Topology of Manifolds and Varieties (Advances in Soviet Mathematics, Vol 18)

Topology of Manifolds and Varieties
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In the first forty or so years of the twentieth century there were two principal strands in algebraic geometry; one was the geometric approach of the Italian school, and the other was the transcendental approach as represented by Lefschetz and Hodge and continued by Kodaira and Spencer. In the last half-century a third strand, the algebraic approach of Weil, Zariski, and Grothendieck was added, and all three strands have now become intertwined. There is no better illustration of this than the Weil conjectures. These purely arithmetical statements were formulated by Weil, who also understood that they could be proved if a ''suitable'' cohomology theory could be developed for varieties defined over a field of finite characteristic.

Such a cohomology theory was introduced by Grothendieck, and as is well known, the Weil program was completed by Deligne, who in effect used an inductive procedure reminiscent of the Lefschetz pencil method to prove an arithmetic analogue of the Hard Lefschetz Theorem.

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When we look at algebraic geometry today, we not only see the intertwining of the historical strands within the field, but equally, we see algebraic geometry intertwined with the rest of mathematics and central to the ongoing developments of the field. One illustration of this is the use of algebraic geometry to generate solutions of special differential equations, both ordinary and partial. Here I mention the work originating from the Russian school that. To some extent it may be said that we have come full circle to the historical roots of algebraic geometry in the study of special transcendental functions arising from abelian integrals, abelian sums, and periods as explained above.

In all of these developments, the topological properties of algebraic varieties, as part of the infrastructure of algebraic geometry, play a central role. I would like to mention a very beautiful recent development that exemplifies both a style and subject that are direct descendents of Lefschetz. This influence is manifest today in his theorems, some of which were stated above, in the intertwining of topology and algebraic geometry, and in his overall approach to mathematics.

Much of Lefschetz' work in topology is concerned with the notion of "fixed point. The first important result of fixed-point theory was proved by L. Brouwer in It asserts that, if E is a closed n -cell, then every mapping of E into itself has at least one fixed point. This result becomes false if E is replaced by a space with a more complicated topological structure.

On the other hand, any mapping of S into itself has a well-defined degree d f [intuitively, d f is the number of times that f maps S around itself; a more precise definition is given in the next paragraph]. If t q is the trace of this matrix, the Lefschetz number of f is the alternating sum L f of the integers t q. The homomorphisms f q depend only on the homotopy class of the map f and therefore the same is true for the Lefschetz number.

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In his first proof of the fixed-point theorem in , 1 , Lefschetz made the additional assumption that X is an orientable closed n -manifold. One can approximate the map f by a map g that has only a finite number of fixed points and that is well-behaved near each fixed point x in the sense that g maps some neighborhood of x homomorphically upon another neighborhood.

The Cartesian square of X is an orientable 2n -manifold Y, and the diagonal D and the graph G of g can be regarded as n -cycles in Y. Their intersection consists of all points x,x such that x is a fixed point of g ; it is a zero-cycle y of Y whose Kronecker index I y is easily seen to equal the sum of the indices of the fixed points of g. While this proof is attractive, it suffers from the disadvantage that it fails to include the Brouwer theorem as a special case. It was to remedy this situation that Lefschetz invented relative homology.

Like many other results of the time, the Lefschetz Duality Theorem was awkward to state because the correct concepts had not yet been developed. Expressed in modern language, it asserts that, if X is a compact, oriented n -manifold with regular boundary A, then the relative ho-. For the purpose of proving the fixed-point theorem, it suffices to know that the homology groups H q X,A and H n-q X are dual vector spaces. This was sufficient to modify the proof above to cover the case of manifolds with regular boundary, and this was done by Lefschetz in , 2.

The importance of the Lefschetz duality theorem was not limited to this application. On the other hand, if A is a subcomplex of a triangulation of the n -sphere S , and if U is a regular neighborhood of A, then S - U is a manifold with regular boundary b U , to which we may apply the Lefschetz duality theorem to conclude that Hq [ S - U,b U ] and H n-q S-U are isomorphic.

Using standard by now! Thus, the Lefschetz duality theorem appears as a unifying factor, connecting two important but apparently unrelated results. Not content with this version of the fixed-point theorem, Lefschetz continued to seek generalizations. In Hopf had proved the theorem for arbitrary compacy polyhedra, but with some restriction on the map f. By Lefschetz had succeeded in removing the latter restrictions ,2 , and by he was able to remove the hypothesis of triangulability of X , requiring instead that X be a compactum which is homologically locally connected in a suitable sense ,4.

The extensions of the fixed point theorem to more general spaces are not simply generalizations for their own sake. Indeed, fixed point theorems often appear in analysis as tools for proving existence theorems. To mention a very simple example, consider the ordinary first-order boundary problem. The correspondence that associates to each function y the function defined by the right-hand side of the latter equation may be regarded as a mapping f of a suitable function space into itself.

And a solution of the equation is nothing but a fixed point of f.

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To be sure, the function spaces appearing here and in other places in analysis are far from being compact, and so the Lefschetz theorem does not apply directly. Nevertheless, this point of view has been a very powerful one in modern analysis. The importance of Lefschetz' work, however, is not limited to the study of fixed-point theorems. The notion of a fixed point of a map of a space into itself can be thought of as a special case of that of a coincidence. This suggested to Lefschetz the idea of defining intersections in an orientable manifold M.

This he succeeded in doing with the result that the graded. All this took place before the discovery of cohomology. While Lefschetz did not define cohomology groups, he introduced pseudo-cycles in ,4. They were not defined intrinsically but, rather, were defined in terms of an embedding of the space in question in a sphere, and were used only as a tool for the proof of one of the versions of his fixed-point theorem. It was not until the late s that the modern treatment of cohomology and cup products was given.

Other of Lefschetz' ideas that by now have thoroughly permeated the subject include singular homology theory and relative homology. While Lefschetz was not the first to use singular chains, his Colloquium Lectures ,1 gave the first formal treatment of the theory. His theory had some mild defects the chain groups turned out not to be free , but these were corrected by Eilenberg in , and the resulting theory has been of the greatest importance. As for relative homology groups, they are principal ingredients in the axiomatic treatment of the homology theory by Eilenberg and Steenrod, which has been so influential in the development of the subject in the last thirty or so years.

Lefschetz was nearly sixty years old when he turned to differential equations, and he devoted the last twenty-five years of his life to the subject. He wrote over forty papers, articles, and books in this field and formed around him a vigorous and distinguished school, guiding and encouraging students and young mathematicians to work on problems of significance.

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In fact, he rekindled interest in a subject that had been nearly totally neglected in the United. States, and he recognized its mathematical importance and practical implications. Although Lefschetz' own contributions to differential equations, control theory, and dynamical systems are not comparable to his great work in algebraic geometry and topology, he nevertheless wrote noteworthy original papers in these areas.

His main interests centered around the theory of dissipative as distinct from conservative dynamical systems, including structural stability, and the resolution of singularities of critical points and bifurcating periodic orbits. Dissipative dynamical systems are important in engineering problems where friction and resistance are essential ingredients. Such dynamical systems can be represented as vector fields on the phase-space manifold. Let S be the set of all C 1 vector fields on a compact differentiable manifold M without boundary and assign to S the C 1 -topology.

Two systems V 1 and V 2 of S are said to be qualitatively equivalent if there is a homomorphism of M mapping the unparameterized solution curves of V 1 onto those of V 2. Structurally stable differential systems are important in applied problems where the parameters of the physical process are known only approximately. Lefschetz stimulated and guided work on these qualitative problems of global analysis. Peixoto proved that the structurally stable systems on a compact surface form an open dense subset of S. Markus proved that, on arbitrary n -dimensional manifolds M, a structurally stable system must necessarily have isolated and elementary critical points and periodic orbits.

Lefschetz was the first person from outside the former Soviet Union to recognize the importance of Liapunov's. He opened up the field of the mathematical theory of control, and in , one of his students, Donald Bushaw, gave the first complete solution of a nontrivial problem in optimal control. Among his other original contributions was his work on the behavior of solutions of analytic differential equations near an isolated singular point. He gave a complete characterization and a constructive procedure for obtaining all the solution curves of a two-dimensional system near an isolated critical point that pass through this critical point ,1.

For a two-dimensional analytic system for which the coefficient matrix of the linear variational equation of an isolated critical point has both roots zero but is not identically zero, he proved that there can be at most a single nested oval of orbits ,1. He gave one of the best treatments of the method of determining the stability of an isolated equilibrium point of an n -dimensional system for which the linear variational equation has some zero roots ,1.

He also studied the existence of periodic solutions of second- and higher-order nonlinear systems of differential equations see ,2; ,2. Phillip Griffiths wrote the section on algebraic geometry, Donald Spencer wrote the sections on personal history and ordinary differential equations, and George Whitehead wrote the section on topology. The date of the accident has been given incorrectly by several authors. The account of it here is taken from a communication by Lefschetz to the Academy dated January 8, , and entitled "A Self Portrait," an unpublished document that was requested by A.

Wetmore on behalf of the Academy. Topology can be described as the study of continuous functions, and it is customary to use the work "map" or "mapping" when referring to such functions. Nebeker and A. Lefschetz, edited by R. Fox, D. Spencer, and A. Tucker, Princeton University Press, , pp. On the V 3 3 with five nodes of the second species in S4. Double curves of surfaces projected from space of four dimensions Bull. Doctoral dissertation, Clark University, Math Note on the n -dimensional cycles of an algebraic n -dimensional variety.

Palermo On the residues of double integrals belonging to an algebraic surface Quart. Pure Appl. Lincei Paris Annali Mat. On the real folds of Abelian varieties. Real hypersurfaces contained in Abelian varieties. Correction, Ann. On certain numerical invariants of algebraic varieties with application to Abelian varieties. Report on curves traced on algebraic surfaces. Paris: Gauthier-Villars.

New edition, Continuous transformations of manifolds. Transformations of manifolds with a boundary. In Memorian N. Lobatschevskii , pp. Manifolds with a boundary and their transformations. The residual set of a complex on a manifold and related questions Proc. On the functional independence of ratios of theta functions. In Selected Topics in Algebraic Geometry , vol.

NRC Bulletin no. Washington, D. A theorem on correspondence on algebraic curves. Duality relations in topology. Colloquium Publications, vol. New York: American Mathematical Society. With W. On the duality theorems for the Betti numbers of topological manifolds Proc. Sbornik With J. On analytical complexes. Princeton, N. Algebraicheskaia geometriia: metody, problemy, tendentsii.

Application of chain-deformations to critical points and extremals Proc. A theorem on extremals. I, II. On critical sets. Duke Math. Locally connected sets and their applications. Part 1. Part 2. On locally connected sets and retracts. Linear degenerations of flag varieties Cerulli Irelli G. Proceedings of the American Mathematical Society. Arnold Mathematical Journal. We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i. Communications in Algebra.

Examples of cylindrical Fano fourfolds Prokhorov Y. Families of disjoint divisors on varieties Bogomolov F. Duke Mathematical Journal. Generalized Weyl modules Makedonskyi I. Generalized Weyl modules and nonsymmetric q-Whittaker functions Feigin E. We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions.

In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules. Generalized Weyl modules for twisted current algebras Feigin E. We introduce the notion of generalized Weyl modules for twisted current algebras.

We study their representation-theoretic and combinatorial properties and connection to the theory of nonsymmetric Macdonald polynomials. As an application we compute the dimension of the classical Weyl modules in the remaining unknown case. Geometric mitosis Valentina Kiritchenko. Geometry and combinatorics of Kostka-Shoji polynomials Finkelberg M. Grassmannians, flag varieties, and Gelfand-Zetlin polytopes Smirnov E. Providence: AMS, Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics Kuznetsov A.

Gushel-Mukai varieties: linear spaces and periods Debarre O. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds Kuznetsov A. Hirzebruch functional equations and complex Krichever genera Netay I. Hopf surfaces in locally conformally Kahler manifolds with potential Ornea L. Research in the Mathematical Sciences. Societe Mathematique de France, Classification of zeta functions of bielliptic surfaces over finite fields Rybakov S. Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry Amerik E.

Complex geometry of moment-angle manifolds Panov T. Comultiplication for shifted Yangians and quantum open Toda lattice Rybnikov L. Series math "arxiv. Construction of automorphisms of hyperkahler manifolds Amerik E. Cylinders in singular del Pezzo surfaces Cheltsov I. Derived categories of curves as components of Fano manifolds Fonarev A. Derived categories of Gushel-Mukai varieties Kuznetsov A. Determinantal identities for flagged Schur and Schubert polynomials Smirnov E. Division polynomials and intersection of projective torsion points Bogomolov F.

Endomorphisms of projective bundles over a certain class of varieties Amerik E. Experimental Mathematics.

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Epub Oct 4. Mat Ob-va v. Floer cohomology of Lagrangian intersections and pseudoholomorphic disks. While this proof is attractive, it suffers from the disadvantage that it fails to include the Brouwer theorem as a special case. Zuber and R.

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. A criterion for left-orthogonality of an effective divisor on a surface Alexey Elagin. Acyclicity of non-linearizable line bundles on fake projective planes Galkin S. Affine cones over smooth cubic surfaces Cheltsov I. Algebraic dimension of complex nilmanifolds Fino A. Algebraic non-hyperbolicity of hyperkahler manifolds with Picard rank greater than one Kamenova L. A simple proof of the non-rationality of a general quartic double solid Prokhorov Y.

Birationally isotrivial fiber spaces Bogomolov F. The Gauss-Manin connection on the periodic cyclic homology Vologodsky V. Oxford University Press, Torsion of elliptic curves and unlikely intersections Bogomolov F. Towards the moduli space of special Bohr - Sommerfeld lagrangian cycles Tyurin N. Transcendental Hodge algebra Verbitsky M. Twisted zastava and q-Whittacker functions Braverman A. Universal spaces for unramified Galois cohomology Bogomolov F. Progress in Mathematics.

Unobstructed symplectic packing by ellipsoids for tori and hyperkahler manifolds Entov M. Weyl modules for osp 1,2 and nonsymmetric Macdonald polynomials Feigin E. Weyl n-Algebras Nikita Markarian. Newton-Okounkov polytopes of flag varieties Valentina Kiritchenko. Non-degenerate locally connected models for plane continua and Julia sets Blokh A.

On contraction of algebraic points Bogomolov F. Bulletin of the Korean Mathematical Society. On M-functions associated with modular forms Lebacque P. Le Centre pour la Communication Scientifique Directe, On monodromy groups of del Pezzo surfaces Serge Lvovski. On the algebra generated by projectors with commutator relation Zhdanovskiy I.

Lobachevskii Journal of Mathematics. Perfect subspaces of quadratic laminations Blokh A. Quasi-simple finite groups of essential dimension 3 Prokhorov Y. Fundamental and Applied Mathematics. Representation theoretic realization of non-symmetric Macdonald polynomials at infinity Feigin E. Rigid hyperholomorphic sheaves remain rigid along twistor deformations of the underlying hyparkahler manifold Verbitsky M. Schubert Quiver Grassmannians Cerulli I. Algebras and Representation Theory. Ergodic complex structures on hyperkahler manifolds: an erratum Verbitsky M.

Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac—Moody algebras Gritsenko V. Transactions of the Moscow Mathematical Society. Families of algebraic varieties and towers of algebraic curves over finite fields Rybakov S. Favourable modules: filtrations, polytopes, Newton—Okounkov bodies and flat degenerations Feigin E. Hilbert-Samuel sequences of homogeneous finite type Loginov K.

Journal of Pure and Applied Algebra. Hilbert's theorem 90 for non-compact groups Rovinsky M.

Complete list of publications

Instanton moduli spaces and V-Algebras Braverman A. Integral Chow motives of threefolds with K-motives of unit type Gorchinskiy S. Kuga-Satake construction and cohomology of hyperkahler manifolds Kurnosov N. Linear degenerations of flag varieties Cerulli I. Kyoto Journal of Mathematics. Minimal del Pezzo surfaces of degree 2 over finite fields Trepalin A. Models for spaces of dendritic polynomials Timorin V. Morrison-Kawamata cone conjecture for hyperkahler manifolds Amerik E. Annales Scientifiques de l'Ecole Normale Superieure. Nef partitions for codimension 2 weighted complete intersections Przyjalkowski V.

Tokyo: The Mathematical Society of Japan, Newton-Okounkov bodies sprouting on the valuative tree Shramov K. Rendiconti del Circolo Matematico di Palermo. Automorphism groups of compact complex surfaces Prokhorov Y. Birationally rigid fano threefold hypersurfaces Cheltsov I. Canonical tilting relative generators Bodzenta A. Categorical measures for finite group actions Bergh D.

Characteristic foliation on non-uniruled smooth divisors on hyperkahler manifolds Amerik E. Combinatorial models for spaces of cubic polynomials Ptacek R. Comptes Rendus Mathematique. Conjecture on theta-blocks of order 1 Valery Gritsenko, Wang H. Cylinders in del Pezzo Surfaces Cheltsov I. Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields Ballet S.

Derived categories of Grassmannians over integers and modular representation theory Efimov A. Diffusion-orthogonal polynomial systems of maximal weighted degree Soukhanov L. Dominant classes of projective varieties Bogomolov F. Algebraically hyperbolic manifolds have finite automorphism groups Verbitsky M. New York Journal of Mathematics. Symmetric Dellac configurations Bigeni A. Threefold extremal curve germs with one non-Gorenstein point Prokhorov Y. Ultraviolet properties of the self-dual Yang-Mills theory Losev A.

Unobstructed symplectic packing by ellipsoids for tori and hyperkahler manifolds Verbitsky M. Unstable polarized del Pezzo surfaces Cheltsov I. Vertex algebras and coordinate rings of semi-infinite flags Feigin E. Journal of the Mathematical Society of Japan. Witt vectors as a polynomial functor Kaledin D. On a conjecture of Tian Cheltsov I. On flops and canonical metrics Cheltsov I. On linear sections of the spinor tenfold. I Kuznetsov A. Izvestiya: Mathematics. On monodromy in families of elliptic curves over C Lvovsky S. On non-rational fibers of del Pezzo fibrations over curves Loginov K.

On residual categories for Grassmannians Kuznetsov A. On stable cohomology of central extensions of elementary abelian groups Bogomolov F. On the characteristic foliation on a smooth hypersurface in a holomorphic symplectic fourfold Amerik E. Prime Fano threefolds of genus 12 with a Gm-action and their automorphisms Kuznetsov A. Mathematische Nachrichten. Pullbacks of hyperplane sections for Lagrangian fibrations are primitive Verbitsky M.

Manifolds

Purely noncommuting groups Bogomolov F. Quotients of del Pezzo surfaces of high degree Trepalin A. Semilinear representations of symmetric groups and of automorphism groups of universal domains Rovinsky M. Some properties of surjective rational maps Karzhemanov I. Stable Higgs bundles over positive principal elliptic fibrations Verbitsky M. Finite groups of birational selfmaps of threefolds Prokhorov Y.

Finite quasisimple groups acting on rationally connected threefolds Prokhorov Y. Flat affine subvarieties in Oeljeklaus-Toma manifolds Verbitsky M. Flat affine subvarieties in Oeljeklaus—Toma manifolds Verbitsky M. Fourier transform on hyperplane arrangements Finkelberg M. Gushel-Mukai varieties: Classification and birationalities Kuznetsov A. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds Prokhorov Y.

Japanese Journal of Mathematics. Hodge complexity for weighted complete intersections Shramov K. Inverse Galois problem for del Pezzo surfaces over finite fields Trepalin A. Kaehler-Einstein Fano threefolds of degree 22 Cheltsov I. MBM loci in families of hyperkahler manifolds and centers of birational contractions Amerik E. Multiplicity of singularities is not a bi-Lipschitz invariant Verbitsky M.

International Mathematical Research Notices. Co-periodic cyclic homology Kaledin D. Coulomb branches of 3-dimensional gauge theories and related structures Finkelberg M. Delta invariants of singular del Pezzo surfaces Cheltsov I. Delta invariants of smooth cubic surfaces Cheltsov I. Derived categories of singular surfaces Kuznetsov A. Dominant classes of projective varieties Buonerba F.

Our publications (2012)

Series Monográficas: Advances in Soviet mathematics QA L Vol. Viro, Oleg Topology of Manifolds and Varieties QA T Vol. 7. Birman, M. Sh. -oriented compact nonsingular real algebraic variety of dimension $n$. If $i:X \ rightarrow Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry, Ergebnisse der Math. vol. MR 96b; Viro, O.: Topology of manifolds and varieties, Advances in Soviet Mathematics, American Mathematical Soc. Vol.

Double covers of quadratic degeneracy and Lagrangian intersection loci Kuznetsov A. Elliptic curves with large intersection of projective torsion points Bogomolov F. Embedding derived categories of Enriques surfaces into derived categories of Fano varieties Kuznetsov A. Fano-Mukai fourfolds of genus 10 as compactifications of C4 Prokhorov Y. Fano threefolds with infinite automorphism groups Shramov K. Finite collineation groups and birational rigidity Cheltsov I. Algebraic dimension of complex nilmanifolds Verbitsky M.

SOLOMON LEFSCHETZ

Alpha-invariants and purely log terminal blow-ups Cheltsov I. Automorphisms of pointless surfaces Shramov K. Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces Cheltsov I. Classification of non-Kahler surfaces and locally conformally Kahler geometry Verbitsky M.

Higher Whitehead products in moment-angle complexes and substitution of simplicial complexes Semyon Abramyan, Panov T. Homomorphisms of multiplicative groups of fields preserving algebraic dependence Bogomolov F. Linear degenerations of flag varieties: partial flags, defining equations, and group actions Cerulli Irelli G. Which quartic double solids are rational? Przyjalkowski V. Journal of Algebraic Geometry. Cornell University, Parabolically connected subgroups Netay I. Cornell University, Remarks on endomorphisms and rational points Bogomolov F. Cornell University, Weakly curved A-infinity algebras over a topological local ring Positselski L.

Refined blowups D. Kaledin, A. Cornell University, Relativistic classical integrable tops and quantum R-matrices A. Cornell University, Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category A. Cornell University, Smart criticality Blokh A. Cornell University, Some applications of p-adic uniformization to algebraic dynamics Amerik E. Cornell University, The ambiguity index of an equipped finite group F.

Cornell University, The conifold point Galkin S. Cornell University, On equivariant triangulated categories Alexey Elagin. Cornell University, On surfaces with zero vanishing cycles Serge Lvovski. Cornell University, On the phenomena of constant curvature in the diffusion-orthogonal polynomials Lev Soukhanov. Cornell University, Exceptional collections on some fake quadrics Lee K. Geometric mitosis Kiritchenko V. Cornell University, Isomonodromic differential equations and differential categories Gorchinskiy S. Cornell University, Combinatorial models for spaces of cubic polynomials Blokh A.

Complete list of publications – Alexander Varchenko

Cornell University, Contraherent cosheaves Positselski L. Cornell University, Derived categories of Grassmannians over integers and modular representation theory Alexander I. Cornell University, Double solids, categories and non-rationality Iliev A. Cornell University, Rational curves on foliated varieties Bogomolov F. Cornell University, Gamma conjecture via mirror symmetry Galkin S. Cornell University, Height of exceptional collections and Hochschild cohomology of quasiphantom categories Kuznetsov A. Cornell University, Merging divisorial with colored fans Altmann K.

Cornell University, Cohomology of exact categories and non- additive sheaves Kaledin D. Cornell University, Complex rotation numbers Buff X. CRC Press, Cornell University, Degenerate twistor spaces for hyperkahler manifolds Verbitsky M. Cornell University, Bokstein homomorphism as a universal object Kaledin D. Cornell University, Two rational nodal quartic 3-folds Cheltsov I. Cornell University, On the Kobayashi pseudometric, complex automorphisms and hyperkaehler manifolds Bogomolov F.

Cornell University, Quadratic-like dynamics of cubic polynomials Blokh A. Cornell University, Irreducible representations of finitely generated nilpotent groups Beloshapka I. Cornell University, Laminations from the main cubioid Blokh A. Cornell University, Hilbert schemes of lines and conics and automorphism groups of Fano threefolds Kuznetsov A.

Cornell University, Hopf surfaces in locally conformally Kahler manifolds with potential Ornea L. Cornell University, Classification of zeta functions of bielliptic surfaces over finite fields Rybakov S. Cornell University, Complex geometry of moment-angle manifolds Panov T. Cornell University, Construction of automorphisms of hyperkahler manifolds Amerik E. Cornell University, Cylinders in rational surfaces Cheltsov I. Shaddock, Encyclopaedia Math. II, , Springer, Berlin, Introduction to homotopy theory joint with D. Fundamental directions, f Vol. Extensions of the Gudkov-Rohlin congruence joint with V.

Kharlamov , Lecture Notes in Math. Some integral calculus based on Euler characteristic, Lecture Notes in Math. Non-diffeomorphic but homeomorphic knottings of surfaces in the 4-sphere joint with S. Finashin and M. Kreck , Lecture Notes in Math. Exotic knottings of surfaces in the 4-sphere joint with S. Kreck , Bull. Progress of the last six years in topology of real algebraic varieties, Uspekhi Mat. Nauk Russian [ ] English translation in Russian Math.

Surveys [ ]. Doklady [ ]. The signature of a branched covering, Mat. Notes Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Lecture Notes in Math. Intersections of loops on two-dimensional manifolds. Free loops joint with V. Turaev , Mat. Progress over the last 5 years in topology of real algebraic varieties, Proceedings of the International Congress of Mathematicians, Aug.

Gluing algebraic hypersurfaces and constructions of curves, Tezisy Leningradskoj Mezhdunarodnoj Topologicheskoj Konferencii , Nauka Russian. AG] [ ]. Complex orientations of real algebraic surfaces, Uspekhi Mat. Nauk 93 Russian. Colored knots, Kvant No.

Analytic Topology

Quantum 8 , no. Doklady 22 , No. Doklady 20 , No. Constructing M-surfaces, Funkts. Generalizing Petrovsky and Arnold inequalities for curves with singularities, Uspekhi Mat.