singularities in the theory of ordinary differential equations, dynamical systems, differential geometry, calculus of variations, and optimal control. Recently, the main theme of the work is singularities of geodesic flows in pseudo-Riemannian spaces with signature varying metrics. In particular, it was established that geodesic lines cannot pass through degenerate points of the metric in arbitrary tangential directions, but only in certain admissible directions. Generically, in 2-dimensional case, the number of admissible directions is finite and varies from one to three, but if the dimension is more than two, the number of admissible directions can be finite or infinite. A brief survey in the two-dimensional case is presented in: https://arxiv.org/pdf/1801.09815.pdf
Main publications:
A.O. Remizov, “Geodesics on 2-surfaces with pseudo-Riemannian metric: singularities of changes of signature”, Sb. Math., 200:3 (2009), 385–403
I.R. Shafarevich, A.O. Remizov, Linear Algebra and Geometry, Springer-Verlag Berlin and Heidelberg GmbH, 2013 (English)
R. Ghezzi, A.O. Remizov, “On a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics”, J. Dyn. Control Syst., 18:1 (2012), 135–158
A.O. Remizov, “On the local and global properties of geodesics in pseudo-Riemannian metrics”, Differential Geometry and its Applications, 39 (2015), 36–58
A.O. Remizov, F. Tari, “Singularities of the geodesic flow on surfaces with pseudo-Riemannian metrics”, Geometriae Dedicata, 185:1 (2016), 131–153
N. G. Pavlova, A. O. Remizov, “Completion of the classification of generic singularities of geodesic flows in two classes of metrics”, Izv. Math., 83:1 (2019), 104–123
7.
A. O. Remizov, “On isomorphisms of pseudo-Euclid spaces”, Math. Ed., 2018, no. 2(86), 15–39
8.
U. Boscain, R. A. Chertovskih, J. P. Gauthier, D. Prandi, A. O. Remizov, “Highly Corrupted Image Inpainting Through Hypoelliptic Diffusion”, J. Math. Imaging Vis., 60:8 (2018), 1231–1245
N. G. Pavlova, A. O. Remizov, “On isomorphisms of pseudo-Euclidean spaces with signature (p,n-p) for p = 2,3”, Linear Algebra Appl., 541 (2018), 60–80
10.
N. G. Pavlova, A. O. Remizov, “A brief survey on singularities of geodesic flows in smooth signature changing metrics on 2-surfaces”, Advances in Singularities and Foliations: Geometry, Topology and Applications (Salvador, Brazil, 2015), Springer Proc. in Math. & Stat., 222, Springer, 2018, 135–155
U. Boscain, R. A. Chertovskih, J. P. Gauthier, D. Prandi, A. O. Remizov, “Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic diffusion”, ESAIM, Proc. Surv., 64 (2018), 37–53
N. G. Pavlova, A. O. Remizov, “A complete classification of generic singularities of geodesic flows on 2-surfaces with pseudo-Riemannian metrics”, Russian Math. Surveys, 72:3 (2017), 577–579
13.
A. O. Remizov, “Geodesics in generalized Finsler spaces: singularities in dimension two”, J. Singularities, 14:1 (2016), 172–193
U. Boscain, J. P. Gauthier, D. Prandi, A. O. Remizov, “Image Reconstruction Via Non-Isotropic Diffusion in Dubins/Reed-Shepp-like Control Systems”, 53-rd IEEE Conference on Decision and Control (Los Angeles, USA, 15-17 Dec. 2014), Proceedings of the IEEE Conference on Decision and Control, 2015, 4278–4283
I. R. Shafarevich, A. O. Remizov, Linear Algebra and Geometry, 2nd edition, IKI Edition, Izhevsk, 2014 , 554 pp. "Mathesis"
18.
U. Boscain, R. A. Chertovskih, J. P. Gauthier, A. O. Remizov, “Hypoelliptic diffusion and human vision: a semidiscrete new twist”, SIAM J. Imaging Sci., 7:2 (2014), 669–698
R. A. Chertovskih, A. O. Remizov, “On pleated singular points of first-order implicit differential equations”, J. Dyn. Control Syst., 20:2 (2014), 197–206
20.
I. R. Shafarevich, A. O. Remizov, Linear Algebra and Geometry, Springer, Heidelberg, 2013 , xxii+526 pp., Translated from the 2009 Russian original by David Kramer and Lena Nekludova, "Springer"
R. Ghezzi, A. O. Remizov, “On a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics”, J. Dyn. Control Syst., 18:1 (2012), 135–158
N. G. Pavlova, A. O. Remizov, “Geodesics on hypersurfaces in Minkowski space: singularities of signature change”, Russian Math. Surveys, 66:6 (2011), 1201–1203
23.
A. O. Remizov, “On geodesics in metrics with singularities of the Klein type”, Russian Math. Surveys, 65:1 (2010), 180–182
24.
A. O. Remizov, “Singularities of a geodesic flow on surfaces with a cuspidal edge”, Proc. Steklov Inst. Math., 268 (2010), 248–257
25.
A. O. Remizov, “Geodesics on 2-surfaces with pseudo-Riemannian metric: singularities of changes of signature”, Sb. Math., 200:3 (2009), 385–403
26.
I. R. Shafarevich, A. O. Remizov, Linear Algebra and Geometry, 1st edition, eds. A. S. Kuleshov, Fizmatlit, Moscow, 2009 , 511 pp. "Fizmatlit"
27.
A. O. Remizov, “Codimension-two singularities in 3D affine control systems with a scalar control”, Sb. Math., 199:4 (2008), 613–627
28.
V. M. Zakalyukin, A. O. Remizov, “Legendre Singularities in Systems of Implicit ODEs and Slow–Fast Dynamical Systems”, Proc. Steklov Inst. Math., 261 (2008), 136–148
29.
A. O. Remizov, “Singularities in three-dimensional affine control systems with scalar control”, Russian Math. Surveys, 62:4 (2007), 821–822
30.
A. O. Remizov, “Multidimensional Poincare Construction and Singularities of Lifted Fields For Implicit Differential Equations”, Journal of Mathematical Sciences, 151:6 (2008), 3561–3602
31.
A. O. Remizov, “On vector fields generated by implicit differential equations”, J. Math. Sci., 126:6 (2005), 1643–1653
32.
A. O. Remizov, “Improper Singular Points of Corank 1 of Systems of Differential Equations Not Solved for the Derivatives”, Differ. Equ., 38:8 (2002), 1122–1131
33.
A. O. Remizov, “On Proper Singular Points of Ordinary Differential Equations Unsolved for Derivatives”, Differ. Equ., 38:5 (2002), 654–662
34.
A. O. Remizov, “Implicit differential equations and vector fields with non-isolated singular points”, Sb. Math., 193:11 (2002), 1671–1690
35.
A. O. Remizov, “Generic singular points of implicit differential equations”, Mosc. Univ. Math. Bull., 57:5 (2002), 9–15
36.
A. O. Remizov, “Vector fields with nonisolated singular points”, Dokl. Math., 65:3 (2002), 437–439
37.
A. O. Remizov, “On typical singularities of implicit differential equations”, Progress in nonlinear science (Nizhny Novgorod, 2001), 1, RAS, Inst. Appl. Phys., 2002, 346–352
A. O. Remizov, “Error estimate for an approximate solution of an ODE system which is not solved with respect to the derivatives”, Mosc. Univ. Comput. Math. Cybern., 2001, no. 2, 30–37
39.
N. G. Pavlova, A. O. Remizov, Introduction to Singularity Theory, MIPT, Moscow, 2022 , 181 pp. "Researchgate"
40.
N. G. Pavlova, A. O. Remizov, “Correction to the article “Smooth functions, formal series and Whitney theorems ””, Math. Ed., 2020, no. 1(93), 69
41.
N. G. Pavlova, A. O. Remizov, “Smooth functions, formal series, Whitney theorems, finished”, Math. Ed., 2017, no. 3(83), 13–27
42.
N. G. Pavlova, A. O. Remizov, “Smooth Functions, Formal Series, and Whitney Theorems”, Math. Ed., 2016, no. 3(79), 49–65
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