Anton Savin
Doctor of Physical and Mathematical Sciences
Professor at the S. Nikolsky Mathematical Institute, s.M. Nikol’skii Mathematical Institute
1997

Graduate of the Department of Nonlinear Dynamic Systems and Control Processes, Faculty of Computational Mathematics and Cybernetics, M.V. Lomonosov Moscow State University. Specialty: "Applied Mathematics".

2000

Defended his PhD thesis: “Elliptic operators in subspaces and their applications”, M.V. Lomonosov Moscow State University, scientific adviser - Professor Boris Sternin.

2007

Pierre Deligne competition for young Russian mathematicians’ winner.

2012

Defended his Doctor of sciences thesis “Index Theory of Nonlocal Elliptic Problems”, Peoples Friendship University of Russia, scientific adviser - Professor Boris Sternin.

2014

Simons Foundation competition for mathematicians, research teachers’ winner.   

Teaching

1. Made a series of training courses, the most significant are the following:

  • "K-theory and Atiyah-Singer index theorem" (University of Potsdam, 2003);
  • “Dirac Operator” (Independent University of Moscow, 2005)
  • "Elements of the calculus of variations in the large" (Independent University of Moscow, 2006)
  • "Differential equations on complex manifolds" (Independent University of Moscow, 2015)
  • “Methods of optimization” (RUDN University, 2016)

2. Conducts the following courses for bachelor and master’s degree students at RUDN University:

  • "Differential equations" (direction "applied mathematics and computer science")
  • "Equations of mathematical physics" (direction "applied mathematics and computer science")
  • "Methods of optimization" (direction "applied mathematics and computer science")
  • "Elements of algebraic topology" (specialization "nonlinear analysis, optimization and mathematical modeling")

Science

  • Research is carried out on the theory of elliptic G-operators associated with group representations. In the case of groups of shifts, new index formulas are obtained; in the case of Lie groups, it was shown that the ellipticity condition must be imposed only on a special subspace of the cotangent bundle (the so-called notion of transversal ellipticity). The results are applied in noncommutative geometry, in particular, a new proof of the Connes index formula on the noncommutative torus is given, and Todd classes of manifolds are defined in the cyclic cohomology of crossed products.
  • The application of methods of noncommutative geometry in the theory of boundary value problems for hyperbolic equations with data on the entire boundary was considered. Such problems are reduced to the boundary. Moreover, the operator on the boundary is studied by methods of the theory of G-operators. The Fredholm solvability of such problems is established. Similar problems have found application in the theory of quantum anomalies in the works of Bär, Strohmeier, Walters and others. Therefore, we can expect applications of results in these issues.
  • A formula is given for the fractional part of the Atiyah-Patodi-Singer eta-invariant for operators with parity conditions in topological terms. Namely, it is shown that the fractional part is equal to the linking index in the K-theory. Also presented are operators with a non-trivial fractional part of this invariant. Thus, the problem posed by the famous American mathematician Peter Gilkey was solved. This problem consisted in presenting even order operators on odd-dimensional manifolds with a non-trivial fractional part of this invariant, or proving that there are no such operators.
  • A homotopy classification of elliptic operators on general stratified manifolds in terms of Kasparov K-homology is obtained. A classification is also obtained for elliptic operators on manifolds with corners (in the sense of Melrose). This provides an important link between analysis and topology and, in particular, enables one to apply topological methods to the study of operators.
  • According to the research results, 3 foreign monographs and more than 70 scientific articles were published. The works are supported by grants from the Russian Foundation for Basic Research, the German Research Society, the German Academic Exchange Service.

Monographs:

  • V.E. Nazaikinskii, A.Yu. Savin, B.-W. Schulze, B.Yu. Sternin, Elliptic theory on singular manifolds, Chapman & Hall/CRC, Boca Raton, FL, 2006 , 356 pp.
  • V.E. Nazaikinskii, A.Yu. Savin, B.Yu. Sternin, Elliptic theory and noncommutative geometry. Nonlocal elliptic operators, Birkhäuser, Basel, 2008 , 224 pp.
  • A. Savin, B. Sternin, Introduction to complex theory of differential equations, Birkhäuser, Basel, 2017, 128 pp.

Scientific interests

  • Index theory of elliptic operators;
  • application of topological methods for the study of non-local elliptic problems, boundary problems, Sobolev problems, equations on manifolds with singularities, etc.
  • application of methods of noncommutative geometry (K-theory, C*-algebras, cyclic homology, etc.) to the problems of elliptic theory;
  • asymptotic methods in the theory of differential equations;
  • differential equations on complex manifolds and their applications in geophysics and radiophysics.
We give a formula for the η-invariant of odd-order operators on even-dimensional manifolds and even-order operators on odd-dimensional manifolds. Second-order operators with nontrivial η-invariants are found. This solves a problem posed by Gilkey.
It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds. As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula for K-groups of algebras of pseudodifferential operators.
We develop an elliptic theory for operators associated with a diffeomorphism of a closed smooth manifold. The aim of the present paper is to obtain an index formula for such operators in terms of topological invariants of the manifold and the symbol of the operator. The symbol in this situation is an element of a certain crossed product. We express the index as the pairing of the class in K-theory defined by the symbol and the Todd class in periodic cyclic cohomology of the crossed product.