (2003),  23–30  mathnet  mathscinet  zmath  elib; Sb. Math., 194:1 (2003), 21–29  isi  scopus
17. 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  mathnet  mathscinet  zmath; Siberian Math. J., 44:2 (2003), 208–212  isi
18. P. M. Akhmet'ev, D. Repovš, I. Maleshich, “On Milnor's Invariants of 4-Component Links”, Mat. Zametki, 71:4 (2002),  496–507  mathnet  mathscinet  zmath; Math. Notes, 71:4 (2002), 455–463  isi  scopus
19. P. M. Akhmet'ev, I. Maleshich, D. Repovš, “A formula for the generalized Sato–Levine invariant”, Mat. Sb., 192:1 (2001),  3–12  mathnet  mathscinet  zmath; Sb. Math., 192:1 (2001), 1–10  isi  scopus
20. 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  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 119:1 (2004), 5–9
21. P. M. Akhmet'ev, “Embedding of compacta, stable homotopy groups of spheres, and singularity theory”, Uspekhi Mat. Nauk, 55:3(333) (2000),  3–62  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 55:3 (2000), 405–462  isi  scopus
22. P. M. Akhmet'ev, S. A. Melikhov, “On isotopic realizability of continuous mappings”, Zap. Nauchn. Sem. POMI, 267 (2000),  53–87  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 113:6 (2003), 759–776
23. P. M. Akhmet'ev, “Generalized Daverman's problem”, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 68 (1999),  5–15  mathnet  mathscinet  zmath; J. Math. Sci. (New York), 105:2 (2001), 1813–1818
24. P. M. Akhmet'ev, P. J. Eccles, “A Geometrical Proof of Browder's Result on the Vanishing of the Kervaire Invariant”, Trudy MIAN, 225 (1999),  46–51  mathnet  mathscinet  zmath; Proc. Steklov Inst. Math., 225 (1999), 40–44
25. P. M. Akhmet'ev, D. Repovš, “A Generalization of the Sato–Levine Invariant”, Trudy MIAN, 221 (1998),  69–80  mathnet  mathscinet  zmath; Proc. Steklov Inst. Math., 221 (1998), 60–70
26. P. M. Akhmet'ev, Yu. V. Muranov, “Obstructions to splitting manifolds with infinite fundamental group”, Mat. Zametki, 60:2 (1996),  163–175  mathnet  mathscinet  zmath; Math. Notes, 60:2 (1996), 121–129  isi
27. 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  mathnet  mathscinet  zmath; Math. Notes, 59:6 (1996), 581–585  isi
28. 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  mathnet  mathscinet  zmath; Sb. Math., 187:7 (1996), 951–980  isi  scopus
29. P. M. Akhmet'ev, “Solution of the problem of realizing a mapping of an $n$-sphere in a Euclidean $2n$-space”, Trudy MIAN, 212 (1996),  37–45  mathnet  mathscinet  zmath; Proc. Steklov Inst. Math., 212 (1996), 32–39
30. P. M. Akhmet'ev, “An elementary proof of the Freedman immersion theorem”, Algebra i Analiz, 7:5 (1995),  93–100  mathnet  mathscinet  zmath; St. Petersburg Math. J., 7:5 (1996), 749–754
31. 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  mathnet  mathscinet  zmath; Sb. Math., 186:12 (1995), 1727–1751  isi
32. P. M. Akhmet'ev, “Smooth immersions of manifolds of low dimension”, Mat. Sb., 185:10 (1994),  3–26  mathnet  mathscinet  zmath; Russian Acad. Sci. Sb. Math., 83:1 (1995), 155–176  isi
33. 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  mathnet  mathscinet  zmath; Math. USSR-Izv., 30:2 (1988), 185–215
34. P. M. Akhmet'ev, Yu. B. Rudyak, “Behavior of Bockstein homomorphisms of infinite cellular spaces”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1985, 5,  31–33  mathnet  mathscinet  zmath

35. 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  mathnet  mathscinet; J. Math. Sci., 172:5 (2011), 759  scopus
36. 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  mathnet  mathscinet  zmath; Russian Math. Surveys, 53:2 (1998), 431–433  isi

Presentations in Math-Net.Ru
1. $M_5$ и $M_3$-invariants for oriented links
P. M. Akhmet'ev
Geometric Topology Seminar
March 19, 2021 17:00   
2. A formula for the $M_3$-invariant of oriented links
P. M. Akhmet'ev
Geometric Topology Seminar
February 26, 2021 17:00   
3. О группах кобордизма стабильно-оснащенных в коразмерности $k$ погружений и об их применении для геометризации теоремы Адамса об инвариантах Хопфа
P. M. Akhmet'ev
Noncommutative geometry and topology
November 5, 2020 17:45   
4. 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 Technologies of Derzhavin Tambov State University
October 12, 2020 15:40   
5. A geometric proof and a strengthening of the easy part of the Adams' theorem on the Hopf invariant
P. M. Akhmet'ev
Geometric Topology Seminar
September 11, 2020 17:00   
6. "Дисперсия коэффициентов зацеплений магнитных линий и уравнение двухмасштабного динамо"
P. M. Akhmet'ev
Seminar on analytic theory of differential equations
May 20, 2020 14:00
7. Invariants of winding numbers in dynamics of flux lines and its visualization
P. M. Akhmet'ev
Geometric Topology Seminar
December 18, 2019 17:00
8. Поток моментов магнитной спиральности для уравнения среднего магнитного поля
P. M. Akhmet'ev
Seminar on analytic theory of differential equations
October 23, 2019 14:30
9. О группах стабильно-оснащенного регулярного кобордизма
P. M. Akhmet'ev
Noncommutative geometry and topology
September 19, 2019 16:45
10. On a new variation of the Arf-invariant, describing a generator of the 2-component of the stable homotopy group of spheres $\Pi_7$
P. M. Akhmet'ev
Geometric Topology Seminar
September 4, 2019 17:00
11. Field line helicity and the second moment of the asymptotic linking number of magnetic lines
P. M. Akhmet'ev
Cohomological geometry of differential equations
May 15, 2019 19:20
12. Invariants of knots in geodesic flows
P. M. Akhmet'ev
Dynamics in Siberia - 2019
March 2, 2019 10:00
13. Асимптотические инварианты зацеплений траекторий в геодезических потоках
P. M. Akhmet'ev
Seminar on analytic theory of differential equations
January 30, 2019 14:30
14. Asymptotic invariants of links
P. M. Akhmet'ev
Geometric Topology Seminar
January 25, 2019 17:00
15. Linking coefficients for geodesic flow and applications for magnetic equilibrium
P. M. Akhmet'ev
Dynamics in Siberia - 2018
March 3, 2018 10:20
16. Вычисление коэффициентов зацепления магнитных линий для некоторых специальных конфигураций МГД равновесия
P. M. Akhmet'ev
Seminar on analytic theory of differential equations
January 31, 2018 14:00
17. Гиперкватернионный Арф-инвариант
P. M. Akhmet'ev
Knots and Representation Theory
September 26, 2017 18:30
18. Quadratic Helicity in MHD
P. M. Akhmet'ev
Geometric Topology Seminar
September 21, 2017 14:00
19. A combinatorial formula for M-invariant
P. M. Akhmet'ev
Geometric Topology Seminar
August 3, 2017 14:00
20. Доказательство теоремы Смейла о погружениях k-сферы в $R^n$, $k<n$, и принцип плотности Хирша
P. M. Akhmet'ev
Noncommutative geometry and topology
March 23, 2017 16:45
21. Теоремы Экклза об инвариантах Хопфа и Кервера в стабильных гомотопических группах сфер и ее обобщение
P. M. Akhmet'ev
Noncommutative geometry and topology
March 9, 2017 16:45
22. О формуле Герберта с локальными коэффициентами и ее приложениях
F. Yu. Popelenskii, P. M. Akhmet'ev
Noncommutative geometry and topology
February 16, 2017 16:45
23. Casson-type asymptotic invariants of knots in the Poincaré sphere
Petr M. Akhmet'ev
International Conference “Anosov Systems and Modern Dynamics” dedicated to the 80th anniversary of Dmitry Anosov
December 22, 2016 17:00
24. 'On polynomial helicities in MHD"
P. M. Akhmet'ev
Seminar on analytic theory of differential equations
October 26, 2016 14:30   
25. Критерий разрешимости почти нормальной монодромной пары в одномерной топологической теории Галуа
P. M. Akhmet'ev
Noncommutative geometry and topology
May 19, 2016 16:45
26. О проблеме Кервера
P. M. Akhmet'ev
Noncommutative geometry and topology
February 25, 2016 16:45
27. О группах кобордизмов скоснащенных погружений
P. M. Akhmet'ev, O. D. Frolkina
Knots and Representation Theory
March 24, 2015 18:30
28. Stable homotopy groups of spheres as differential relations
P. M. Akhmet'ev
International Conference on Differential Equations and Dynamical Systems
July 6, 2014 12:00
29. Об инварианте Хопфа 4-кос на сфере
P. M. Akhmet'ev
Knots and Representation Theory
February 11, 2014 18:30
30. О непогружаемости ${\mathbb R}P^{10}$ в ${\mathbb R}^{15}$
P. M. Akhmet'ev, O. D. Frolkina
Knots and Representation Theory
October 29, 2013 18:30
31. On applications of homotopy groups of spheres in three-dimensional problems
P. M. Akhmet'ev
Algebraic topology and its applications. Postnikov memorial seminar
October 22, 2013 16:45
32. On non-immersibility of $RP^{10}$ to $R^{15}$
P. M. Akhmet'ev, O. D. Frolkina
Geometric Topology Seminar
June 14, 2012 14:30
33. On higher asymptotic analogs of the helicity invariant in MHD
P. M. Akhmet'ev
Knots and Representation Theory
May 3, 2011 18:30

Contact us:
 Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2021