Akhmet'ev Petr Mikhailovich

Statistics Math-Net.Ru
Total publications: 30
Scientific articles: 28
Presentations: 19

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Doctor of physico-mathematical sciences (2000)
Speciality: 01.01.04 (Geometry and topology)
Birth date: 2.02.1963
Phone: +7 (4967) 51 09 12
Fax: +7 (495) 334 01 24
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. On properties of skew-framed immersions cobordism groups
P. M. Akhmet'ev, O. D. Frolkina
Fundam. Prikl. Mat., 21:5 (2016),  19–46
2. Quadratic helicities and the energy of magnetic fields
P. M. Akhmet'ev
Tr. Mat. Inst. Steklova, 278 (2012),  16–28
3. Hypermagnetic helicity flux in the nuclei of a new phase in the electroweak phase transition
P. M. Akhmet'ev, V. B. Semikoz, D. D. Sokolov
Pis'ma v Zh. Èksper. Teoret. Fiz., 91:5 (2010),  233–236
4. Integral Formula for a Generalized Sato–Levine Invariant in Magnetic Hydrodynamics
P. M. Akhmet'ev, O. V. Kunakovskaya
Mat. Zametki, 85:4 (2009),  524–537
5. Remark on the dissipation of the magnetic helicity integral
P. M. Akhmet'ev, O. V. Kunakovskaya, V. A. Kutvitskii
TMF, 158:1 (2009),  150–160
6. Geometric approach to stable homotopy groups of spheres. Kervaire invariants. II
P. M. Akhmet'ev
Fundam. Prikl. Mat., 13:8 (2007),  17–41
7. Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants
P. M. Akhmet'ev
Fundam. Prikl. Mat., 13:8 (2007),  3–15
8. Some Algebraic Properties of Cerf Diagrams of One-Parameter Function Families
P. M. Akhmet'ev, D. Repovš, M. Cencelj
Funktsional. Anal. i Prilozhen., 39:3 (2005),  1–13
9. Classification of Harmonic Functions in the Exterior of the Unit Ball
P. M. Akhmet'ev, A. V. Khokhlov
Mat. Zametki, 75:2 (2004),  182–191
10. A Remark on the Realization of Mappings of the 3-Dimensional Sphere into Itself
P. M. Akhmet'ev
Tr. Mat. Inst. Steklova, 247 (2004),  10–14
11. $K_2$ for the simplest integral group rings and topological applications
P. M. Akhmet'ev
Mat. Sb., 194:1 (2003),  23–30
12. On the Euler characteristic of multiple selfintersection points of immersed manifolds
P. M. Akhmet'ev, I. Maleshich, D. Repovš
Sibirsk. Mat. Zh., 44:2 (2003),  256–262
13. On Milnor's Invariants of 4-Component Links
P. M. Akhmet'ev, D. Repovš, I. Maleshich
Mat. Zametki, 71:4 (2002),  496–507
14. A formula for the generalized Sato–Levine invariant
P. M. Akhmet'ev, I. Maleshich, D. Repovš
Mat. Sb., 192:1 (2001),  3–12
15. A higher-order analog of the helicity number for a pair of divergent-free vector fields
P. M. Akhmet'ev
Zap. Nauchn. Sem. POMI, 279 (2001),  15–23
16. Embedding of compacta, stable homotopy groups of spheres, and singularity theory
P. M. Akhmet'ev
Uspekhi Mat. Nauk, 55:3(333) (2000),  3–62
17. On isotopic realizability of continuous mappings
P. M. Akhmet'ev, S. A. Melikhov
Zap. Nauchn. Sem. POMI, 267 (2000),  53–87
18. Generalized Daverman's problem
P. M. Akhmet'ev
Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 68 (1999),  5–15
19. A Geometrical Proof of Browder's Result on the Vanishing of the Kervaire Invariant
P. M. Akhmet'ev, P. J. Eccles
Tr. Mat. Inst. Steklova, 225 (1999),  46–51
20. A Generalization of the Sato–Levine Invariant
P. M. Akhmet'ev, D. Repovš
Tr. Mat. Inst. Steklova, 221 (1998),  69–80
21. Obstructions to splitting manifolds with infinite fundamental group
P. M. Akhmet'ev, Yu. V. Muranov
Mat. Zametki, 60:2 (1996),  163–175
22. $\operatorname{Prem}$-mappings, triple self-intersection points of oriented surfaces, and the Rokhlin signature theorem
P. M. Akhmet'ev
Mat. Zametki, 59:6 (1996),  803–810
23. On isotopic and discrete realizations of maps of an $n$-dimensional sphere in Euclidean space
P. M. Akhmet'ev
Mat. Sb., 187:7 (1996),  3–34
24. Solution of the problem of realizing a mapping of an $n$-sphere in a Euclidean $2n$-space
P. M. Akhmet'ev
Tr. Mat. Inst. Steklova, 212 (1996),  37–45
25. An elementary proof of the Freedman immersion theorem
P. M. Akhmet'ev
Algebra i Analiz, 7:5 (1995),  93–100
26. Smooth immersion of manifolds of small dimension. II. Cobordism group of critical points of multiparameter families of functions
P. M. Akhmet'ev
Mat. Sb., 186:12 (1995),  37–62
27. Smooth immersions of manifolds of low dimension
P. M. Akhmet'ev
Mat. Sb., 185:10 (1994),  3–26
28. Splitting homotopy equivalences along a one-sided submanifold of codimension one
P. M. Akhmet'ev
Izv. Akad. Nauk SSSR Ser. Mat., 51:2 (1987),  211–241

29. 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))
P. M. Akhmet'ev
Fundam. Prikl. Mat., 15:5 (2009),  211
30. Mikhail Mikhailovich Postnikov (on his 70th birthday)
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
Uspekhi Mat. Nauk, 53:2(320) (1998),  183–184

Presentations in Math-Net.Ru
1. Linking coefficients for geodesic flow and applications for magnetic equilibrium
P. M. Akhmet'ev
Dynamics in Siberia - 2018
March 3, 2018 10:20
2. 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
3. Гиперкватернионный Арф-инвариант
P. M. Akhmet'ev
Knots and Representation Theory
September 26, 2017 18:30
4. Quadratic Helicity in MHD
P. M. Akhmet'ev
Geometric Topology Seminar
September 21, 2017 14:00
5. A combinatorial formula for M-invariant
P. M. Akhmet'ev
Geometric Topology Seminar
August 3, 2017 14:00
6. 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
7. 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
8. 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
9. 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
10. 'On polynomial helicities in MHD"
P. M. Akhmet'ev
Seminar on analytic theory of differential equations
October 26, 2016 14:30   
11. Критерий разрешимости почти нормальной монодромной пары в одномерной топологической теории Галуа
P. M. Akhmet'ev
Noncommutative geometry and topology
May 19, 2016 16:45
12. О проблеме Кервера
P. M. Akhmet'ev
Noncommutative geometry and topology
February 25, 2016 16:45
13. О группах кобордизмов скоснащенных погружений
P. M. Akhmet'ev, O. D. Frolkina
Knots and Representation Theory
March 24, 2015 18:30
14. 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
15. Об инварианте Хопфа 4-кос на сфере
P. M. Akhmet'ev
Knots and Representation Theory
February 11, 2014 18:30
16. 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
17. 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
18. On non-immersibility of $RP^{10}$ to $R^{15}$
P. M. Akhmet'ev, O. D. Frolkina
Geometric Topology Seminar
June 14, 2012 14:30
19. On higher asymptotic analogs of the helicity invariant in MHD
P. M. Akhmet'ev
Knots and Representation Theory
May 3, 2011 18:30

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