Alexander Ivanov

Doctor of physical and mathematical sciences (habilitation), professor at Faculty of Mechanics and Mathematics of Lomonosov Mosco­w State University, deputy head of the department of Differential Geometry and Applications, academic secretary of the dissertation committee at Faculty of Mechanics and Mathematics, MSU, professor at the department of Mathematical Modeling at Bauman Moscow Technical University.

A.O. Ivanov was born on August 14, 1963, at Opochka, Pskov region. In 1985 he graduated with honours Faculty of Mechanics and Mathematics of MSU, in finished his graduate studies, and in 1990 defended PhD thesis “Minimal surfaces and calibration forms” (scientific advisor academician A.T. Fomenko). The dissertation was devoted to investigation of minimal surfaces with symmetries. A.Ivanov succeeded in efficient application of so-called calibration forms method created by American specialists R. Harvey and H.B. Lowson to the case of manifolds with symmetries. New examples of globally minimal and stable cones of large codimension were constructed, that attracted interest of specialists in multidimensional Plateau problem.

After that, scientific interests of A. Ivanov concentrated in the branch of well-known Steiner problem (one-dimensional Plateau problem) and its various generalizations. A. Ivanov is one of the creators of the theory of branching extremals of one-dimensional variational problems. That Theory appeared at a junction of Differential Geometry, Variational Calculus, and Graph Theory under attempts to find a geometrical approach to Steiner problem solution. The theory gives an opportunity to find some new geometrical effects, to find relations between geometry of boundary setsand topology of extreme networks, between structure of extreme networks and metrical and/or topological characteristics of ambient manifolds. Recently, theory of extreme networks in metric spaces of more general form is developed actively, for example, in normed spaces, in manifolds with singularities, in the Alexandrov spaces with bounded curvature, etc. Among possible applications of the branching extremals theory are optimal connection problems, in particular, transportation problems, molecular biology, evolution theory, mathematical economy, etc. Also, recently theory of minimal fillings of finite metric spaces appears, that turns out to be related closely to Steiner Problem.

In 1998, A. Ivanov defended Doctor Dissertation (habilitation “Geometry of Minimal Networks in Riemannian Manifolds”. For the works concerning branching extremals theory A. Ianov and A. Tuzhilin got Shuvalov Prize of 1st degree 2001.

At present A. Ivanov is a well-known specialist in topological variational problems, extreme networks theory, in particular, in branching geodesics and Steiner problem, graph theory, metric geometry and computer geometry. He has more than 90 scientific publications, among them 4 monographs and 2 textbooks. A. Ivanov many times presented his results in international seminars, conferences and workshops, he is one of the creators and advisors of scientific seminar on the theory of extremals of geometrical variational problems, working at Faculty of Mechanic and Mathematics more than 20 years. A.Ivanov is a scientific advisor of graduate and post graduate students. Under his advising 3 PhD Theses were defended.

Current Scientific Interests:
Interior minimal trees in immersed polygons and/or Alexandrov surfaces, minimal fillings of finite metric spaces, metric halls, Gilbert-Pollack conjecture, computer geometry. Ten Recent Publications:
1. Ivanov A.O., Tuzhilin A.A, and A.T.Fomenko, Computer modeling of curves and surfaces, Fundament. i Prikl. Matem, 2009, 15:5, 63–94 (Journal of Mathematical Sciences (New York), 2011, 172:5, 663–689).
2. Ivanov A.O., Tuzhilin A.A. The Length of a Minimal Tree With a Given Topology: generalization of Maxwell Formula. arXiv:1101.2117v1; Vestnik MGU, Ser. Math., no.3, pp.7-14 (2010).
3. Ivanov A.O., Tuzhilin A.A. Minimal fillings in the sense of M.Gromov for finite metric spaces, International Conference “Metric Geometry of surfaces and polyhedra” dedicated to 100th anniversary of N.V.Efimov, 2010, p. 74.
4. Ivanov A.O., Il’utko D.P., Nosovskii G.V., Tuzhilin A.A, and A.T.Fomenko, Computer Geometry: Practical Training, Internet Univ. Electr.Technol., Binom, Moscow, 2010 [in Russian].
5. Ivanov A.O., Tuzhilin A.A. One-dimensional Gromov minimal filling. arXiv:1101.0106v2 [math.MG] (http://arxiv.org), Matem. Sbornik, 203 (5), 65–-118 (2012).
6. Ivanov A.O., Ovsyannikov Z.N., Strelkova N.P., Tuzhilin A.A. One-dimensional minimal fillings with negative edge weights. arXiv:1101.3014v1; Vestnik MGU, Ser Matem., Mekh., 2012, to appear.
7. Bozhenko V.K., Ivanov A.O., Mishchenko A.S., Tuzhilin A.A., Shishkin A.M. Determination of Different Biological Factors on the Base of Dried Blood Spot Technology, arXiv:1101.2576v1.
8. Ivanov A.O., Tuzhilin A.A. The Steiner Ratio Gilbert–Pollak Conjecture Is Still Open. Clarification Statement. Algorithmica, 2011, DOI: 10.1007/s00453-011-9508-3.
9. Ivanov A.O., Tuzhilin A.A., “The geometry of inner spanning trees for planar polygons”, Izvestiya: Mathematics, 76(2):215, 2012.
10. Ivanov A.O, S'edina O.A., and Tuzhilin A.A., “The structure of minimal Steiner trees in the neighborhoods of the lunes of their edges”, Mathematical Notes, 91:339–353, 2012.

Some Basic Results in Extreme Networks Theory obtained by A. Ivanov and A. Tuzhilin:

• Classification of planar locally minimal binary trees with convex boundaries. An original language of so-called tilings is elaborated which gives an opportunity to obtain an efficient description of possible locally minimal binary trees structure. That, in turn, gives an opportunity to prove several non-trivial results on geometry of minimal binary trees with convex boundaries.
• Local structure of locally minimal networks in Riemannian manifolds description. Two basic classes of admissible deformations of the networks are considered (so-called parametric networks and networks traces), and for each of them the local structure f the corresponding extremals is described.
• Existence theorems for extreme networks in Riemannian manifolds are proved.
• Classification of closed locally minimal networks in closed surfaces of non-negative curvature is obtained. It turns out that in the case of flat tori the types of such networks can be classified by the quotient of the space integer matrices with respect to some 6 order group action. In particular, the results obtained, give opportunity to list all the topologies of closed locally minimal networks in each special surface from the class under consideration.
• Restrictions on possible topologies of planar locally minimal binary trees in terms of the number of the convexity levels of their boundary sets are obtained. Moreover, it turns out that these results can be generalized to the case of arbitrary planar linear trees (without any additional assumptions on some extreme properties) and their geometrical boundaries.
• A description of the space of all locally minimal networks with fixed topology and boundary in multidimensional space is obtained. It is shown that this space is a polyhedral set, and the dimension of this set is calculated in terms of the structure of the network.
• The structure of extreme networks in Manhattan plane and in some other normed spaces is described. It turns out that in the case of normed spaces the class of locally minimal networks does not coincide with the class of extreme networks. For the Manhattan plane a criterion of extremality for locally minimal networks is obtained.
• Theory of branching extremals for Lagrangian type functional is constructed. For such extremals, the local structure is described, and existence theorems for two main admissible deformation classes are proved. In particular, it is demonstrated that the presence f reach geometry of extremals is caused by the singularities of Lagrangians under consideration.
• An estimate for Steiner ratio of an arbitrary Riemannian manifold in terms of the Steiner ratio of the Euclidean space with the same dimension is obtained.
• It is shown that the Steiner ratios of flat tori, flat Klein bottles, and the standard Euclidean plane are equal to each other. Also there are equal the Steiner ratios of the standard sphere and the projective plane.
• The directional differentiability of the lengths of minimal spanning tree and shortest tree considered as the functions on the boundary sets is proved. The explicit formulas for the derivatives of these functions and of the Steiner ratio are found. An extremality criterion for a boundary set with respect to the Steiner ratio function is obtained.
• Immersed plane polygons considered as piecewise affine mappings of standard plane polygons into the plane are investigated. It is shown that each such immersed polygon possess a diagonal triangulation. It is also shown that the closure of any monotonical polygonal line is a boundary of an immersed polygon.
• Countable subsets of metric spaces permitting connection by finite length trees are described. It is shown that the length of a minimal spanning tree can be calculated as an integral of a real-valued function constructed by the boundary tree.
• The uniqueness of a shortest tree in the Euclidean plane for the boundary sets in general position is proved.
• A stabilization theorem is proved, saying that any locally minimal tree can be transformed into a shortest tree by adding boundary vertices of degree two but without changing the tree as a subset of the ambient Euclidean space.
• A method for modeling and describing biopolymers is suggested, based on discrete analogues of curvature and torsion of the polygonal lines describing the structure of the molecules under consideration.
• Possible structure of shortest trees in the neighborhoods of the lunes of their edges is described. These results amplify classical wedge and lune rules for the shortest trees, discovered in 60th.
• Geometry of interior spanning trees for plane polygons is investigated. Analogues of classical results concerning the relations between minimal spanning trees, Voronoi diagrams, and Delaunay triangulations are obtained. It is described the possible structures of the faces of the Delaunay graph.
• Theory of Gromov minimal fillings for finite metric spaces is elaborated. As the fillings, connected weighted graphs are considered (one-dimensional stratified manifolds). General existence theorems are proved, possible structures of minimal fillings are described, minimal fillings for some special classes of metric spaces are found, for example, for additive spaces that are often appear in applications. It turns out that minimal fillings theory is closely related to extreme networks theory and Steiner ratio calculating for different metric spaces.

A. Ivanov is also interested in molecular biology and bioinformatics. In collaboration with Professor A.S. Mishchenko and professor A.A. Tuzhilin, he organized a special course on geometrical applications in bioinformatics for students of Mechanical and Mathematical Faculty. Several years A.Ivanov leads RFBR project directed to investigation of proliferation and apoptosis regulations by methods of computer simulations. In that project the specialists from the Mechanical and Mathematical Faculty worked in collaboration with specialists from Russian Scientific Center of Radiology and Nuclear Medicine.

A. Ivanov is one of the initiators organizing the scientific and educational center “Mathematical and Computer Methods” at Mechanical and Mathematical Faculty of MSU. The aims of the center are to elaborate new mathematical and computer methods in geometry and analysis, to implement these methods in educational process and in practical applications, and to attract students to scientific and pedagogical work.