\begin{document} \begin{center} \Large\bf Biographical sketch\\ \end{center} \noindent{Alexander Volberg} \medskip {\bf personal homepage:} http://www.math.msu.edu/$\widetilde{}$volberg \medskip {\bf DEGREES:} \begin{itemize} \item Ph. D., V.A.Steklov Math Inst., May 1982 \item Doctor of Science, V.A. Steklov Math. Institute, 1989 \end{itemize} {\bf POSITIONS:} \begin{itemize} \item 1977 -- 1988; Leningrad Institute of Electrotechnical Engineering (Assist.prof.). \item 1988 -- 1989; Leningrad branch of V.A.Steklov Math.Institute (Researcher). \item September 1991 -- now; Full Professor - Michigan State University. \item Spring Quarter, 1997 ; Visiting Professor - Caltech. \item Fall semester, 1997; Research Professor - MSRI, Harmonic Analysis Program. \item June of 1999, Visiting professor, Schr\"odinger Inst. for Math. Physics, Vienna. \item June of 2000, Visiting professor, Univ. Bordeaux I. \item May of 2001; Visiting professor - Stanford University.\item Sept. 2001--Sept 2003. Professor, Univ. Paris VI. \item Spring 2004, member of the IAS, Princeton.\item Fall 2004, visiting researcher IPAM, UCLA\item Spring 2005, visiting research professor, MSRI, Berkeley. \end{itemize} {\bf DISSERTATIONS:} \begin{itemize} \item Density of the systems of rational fractions in weighted $L^p$ spaces of the circle. 20.5.1982, Leningrad, Leningrad branch of V.A. Steklov Math. Institute (LOMI). \item Asymptotically holomorphic functions and their applications. 30.11.1989, Leningrad, Leningrad branch of V.A. Steklov Math. Institute \end{itemize} {\bf HONORS:} \begin{itemize} \item Prix Salem: Le prix international de math\'{e}matiques, en hommage \`{a} la memoire de Raphael Salem et \`{a} son oevre math\'{e}matiques, a \'{e}t\'{e} decern\'{e} en 1988 \`{a} A. Volberg. Le jury \'{e}tait compos\'{e} de MM. les Professeurs V. Havin, Y. Katznelson,Y. Meyer, E. Stein. \item invited speaker at ICM-90 in Kyoto, Japan. The talk: ``Asymptotically holomorphic functions and their use in analysis". \item invited 1 hour address, AMS Central Region Meeting, Detroit, MI, May 1997. \item invited Karl Str\"omberg lecture, Kansas State Univ., Manhattan, Kansas, Sept. 1998. \item Distinguished Faculty Award, College of Natural Science, MSU, 2000. \item LARS ONSAGER professorship, Norwegian Institute of Technology, 2005. \end{itemize} {\bf INVITED SPEAKER OR RESEARCHER RECENTLY:} \begin{itemize} \item Invited speaker to Conference in Math. Analysis and Applications honoring Lars Hedberg, Linkoping, Sweden, 1996. \item Visiting Professor, Caltech, Spring quarter, 1997. \item Research professorship, Harmonic analysis semester, Berkeley, MSRI, Fall 1997. \item Invited speaker at Satellite to ICM: Geometric methods in Fourier and Functional Analysis, Kiel, Germany, August 1998. \item Invited speaker at Meeting of London Math. Soc., London, Oct. 16-17, 1998. \item Visiting researcher, Erwin Schr\"{o}dinger Institute, Vienna, Austria, June 1999, April 2003. \item Invited speaker at International Workshop on Operator Theory, Bordeaux, France, June 2000. \item Invited lecturer at Satellite conference to European Congress of Mathematicians, El Escorial, Spain, August 2000. \item Invited lecturer at Paseky summer school in Analysis, Czech Republic, June 2000. \item Invited speaker at International Conference on Harmonic Analysis, Heat kernel and PDE, Paris, 2001. \item Invited speaker at International Workshop on Operator Theory, Blacksburg, VA, 2002. \item Invited lecturer at NSF CBMS Conference at Chapel Hill, NC, 2002. \item invited speaker, 19 Nevanlinna Colloquium, Jyvaskyla, Finland, June 2003. \item invited lecturer at Harmonic Analysis and PDE trimester, Scuola Normale Superiore de Pisa, September 2003 and June 2004.\item invited lecturer at Harmonic Analysis conference, Osaka Nov. 2004. \end{itemize} {\bf INVITED to AMS meetings:} \begin{itemize} \item 1994: Manhattan,Kansas; \item 1994: Cincinnati, Ohio; \item 1995: San Fransisco, Ca. \item 1997: Milwaukee, Wisconsin; \item 1997: Albuquerque, New Mexico; \item 1997: invited 1 hour address, AMS Central Region Meeting, Detroit, MI. \item 1999: Austin, Texas; \item 2000: Washington, DC. \item 2001: Lawrence, KS. \item 2002: Madison, WI.\item 2004: Los Angeles, CA. \end{itemize} {\bf GRADUATED STUDENTS:} \begin{itemize} \item Zoltan Balogh, graduated in Aug.1995. Dissertation "Metric properties of semihyperbolic repellers with application to harmonic measure". Now: Full Professor, Univ. of Bern, Switzerland. \item Irina Popovici, graduated in 1997. \item First job at UCLA (Univ. of California at Los Angeles), Hedrick assistant professorship. \item Now: Assoc. Professor, US Navy Academy, Annapolis, MD. \item Stefanie Petrmichel, finished in June 2000. \item First job at Princeton (IAS) followed by Tamarkin Assist. Prof. at Brown Univ. \item Oliver Dragicevic, finished in 2003. First job: Postdoctoral position at Scuola Normale Superiore de Pisa. \item Leonid Slavin, finished May 2004, Dissertation ``Bellman functions and BMO". First job: PostDoc at the Univ. of Connecticut. \end{itemize} {\bf NSF GRANTS:} \begin{itemize} \item 1991-1992. DMS 9101788. Asymptotically holomorphic functions. Joint with J.E.Brennan. \item 1993-1995. DMS 9302728. Three measures on fractals. \item 1996-1998. DMS 9622936. Operator approach to problems in analysis and probability: Matrix Muckenhoupt weights, Hankel and Toeplitz operators, the angle between past and future. Joint with S.R.Treil. \item 1999-2001. DMS 9970395. Nonhomogeneous harmonic analysis with applications to probability, analytic capacity and Geometric Measure Theory. Joint with S. Treil, F. Nazarov. \item 2002-2004. DMS 0200713. Multidimensional and Non-Homogeneous Harmonic Analysis: Bellman Functions, Pertubations of Normal Operators and Two Weight Estimates of Singular Integrals. Joint with S. Treil, F. Nazarov. \end{itemize} {\bf OTHER GRANTS:} \begin{itemize} \item Binational Israeli-USA grant, 1997-1999, Hilbert transform and approximation in Analysis and Probability. Joint with A.Atzmon, V.Matsaev, F.Nazarov, M.Sodin, S.R.Treil. \end{itemize} {\bf PUBLICATIONS of Alexandre Volberg.} \vspace{.2in} {\bf 1978-1983 publications} \bigskip 1. Mean square completeness of polynomials beyond the scope of a theorem of Szego.-Dokl. Akad. Nauk SSSR, 1978, v.241, 3, 521-524 (in Russian). English translation in Sov. Math. Dokl., 1978, 19, 4, 877-881. 2. Simultaneous approximation by polynomials on the circle and in the disc.-Zap. nauchn. semin. LOMI, 1979, v.92, 60-84 (in Russian). 3. Thin and thick families of rational fractions.-Lect. Notes in Math. 1981, v.864, pp. 440-480. 4. Density of rational fractions in weighted $ L sub p $ spaces on the circel. Funk. Anal. i ego Pril., 1981, v.15, 2, 69-70 (in Russian). English translation in Funct. Anal. and its Appl., 1981, 15, 2, 130-131. 5. The comparison of integral norms on the subspaces of pseudoextendible functions.Uspekhi Matem. Nauk, 1981, v.36, 6, 205-206 (in Russian) 6. Two remarks concerning the theorem of S. Axler, S.-Y. Chang and D. Sarason. J. of Operator Theory, 1982, v.8, pp. 209-218. 7. (with S.V. Khruschev). A generalization of Koosis-Lax interior compactness theorem. J. of Operator Theory, 1982, v.8, pp. 197-208. 8. The logarithm of an almost analytic function is summable (in Russian). English translation in Soviet Math. Dokl., 1982, v.26, 1, pp. 238-243. \bigskip {\bf 1983-1987 publications} \bigskip 9. Denseness of the polinomials on the system of rays (in Russian). English translation in Soviet Math. Dokl., 1984, v.29, 2, pp. 342-347. 10. (with S.V. Konyagin). On every compact in $R^n$ there is a homogeneous measure (in Russian). Dokl. Akad. Nauk, SSSR, 1984, v.278, 4, 783-788. English translation in Sov. Math. Dokl., 1984, 30, 2, 453-457. 11. (with V.A. Tolokonnikov). Some remarks concerning the multipliers of the Cauchy type integral and the algebras of Sarason. Funk. Anal. i ego Pril., 1984, v. 18, 2, 61-62. English translation in Funct. Anal. and its Appl., 1984, 18, 2, 137-138. 12. A constructive proof of a Marshall-Chang theorem (in Russian). Zap. Nauchn. Semin. LOMI. 1985, v. 141, 149-153. 13. (with V.A. Tolokonnikov). Hankel operators and the problems of best approximation of unbounded functions (in Russian)., Zap. Nauchn. Semin. LOMI, 1985, v.141, 5-17. 14. (with S.R. Treil). Embedding theorems for the invariant subspaces of the backward shift operator. Zap. Nauchn. Semin. LOMI, 1986, v.149, 38-51 (in Russian) 15. (with N.G. Makarov). On the harmonic measure of discontinuous fractals. LOMI preprint, E-6-86, pp. 1-34. \bigskip {\bf 1987-1990 publications} \bigskip 16. (with O.V.Ivanov) Membership of the product of two Hankel operators in Schatten -von Neuman class. Dokl. Acad. Nauk. Ukrain, Ser. A, 1987, No. 4, pp.3-6. 17. (with B. Joricke). The summability of logarithm of an almost analytic function and a generalization of the Levinson-Cartwright theorem. Matem. Sb., 1986, v.130, 3, 335-348 (in Russian). English translation in Math. USSR Sb., 1987, 58, 2, 337-349. 18. (with S.V. Konyagin). On the meausres with doubling condition. Izv. Akad. Nauk. SSSR, 1987, v.51, 3, 666-675 (in Russian). English translation Math. USSR-Izv. 30 (1988), no. 3, 629--638 19. (with B.M. Solomyak). Multiplicity of spectrum of Toeplitz operators, weighted cocycles and the vector Riemann-Hilbert problem. Funk. Anal. i ego Pril., 1987, v. 21, 3, 1-10 (in Russian). English translation in Funct. Anal. and its Appl., 1987, 21, 3, 175-182. 20. (with B.M. Solomyak). Multiplicity of spectrum of Toeplitz operators. Oper. Theory: Adv. Appl., 1989, v. 42, pp 87-192. 21. The Lojasiewicz inequality for very smooth functions. Soviet Math. Dokl., 1990, v.41, 1, pp.170-174. 22. A criterion on a subdomain of the disc to have its harmonic measure comparable with Lebesgue measure. Proc. Amer. Math. Soc., 1991, v.112, 1, pp. 153-162. 23. How to break in a prescribed contour? LOMI preprint, P-1-89 (in Russian). 24. Weighted polynomial approximation in simply connected domains. LOMI preprint, P-2-89 (in Russian). 25. (with A.A. Borichev). Uniquiness theorems for almost analytic functions. Algebra and Analysis, 1989, v.1, 1, 146-177 (in Russian). English Translation in Leningrad J. of Math., 1990, v.1, No. 1. 26. Un theoreme de Dulac-Ecalle-Illyashenko-Martinet-Moussu-Ramis etendu aux fonctions quasianalytiques. Publ. Mathematique d'ORSAY, Semin. d'Analyse Harmonique, 1990 (in French). 27. (with N.K. Nikolski) Tangential and approximate free interpolation. Analysis and PDE. Lecture Notes in Pure and Applied Math. v. 122, pp. 277-299 Dekker, Ne York, 1990. \bigskip {\bf 1991-1994 publications} \bigskip 28. (with V. Peller, Dm. Yakubovich). A brief excursion to a theory of hyponormal operators. Algebra and Analysis, 1990, v.6. (in Russian). English Translation in Leningrad Math. J. 1991, v. 2, No. 2, pp. 211-243. 29. On the harmonic measure of self-similar sets on the plane. in Harmonic Analysis and Potential Theory, ed. M.A.Picardello, Plenum Press; New York, London 1992. 30. (with M.Yu. Lyubich). A comparison of harmonic and maximal measure for rational functions. Proc. of the NATO Research Workshop on ``Approx. by Solutions of Partial Differential Equations, and Related Topics", ed. B.Fuglede, M.Goldstein, W.Haussmann, W.K. Hayman, L.Rogge Kluwer Ac. Publ., NATO ASI Series, Ser C, Vol. 365, 1992. 31. On the dimension of harmonic measure of Cantor repellers. Michigan Math. J., 1993, v. 40, pp. 239-258. 32. Asymptotically holomorphic functions and their applications. Proc. of the International Congress of Mathematiciens 1990, Kyoto, Japan. 33. (with P.P. Kargaev). Three results concerning the support of functions and thir Fourier transforms. Indiana Univ. Math .J, 1992, v.41, No. 4, pp. 1143-1164. 34. Review of the books J.W.Helton . OPERATOR THEORY, ANALYTIC FUNCTIONS, MATRICES,and ELECTRICAL ENGINEERING. CBMS Reg. Conf. Series in Math, V.68, Amer Math Soc, 1987. and Bruce A. Frasncis . A COURSE in $H^{\infty}$ CONTROL THEORY. Lect. Notes in Control and Information Sciences. Vol. 88, Springer, 1987.-in Leningrad Math. J., 1992, v. 3,No. 3. 35.Rapidly growing functions with empty spectrum and a gap in the support. St. Petersburg Math. J., 1993,v.5,3, pp.77-99. 36.(with L. Bialas) Markov's inequality on Cantor-like sets. Studia Math., 1993,104, 3, pp.259-268. 37.(with S.Treil) Nehari-type theorems in weighted $l^2$-spaces via fixed point theorems. Operator Theory: Advances and Applications, v.71, 1994, pp.165-186. 38.(with Zoltan Balogh) Principe de Harnack a la frontiere pour des repulseurs holomorphes non recurrents. C. R. Acad.Sci. Paris, v.319, 1994, p. 311-314. \bigskip {\bf 1995-2004 publications and preprints} \bigskip 39. Markov's inequality for Cantor repellers: Topics on Complex analysis. {\em Banach Center Publ.,} {\bf 31}, (1995), pp. 383-391. Proceedings of the conference on Complex Dynamics. 40. (with M.Yu. Lyubich). A comparison of harmonic and maximal measures on Cantor repellers. {\em J. of Fourier Analysis and Applications,} {\bf 1}, (1995), pp.379-399. Volume in honor of J. P. Kahane. 41.(with R.Younis and D.Zheng) Subalgebras of $C(M(H^{\infty}))$. {\em Proc. AMS}, {\bf 123}, (1995), no. 2, 367-371. 42.(with A.A. Borichev) Finiteness of limit cycles and uniqueness for asymptotically holomorphic functions. {\em St. Petersburg Math. J.,} {\bf 7}, (1996), no. 3, pp.343-368. 43.(with Zoltan Balogh) Normalization of almost conformal parabolic germs. {\em Ann. Acad. Sci. Fenn.}, Ser. A, Math., {\bf 20}, (1995), No.1, 109-121. 44.(with Zoltan Balogh) Geometric localization, uniform Johnpropertoes, and separated semihyperbolic dynamics. {\em Arkiv f\"{u}r Mat.,} {\bf 34} (1996), no. 1, 21-49. 45.(with Zoltan Balogh) Boundary Harnack principle on separated semihyperbolic repellers. Harmonic measure applications. {\em Revista Mathematica Iberoamericana}, {\bf 12} (1996), 299-336. 46.(with Z.Balogh and I. Popovici) Conformally maximal polynomiallike dynamics nd invariant harmonic measure. {\em Ergodic Teory and Dynamical Systems.}, {\bf 17} (1997), 1-27. 47.(with S.R.Treil) Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator. {\em St. Petersburg Math.J.} {\bf 7} (1995), 205-226. 48.(with M.Urbanski) A rigidity result for holomorphic dynamics. {\em Progress in Probability,} {\bf 37}, (1995), 180-187, Birkhauser Verlag. 49.(with I.Popovici) Rigidity of harmonic measure. {\em Fundamenta Math.}, {\bf 150}, (1996), 237-244. 50.(with I.Popovici) Dimension of harmonic measure on the Julia set of one-petal Blaschke products. Algebra i Analysis {\bf 9} (1997), no. 3, 150-197. In English {\em St. Petersburg Math. J.} {\bf 9} (1998), 130-180. 51.(with D.Zheng and S.Treil) Hilbert transform, Toeplitz operators and Hankel oerators, and invariant $A_{\infty}$ weights. {\em Revista Mat. Iberoamericana,} {\bf 13}, (1997), no.2, 319-360. 52.(with S.R. Treil) Wavelets and the angle between past and future. {\em J. of Funct. Analysis,} {\bf 143} (1997), no. 2, 269-308. 53.Matrix $A_p$ condition via $S$-functions. {\em J. of Amer. Math.Soc.,} {\bf 10} (1997), no.2, 445-466. 54.(with F.Nazarov and S.Treil) Counterexample to infinitedimensional Carleson embedding theorem. {\em Comptes Rendus Ac. Sci. Paris}, S\'{e}r. I Math., 1997, {\bf t. 325}, no. 4, 383-388. 55.(with F.Nazarov and S.Treil) Cauchy integral and Calderon-Zygmund operators on nonhomgeneous spaces. {\bf IMRN} {\em Intern. Math. Res. Notes.,}{\bf 1997}, no. 15, 703-726. 56.(with S.Treil) Continuous frame decomposition and vector Hunt--Muckenhoupt--Wheeden theorem. {\em Arkiv f\"{u}r Math. }, {\bf 35} (1997), no. 2, 363-386. 57.(with I.Popovici) A. Boundary Harnack principle for Denjoy domains. {\em Complex Variables Theory Appl.} {\bf 37}, (1998), no. 1-4, 471--490. 58.(with F. Nazarov, S.Treil) Weak type estimates and Cotlar inequalities for Calder\'{o}n-Zygmund operators in nonhomogeneous spaces. {\bf IMRN} {\em Intern. Math. Research Notices}, {\bf 1998}, no. 9, 463-487. 59.(with S.Treil) Completely regular multivariate stationary processes and Muckenhoupt condition. {\em Pacific J. Math.,} {\bf 190}, (1999), no. 2, pp. 361-382. 60.(with K.Baranski and A.Zdunik) The solution of Brennan's conjecture for Fatou sets of quadratic polynomials. {\bf IMRN} {\em Intern. Math. Research Notices}, {\bf 1998}, no. 12, 589-600. 61.(with F. Nazarov, S. Treil) Bellman function and two-weight inequality for martingale transform. {\em J. of Amer. Math.Soc.}, {\bf 12}, (1999), no. 4. 62.(with J. Esterle) Analytic left invariant subspaces and asymptotically holomorphic functions. {\em Comptes Rendus Acad. Sci. Paris}, S\'{e}rie I, {\bf 326}, 1998, p. 295-300. 63.(with J. Esterle) Asymptotically holomorphic functions and translation invariant subspaces of weighted Hilbert spaces of sequences, I. { J. of Operator Theory} 2000. 64.(with J. Esterle) Asymptotically holomorphic functions and translation invariant subspaces of weighted Hilbert spaces of sequences, II. {\em Ann. Sci. Ecole Norm. Sup.}, 2002. 65. (with S. Hukovic, S. Treil) Bellman functions and sharp weighted estimates for the square functions. Complex analysis, operators, and related topics (in memory of S.A.Vinogradov), pp. 97--113, {\em Oper. Theory Adv. Appl.}, {\bf 113}, Birkhauser, Basel, 2000. 66. (with A. Gillespie, S. Pott, S. Treil) Logarithmic growth for matrix martingale transform. Preprint Schr\"{o}dinger Institute, 719 (1999) pp.1-12. See the homepage: http://www.math.msu.edu/$\widetilde{}$volberg {J. London Math. Soc.} (2001) 67. (with A. Gillespie, S. Pott, S. Treil) A transference approach to estimates of vectorial Hankel operators. {\em St. Petersburg Math. J.}, {\bf 12}, (2001), no. 6. 68. (with G. Pisier, F. Nazarov, S. Treil) Sharp estimates for noncommutative Carleson embedding theorem and vector Hankel operators. {\em J. f\"ur die reine und angew. Math.}, v. 542, 2002. 69. (with St. Petermichl) Heating the Beurling operator: weakly quasiregular maps on the plane are quasiregular. See the homepage http://www.math.msu.edu/$\widetilde{}$volberg {\em Duke Math. J.}, v. 112, No. 2, 2002. 70. (with F. Nazarov and S. Treil) Accretive system $Tb$ theorems for nonhomogeneous spaces. See the homepage http://www.math.msu.edu/$\widetilde{}$volberg {\em Duke Math. J.}, v. 113, No. 3, 2002, pp. 259-312. 71. (with F. Nazarov and S. Treil) Bellman function in Stochastic optimal Control and Harmonic Analysis (how our Bellman function got its name.) See the homepage http://www.math.msu.edu/$\widetilde{}$volberg {\em Oper. Theory Adv. Appl.}, {\bf 129}, Birkhauser, Basel, 2002. 72. (with F. Nazarov) Bellman function, two weight Hilbert transform and imbedding for the model space $K_{\theta}$. Volume in the memory of Tom Wolff. {\em J. d'Analyse Math.}, v. 87, 2002, 385-412, 73. (with P. Yuditskii) On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, several intervals or a Cantor set of positive length. {\em Commun. in Math. Physics,} v. 226, (2002), pp. 567-605. 74. (with C. Finet and H. Queffelec) Composition operators in Dirichlet spaces, their smoothness and their numerical range. {\em Comptes Rendus de l'Acad. des Sciences,} v. 335, No. 4, (2002), 325-328. 75. (with S. Petermichl and S. Treil) Riesz transforms are averaging of dyadic shifts. {\em Publ. Mat.}, 2002, pp. 209-229, Proc. of the 6th Conference on Harmonic Analysis and Partial Differential Equations at El Escorial, ed. Patricio Cifuentes, Jose Garcia-Cuerva, Eugenio Harnandez, Fernando Soria, Jose Luis Torrea, and Anna Vargas. 76. (with F. Nazarov, M. Sodin) The geometric Kannan-Lov\'asz-Simonovits lemma, dimension-free estimates for volumes of sublevel sets of polynomials, and distribution of zeroes of random analytic functions. I. St. Petrsburg Math. J. {\bf 14} (2003) no. 2, 1-15. 77. (with F. Nazarov, M. Sodin) The geometric Kannan-Lov\'asz-Simonovits lemma, dimension-free estimates for volumes of sublevel sets of polynomials, and distribution of zeroes of random analytic functions. II. Israel J. Math., (2003), no. 2. 78. (with F. Nazarov) Heating the Beurling operator and estimates of its norms. St. Petrsburg Math. J. {\bf 14} (2003) no. 3. 79. (with O. Dragicevic) Sharp estimates of the Ahlfors-Beurling operator via averaging of Martingale transform. Michigan Math. J. {\bf 51} (2003), 415-435. 80. (with A. Borichev and H. Hedenmalm) Zero free invariant subspaces in large Bergman spaces. J. Funct. Analysis, {\bf 207}, (2004), 111--160. 81. (with F. Nazarov and S. Treil) $Tb$ theorems for nonhomogeneous spaces. See the homepage http://www.math.msu.edu/$\widetilde{}$volberg Acta Mathematica, v. 190 (2003), pp. 151-239. 82.Factorization of polynomials with estimates of norms. Operator Theory: Advances and Applications, v. 149, 2004, pp. 277-295. 83.(with F. Nazarov, F. Peherstorfer, P. Yuditskii)On generalized sum rules on Jacobi matrices.Preprint: arXiv:math.SP/0311469 v1 26 Nov 2003. Accepted to Intern. Math. Research Notices. \bigskip \noindent{\bf A small book}\bigskip84. ``Calder\'on-Zygmund Capacities and Operators on Nonhomogeneous Spaces"CBMS Regional Conference Series in Mathematics, v. 100, 2003, pp. 1-167. \bigskip {\bf Preprints} \bigskip 85. (with F. Nazarov and S. Treil) $Tb$ theorem that proves Vitushkin's conjecture. Preprint no. 519, CRM, Univ. Auton. de Barcelona, 2002, pp. 1-84. 86. (with S. Treil) A simple proof of Hunt-Muckenhoupt-Wheeden theorem. Preprint, 1995, 1-5. 87. (with S. Treil) Two weights inequalities for Bergman operator. Preprint, 1997, pp. 1-20. 88. (with F. Nazarov) On analytic capacity of portions of continuum and a theorem of Guy David. Preprint Schr\"{o}dinger Institute, 718 (1999) pp.1-11. See the homepage: http://www.math.msu.edu/$\widetilde{}$volberg 89. (with O. Dragicevic) Bellman function and dimensionless estimates of classical and Ornstein-Uhlenbeck Riesz transforms. Preprint, 2002, MSU, pp. 1-26. 90. (with F. Nazarov, S.Treil) Two weight $T1$ theorems for the Hilbert transform: the case of doubling measures. Preprint, 2004, pp. 1--40.91. (with F. Nazarov, S.Treil) Two weight estimate for the Hilbert transform and corona decomposition for non doubling measures. Preprint, 2004, pp. 1--38.92. (with F. Nazarov, S.Treil) Two weight estimate for theindividual Martingale transform and and othe well localized operators. Preprint, 2004, pp. 1--14.93. (with P. Yuditskii) Noncomutative Perron-Frobenius-Ruelle theorem and two weight Hilbert transform, and almost periodicity. Preprint: arXiv:math.SP/0408257 94. (with P. Yuditskii and F. Peherstorfer) Limit periodic Jacobi matrices with a prescribed p-adic hull and a singularly continuous spectrum . Preprint, 2004, 1--8. Preprint: arXiv:math.SP/0408259 \bigskip {\bf Preprints in preparation} \bigskip 95. Riesz capacities (semiadditivity) without curvature. %\begin{tabbing} %\hspace{3cm} \= \hspace{4cm} \= \hspace{2cm} \= \kill % \> midterms \> 33 \% \> (3 exams)\\ % \> homework and quizzes \> 33 \% \> (collected weekly) \\ % \> final \> 33 \% \> \ %\end{tabbing}