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Mathematical Life Iskander Asanovich Taimanov (on his 60th birthday) A. V. Bolsinov, V. M. Buchstaber, A. P. Veselov, P. G. Grinevich, I. A. Dynnikov, V. V. Kozlov, Yu. A. Kordyukov, D. V. Millionshchikov, A. E. Mironov, R. G. Novikov, S. P. Novikov, A. A. Yakovlev
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     The outstanding mathematician Iskander Asanovich Taimanov observed his sixtieth birthday on December 20, 2021. He was born in Akademgorodok, Novosibirsk, in a family of scientists. His father, Asan Dabsovich Taimanov, was a member of the Academy of Sciences of the Kazakh Soviet Republic and a prominent expert in set theory and mathematical logic. Asan Taimanov participated in the WWII, by volunteering for the acting army in the summer of 1941 and finishing the war in Germany in 1945. For several decades he worked in the Institute of Mathematics of the Siberian Branch of the USSR Academy of Sciences and Novosibirsk State University. He raised several generations of brilliant specialists in mathematical logic. The mother, Olga Ivanovna Taimanova, graduated from the Faculty of Physics at Moscow State University. She taught physics in the physical and mathematical boarding school associated with Novosibirsk University for many years. After finishing school no. 130 (now the Lavrentyev Lyceum) in Akademgorodok, Iskander Taimanov enrolled the Faculty of Mechanics and Mathematics of Moscow State University. He chose geometry and topology as his major, where the outstanding mathematician academician Sergei Novikov became his scientific advisor. After completing his postgraduate studies in 1986, he returned to his native Akademgorodok and, since January 1987, has been working in the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. In 2003 he created the Laboratory of Dynamical Systems at the institute and continues to be its head now. Four members of the Russian Academy of Sciences and a number of talented young researchers work in the laboratory at the moment. Since 1991 Taimanov has been teaching at the Department of Geometry and Topology of Novosibirsk State University, and since 2005 he has been the head of the department. Since 2000 the seminar “Geometry, topology and their applications” founded by Taimanov has been organized at the department. This seminar is well known both in Russia and in other countries. Foreign experts regularly give talks at the seminar. A distinctive feature of Iskander Taimanov is his capability to explain complicated facts in a clear and accessible way. Thanks to this, many discussions with him live a lasting memory. This talent of his has brightly manifested in the courses he has created and books he has published. He developed a wonderful course on differential geometry and, on its basis, wrote the book Lectures on differential geometry [14], which is quite popular among students and experts alike. The book Modern geometric structures and fields [16], written jointly with Novikov, is also very popular. The first paper of Taimanov [1], published when he was a student, was devoted to the development of the method of throwing out cycles, proposed by Novikov in the early 1980s. This method generalizes the Lyusternik–Schnirelmann–Morse theory to the case of multivalued or not everywhere non-negative action functionals. Here we are talking about Lagrangian systems with Lagrangian that locally has the form
$$
\begin{equation}
L(x,\dot x) =\frac12g_{ij}(x)\dot x^i\dot x^j+A_i(x)\dot x^i-U(x),
\end{equation}
\tag{1}
$$
where the $1$-form $(A_i)$ is locally defined and only the form $F=dA$ is defined globally. Among the examples of physical systems leading to such a setting, Novikov pointed out Kirchhoff’s equations describing the motion of a rigid body in an ideal fluid and the motion of a top in an axisymmetric gravitational field, as well as the equations of the A- and B-phases of superfluid helium-3. The problem is to describe the closed trajectories of such Lagrangian systems.
According to Maupertuis’s principle, the trajectories of the Lagrangian system with Lagrangian (1) at a fixed energy level $E$ coincide with the extremals of the (also, generally speaking, multivalued) functional
$$
\begin{equation}
\ell(\gamma) =\int_\gamma\bigl(\sqrt{(E-U)g_{ij}\dot\gamma^i\dot\gamma^j} +A_i\dot\gamma^i\bigr) \,dt,
\end{equation}
\tag{2}
$$
of which each one-point curve is a local minimum. The method of throwing out cycles involves constructing cycles in the space of closed curves in the phase space relative to the subset where $\ell\leqslant0$. Under ‘gradient descent’ such cycles should ‘hang’ on closed extremals. The term ‘throwing out’ itself was originally proposed in [1], where it was shown that, for single-valued functionals, when the magnetic field $F$ is given by an exact form, the whole manifold of one-point curves can be ‘thrown out’ to the domain where $l<0$. In the process any $k$-dimensional cycle in the configuration space produces a $(k+1)$-dimensional relative cycle in the space of closed contractible curves.
Applications of the method of throwing out cycles face some difficulties (pointed out by S. V. Bolotin and absent in the classical Lyusternik–Schnirelmann–Morse theory), which are caused by the possible non-compactness of the space of curves on which $\ell\leqslant\mathrm{const}$. Overcoming these requires additional arguments in each particular case. In [5] Taimanov proved (for the two-dimensional sphere, generalizing this result subsequently to all closed surfaces [6]) the existence of closed non-selfintersecting curves that are local minima of a functional of the form
$$
\begin{equation*}
\ell(\gamma) =\int_\gamma\bigl(f(\gamma,\dot\gamma)+A_i(\gamma)\dot\gamma^i\bigr)\,dt,
\end{equation*}
\notag
$$
where $f(x,\dot x)$ is an arbitrary Finsler metric on $\mathbb S^2$, under certain conditions, which can be satisfied by replacing $F=dA$ by $\lambda F$ for sufficiently large $\lambda$ (again, the form $A$ can be defined only locally). To justify this method, instead of the space of closed curves he considered the space of two-dimensional membranes with boundary, on which the functional becomes single-valued:
$$
\begin{equation*}
\ell(\Pi) =\int_{\partial\Pi}f(x,\dot x)\,dt+\int_\Pi F.
\end{equation*}
\notag
$$
The method of throwing out cycles was widely developed in subsequent works by Taimanov and other authors. In [11] Bahri and Taimanov proved the existence of closed extremals of the functional
$$
\begin{equation*}
S(\gamma)=\int_\gamma\bigl(\sqrt{g_{ij}\dot\gamma^i\dot\gamma^j}+A_i\dot\gamma^i\bigr)\,dt
\end{equation*}
\notag
$$
in the case of an arbitrary closed Riemannian manifold endowed with a metric $(g_{ij})$ and a single-valued $1$-form $A$ such that, for all unit tangent vectors $v$, the following condition holds: where $H$ is the following $1$-form:
$$
\begin{equation*}
H_j=g^{ik}\nabla_kF_{ij}, \qquad F_{ij}=\partial_i A_j-\partial_j A_i,
\end{equation*}
\notag
$$
and $\operatorname{Ric}$ is the Ricci tensor.
In the joint paper [23] by Taimanov with A. Abbondandolo, L. Asselle, G. Benedetti, and M. Mazzucchelli the method of throwing out cycles was justified for non-exact magnetic fields on the two-dimensional sphere for almost all energy levels which are lower than a certain constant. In this case the magnetic field is ‘strong’: according to [5], there exists a locally minimal closed extremal for it, which implies the existence of infinitely many geometrically distinct closed extremals. We also note that one of the two known ways to the rigorous justification of the classical Lyusternik–Schnirelmann theorem on the existence of three closed non-selfintersecting geodesics on the two-dimensional sphere was developed by Taimanov [7]. The description of the homotopy groups of the spaces of non-contractible (to a point) curves on manifolds, which Taimanov presented in [2], has been used to prove the existence of closed geodesics (see, for instance, [31]). Taimanov’s paper [4] became a fundamental contribution to the theory of topological obstructions to the integrability of geodesic flows on manifolds. Its main result is a far-reaching generalization of Kozlov’s theorem on the non-existence of geodesic flows on surfaces of genus $g>1$ that are analytically integrable in the sense of Liouville. Taimanov’s theorem states that a necessary condition for the existence of such a flow on a closed manifold $M^n$ is that its fundamental group $\pi_1(M^n)$ is almost commutative. Moreover, he proved that such flows do not exist on manifolds with first Betti number greater than the dimension of the manifold. The key role in that paper is played by the notion of geometric simplicity of an integrable geodesic flow, a property which holds in the analytic category in view of results due to A. M. Gabrielov. This property allowed Taimanov to combine various ideas underlying the proof of his remarkable theorem. These results aroused great interest among experts in the theory of dynamical systems on manifolds. In particular, G. Paternain proposed a new approach to this problem, which was based on the notion of topological entropy, suggesting that the integrability of a geodesic flow implies the polynomial growth of the fundamental group of the manifold. This conjecture was disproved by A. V. Bolsinov and Taimanov [13], who constructed the first example of an integrable geodesic flow on an analytic Riemannian manifold with fundamental group of exponential growth. This example also showed that in the case of integrable systems with smooth first integrals the topological entropy can be positive, although its positivity had previously been considered as one of the characteristic properties of chaotic systems. Note that the existence of such systems in the analytic category is still an open question. Another interesting result in this direction was obtained by Taimanov jointly with A. Knauf in [15], where, in particular, they proved that the $n$-centre problem in three-dimensional space is integrable in the smooth category at sufficiently high energy levels, but is not integrable in the analytic category for any number of centres in a general position. In 1998–2001 a series of joint papers of Taimanov and I. K. Babenko (see [10], [12], and the references there) devoted to applications of rational homotopy type theory to symplectic manifolds appeared. The well-known Deligne–Griffiths–Morgan–Sullivan theorem (1975) states that simply connected closed Kähler manifolds are formal, which, in particular, means that all Massey products in the rational cohomology of such manifolds are trivial. Babenko and Taimanov showed that this cannot be generalized to symplectic manifolds [10]. They constructed simply connected symplectic non-formal closed manifolds in even dimensions $\geqslant 10$. Their construction is based on the analysis of the behaviour of non-trivial Massey products under symplectic blow-ups. These papers by Babenko and Taimanov immediately became popular with experts in the area. This was largely due to a new and extremely successful definition of Massey products in terms of the formal connection matrix and the generalized Maurer–Cartan equation which was proposed in [12]. The authors themselves attributed this definition to P. May, although most authors in the field still do not understand the reasons underlying this decision and refer to [12] as a source of a short and understandable definition of Massey products. Another important area of research where Taimanov obtained fundamental results is the theory of soliton equations and their algebro-geometric solutions, and its applications to the geometry of surfaces, where much attention was paid to the theory of two-dimensional operators and equations in $2+1$ dimensions. In 1985 Taimanov carried out effectivization of finite-gap solutions of the Veselov–Novikov equations (by calculating the period vectors of meromorphic differentials in terms of the Prym matrix), which he applied subsequently to the problem of the characterization of Prymians. In 1987 Taimanov [3] obtained an important result on the Schottky–Prym problem of the characterization of Prymians. As is well known, for curves of genus $g\geqslant 4$ the dimension of the space of Jacobians turns out to be lower than that of the space of symmetric $ g\times g $ matrices with non-negative imaginary part. The problem of distinguishing the Riemann matrices of curves in this space was posed by F. Schottky in 1882 and, for arbitrary $g$, resisted the efforts of mathematicians for quite a long period of time. In 1979 Novikov formulated the conjecture that a matrix is a Riemann matrix if and only if a certain function $u(x,y,t)$ constructed from it satisfies the Kadomtsev–Petviashvili equation. A proof that the Jacobians form a connected component of the variety distinguished by this condition was presented by B. A. Dubrovin in 1981, and the complete solution was given by T. Shiota in 1986. If a curve is a double covering of another curve without branch points or with two branch points, then odd abelian differentials generate a principally polarized abelian variety, called the Prymian. An analogue of the Schottky problem for Prymians was considered to be a deeply non-trivial problem. Using the fact that solutions of the Veselov–Novikov equation are expressed in terms of the Prymians of curves with two branch points, Taimanov proposed an analogue of Novikov’s conjecture for Prymians and proved that a substitution into the Veselov–Novikov equation defines a variety of which the Prymians form an irreducible component. The Schottky problem for such Prymians was completely solved by I. M. Krichever twenty years later, who also used methods of soliton theory. In 1995 Taimanov [8] applied B. G. Konopelchenko’s results on constructing surfaces (locally) in terms of spinors belonging to the kernel of the two-dimensional Dirac operator with potential and on deforming these surfaces using the modified Veselov–Novikov equation (mVN) to the construction of a global representation of two-dimensional surfaces in terms of such spinors [8]. He showed in [8] that any closed surface in three-dimensional space has such a representation, and the Dirac operator acts on the sections of spinor representations over surfaces. The quadratic $L_2$-norm of the potential coincides, up to a factor, with the value of the Willmore functional at the closed surface. Moreover, Taimanov established a connection of such a representation with the conformal geometry of surfaces, and, in particular, he showed that the mVN-deformation admits a global definition as a deformation of tori preserving the conformal class and the Willmore functional. Embeddings of tori are described using doubly periodic Dirac operators. For these operators Taimanov proved a theorem on the existence of the spectral curve (complex Fermi curve); in contrast to the one-dimensional case, this theorem turned out to be deeply non-trivial and required the use of delicate analytic methods [9]. Note that this proof is based on Keldysh’s theorem on the regularized determinant of pencils of compact operators. In the mid-1980s Taimanov, with the help of this method, established the existence of Fermi curves for the two-dimensional Schrödinger and heat operators; this result was used by Krichever at the same time, but for the first time it was also presented in [9]. A similar theory was also developed for conformal embeddings of surfaces in $\mathbb{R}^4$, where the potential of the Dirac operator becomes complex and the focusing Davey–Stewartson II equation arises. Note that, since a generalized Weierstrass representation is not unique, defining a dynamics of tori preserving their conformal class required of Taimanov an additional analysis. One of the important properties of the Willmore functional is its invariance under conformal transformations of the ambient space. Taimanov put forward the conjecture that the higher Willmore functionals corresponding to higher first integrals of soliton equations are also conformally invariant. This conjecture was proved by P. G. Grinevich and M. U. Schmidt for embeddings in $\mathbb{R}^3$, and by Taimanov and Grinevich for $\mathbb{R}^4$, who used the fact that generators of conformal transformations correspond to Melnikov-type equations which preserve the dispersion relation. However, as shown in [20], transformations of Dirac operators corresponding to conformal transformations are not isospectral, since Melnikov-type equations can generate or annihilate a double point on the spectral curve in finite time. The fact that spectral curves with double points arise naturally in the theory of two-dimensional operators and the geometry of surfaces required the development of a finite-gap theory for this range of problems, which was performed by Taimanov. We note that such double points can correspond to unstable modes of soliton equations. On the basis of these observations Taimanov proposed an approach to the proof of Willmore’s conjecture, which was based on the fact that the spectral curve of a minimal torus must be stationary for all equations of the mVN hierarchy. This approach has not been implemented; a discussion of it and some results on the generalization of the spinor representation to surfaces in three-dimensional Lie groups were presented in [17]. Starting in 2007, Taimanov (in part, in collaboration with S. P. Tsarev, R. G. Novikov and R. M. Matuev) published a series of works that made a fundamental contribution to the application of Darboux–Moutard type transformations to the spectral theory of two-dimensional differential operators and soliton theory in $2+1$ dimensions (see [28] and the references there). The results in these papers include explicit examples of two-dimensional Schrödinger operators with regular potentials for which the zero energy or some positive energy is a point in discrete spectrum, which can even be multiple. Moreover, in the case of the zero energy the potentials decay as $1/|x|^n$ at infinity, where $n=6$ and $n=8$ appeared in the first examples. At a positive energy these potentials (called Wigner–von Neumann type potentials) decay as $O(1/|x|)$ at infinity. It was also shown in those papers that Moutard-type transformations make it possible to construct simple explicit examples of solutions of $(2+1)$-dimensional soliton equations whose initial data are regular and decrease sufficiently rapidly at infinity, but whose singularities appear in finite time. Examples of such type were given, in particular, for the Veselov–Novikov equation [21], the modified Veselov–Novikov equation, and the Davey–Stewartson equation. Note that for soliton equations in $1+1$ dimensions, the opposite situation is more typical, when solutions that are regular for some value of time remain regular throughout. In addition, a beautiful geometric explanation of the arisal of these singular solutions was found. If a two-dimensional surface in $\mathbb{R}^n$, where $n=3$ or $n=4$, is given by a generalized Weierstrass representation, then the modified Veselov–Novikov equation for $n=3$ and the Davey–Stewartson equation for $n=4$ generate a dynamics of such surfaces. One result in this series of papers was that inversions of the ambient space correspond to Moutard transformations of the Dirac operator. Thus, if a family of surfaces passes through the centre of an inversion, then, after the corresponding Moutard transformation, the new family of surfaces and the corresponding solution of the above equations become singular. Moreover, the results in these papers include formulae describing the action of Darboux–Moutard type transformations on the Poincaré–Steklov boundary operators. Such formulae have applications to coefficient inverse problems in a bounded domain. Theta-functional formulae for finite-gap solutions of soliton equations are quite complicated in the case of smooth spectral curves, but if the spectral curves degenerate to singular ones, then the formulae for exact solutions can be expressed in terms of simpler functions. For non-singular solutions of soliton wave equations the spectral curves are smooth, but this approach can be applied successfully to other problems. In [18] the degenerate case of Krichever’s construction of orthogonal curvilinear coordinates in Euclidean space was considered. In that construction the spectral data corresponding to curvilinear coordinates include a smooth spectral curve. In [18] the case when the spectral curve becomes reducible and each irreducible component is isomorphic to $\mathbb{C}\mathbb{P}^1$ was treated. In this case coordinate functions are expressed in terms of elementary functions. This enables one to find the spectral data for the polar and spherical coordinate systems. The same method was used in [19] to construct explicitly Frobenius manifolds corresponding to reducible singular spectral curves. In a series of joint papers with Yu. A. Kordyukov spectral problems for the magnetic Laplacian were considered when the $2$-form defining the magnetic field is not exact, that is, in the case of magnetic monopoles. In the theory of dynamical systems a model example for the notion of the Mané level is the magnetic geodesic flow on a hyperbolic surface, which is qualitatively different at energy levels lower or higher than the Mané level. In [26] and [30] the authors described how the trace formula for the magnetic Laplacian (Guillemin–Uribe formula), which depends on the energy level $E$, changes analytically as $E$ goes over the Mané level. In [27] the multidimensional WKB method was extended to the construction of eigenfunctions in the case of magnetic monopoles. In this case, these functions (quasi-classical magnetic harmonics) take values in sections of non-trivial line bundles over the configuration space, and the small parameter $h$ (‘Planck constant’) is quantized. Taimanov pays much attention to applied and computational problems (see [22], [24], [25], and [29]). He made a significant contribution to the development of methods for forecasting oil and gas production from deposits whose reservoir properties are characterized by strong spatial non-stationarity and anisotropy. He proposed and developed numerical methods for taking account of dynamic data from the normal operation of wells, logging data, and core analysis. The results of this work were used in practice in the development of hard-to-recover reserves of the Priobskoye oil field. Taimanov is currently active as a researcher, an expert, and an organizer of science. He is a deputy editor-in-chief of the journal Uspekhi Matematicheskikh Nauk1, a member of the editorial boards of the journals Regular and Chaotic Dynamics and Sibirskii Matematicheskii Zhurnal2. Every year he takes an active part in the organization of many conferences, including two large annual conferences in Novosibirsk, “Geometry Days in Novosibirsk” and “Dynamics in Siberia”. He is a member of the Praesidium of the Russian Academy of Sciences, the Praesidium of the Siberian Branch of the Russian Academy of Sciences, and the Bureau of the Department of Mathematics of the Russian Academy of Sciences. He was a scientific advisor of 10 PhD students, two of whom are now Doctors of Sciences, including one who is a corresponding member of the Russian Academy of Sciences. In 2003 Taimanov was elected a corresponding member, and in 2011 an acting member of the Russian Academy of Sciences. In 2022 he was an invited speaker at the International Congress of Mathematicians. He was awarded the Kovalevskaya prize of the Russian Academy of Sciences and a Medal of the Order For Merit to the Fatherland, class 2. Iskander Taimanov is an amazingly sensitive person with extraordinary erudition, and communicating with him is a pleasure for everyone. We congratulate Iskander Asanovich Taimanov sincerely on his 60th birthday and wish him good health, happiness, and new scientific achievements. 
Citation in format AMSBIB
\Bibitem{BolBucVes22} |
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