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 study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without boundary and does not preserve the manifold with boundary on which the problem is stated. In particular, the group action does not map the boundary into itself. The orbits of the boundary under the group action split the manifold into subdomains, and this decomposition, being combined with the C∗-algebra techniques, plays an important role in our approach to the analysis of the problem.
η -invariants for a class of parameter-dependent nonlocal operators associated with an isometric action of a discrete group of polynomial growth on a smooth closed manifold are studied. The η-invariant is defined as the regularization of the winding number. The formula for the variation of the η-invariant when the operator changes is obtained. The results are based on the study of asymptotic expansions of traces of parameter-dependent nonlocal operators.
On a closed smooth manifold, we consider operator families being linear combinations of parameter-dependent pseudodifferential operators with periodic coefficients. Such families arise in studying nonlocal elliptic problems on manifolds with isolated singularities and/or with cylindrical ends. The aim of the work is to construct the ????-invariant for invertible families and to study its properties. We follow Melrose’s approach who treated the ????-invariant as a generalization of the winding number being equal to the integral the trace of the logarithmic derivative of the family. At the same time, the Melrose ????-invariant is equal to the regularized integral of the regularized trace of the logarithmic derivative of the family. In our situation, for the trace regularization, we employ the operator of difference differentiating instead of the usual differentation used by Melrose. The main technical result is the fact that the operator of difference differentiation is an isomorphism between the spaces of functions with conormal asymptotics at infinity and this allows us to determine the regularized trace. Since the obtained regularized trace can increase at infinity, we also introduce a regularization for the integral. Our integral regularization involves an averaging operation. Then we establish the main properties of the ????-invariant. Namely, the ????-invariant in the sense of this work satisfies the logarithmic property and is a generalization of Melrose’s ????-invariant, that is, it coincides with it for usual parameter-dependent pseudodifferential operators. Finally, we provide a formula for the variation of the ????-invariant under a variation of the family.
We define η-invariants for periodic pseudodifferential operators on the real line and establish their main properties. In particular, it is proved that the η-invariant satisfies logarithmic property and a formula for the derivative of the η-invariant of an operator family with respect to the parameter is obtained. Furthermore, we establish an index formula for elliptic pseudodifferential operators on the real line periodic at infinity. The contribution of infinity to the index formula is given by the constructed η-invariant. Finally, we compute η-invariants of differential operators in terms of the spectrum of their monodromy matrices.
We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov’s theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of abelian groups.
Given a pair (M,X), where X is a smooth submanifold in a smooth manifold M, we consider complexes of operators associated with this pair. We describe the notion of ellipticity in this situation and prove the Fredholm property for elliptic complexes. As applications, we consider the relative de Rham complex and Dolbeault complex.
For a smooth pair (M,X) consisting of a manifold M and its submanifold X, there is a trace — taking operation that matches each operator on the enclosing manifold with its trace-some operator on the submanifold. In this paper, we investigate traces of operators associated with actions of compact Lie groups on a variety M. We establish that the traces of such operators are concentrated on special submanifolds in X, and investigate the structure of the trace in the neighborhood of these submanifolds.
For a pair of smooth transversally intersecting submanifolds in some enclosing smooth manifold, we study the algebra generated by pseudodifferential operators and (co)boundary operators corresponding to the submanifolds. It is established that this algebra has 18 types of generating elements. For operators from this algebra, the concept of a symbol is defined and the composition formula is established.
We consider Sobolev problems (problems for an elliptic operator on a closed manifold with conditions on a closed submanifold) for the case in which these conditions are of nonlocal nature and include weighted spherical means of the unknown function over spheres of a given radius. For such problems, we establish a criterion for the Fredholm property and, in some special cases, obtain index formulas.
In this paper, we consider the problem of computing a group of stable homotopy classes of pseudodifferential elliptic boundary value problems. This problem is investigated in terms of topological K-groups of certain spaces in the following situations: for boundary value problems on a manifold with an edge, for conjugation problems with conditions on a closed submanifold of codimension one, and for non-local problems with contractions.
In this paper, we study elliptic operators on manifolds with singularities in the situation when a discrete group G acts on the manifold. As is usual in elliptic theory, the Fredholm character of an operator is determined by the main character. We show that in this situation the symbol is a pair consisting of a symbol on the main stratum (inner symbol) and a symbol at the conic point (conormal symbol). The Fredholm property of elliptic elements is established.
We give a statement of dilation-contraction boundary value problems on manifolds with boundary in the scale of Sobolev spaces. For such problems, we introduce the notion of symbol and prove the corresponding finiteness theorem.
The aim of this book is to explain the theory of complex differential equations on a complex manifold.
We study boundary value problems with dilations and contractions on manifolds with boundary. We construct a C∗-algebra of such problems generated by zero-order operators. We compute the trajectory symbols of elements of this algebra, obtain an analog of the Shapiro-Lopatinskii condition for such problems, and prove the corresponding finiteness theorem.
For the action of a discrete group G on a smooth compact manifold M with an edge, we consider a class of operators generated by pseudo-differential operators on M and shift operators associated with the action of the group. For elliptic operators from this class, we establish a classification up to stable homotopies and show that the group of stable homotopy classes of such problems is isomorphic to the K-group of the crossed product of the algebra of continuous functions on the cotangent bundle of the interior of the manifold and the group G acting on this algebra by automorphisms.
We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely, differential operators with shifts induced by the action of a (not necessarily periodic) isometric diffeomorphism. The key to the solution is the method of uniformization. To the nonlocal problem we assign a pseudodifferential operator, with the same index, acting on the sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.
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.
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 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.