Melkana Brakalova-Trevithick

Melkana Brakalova-Trevithick

Professor of Mathematics
Phone: 718-817-5843
Email: [email protected]
Office: JMH 417

  • Ph.D. (Geometric Function Theory)
    Thesis: On the Asymptotic Behavior of Quasiconformal and Conformal Strip Mappings, Sofia University, Bulgaria, 1988.
    Advisor: Academician Lyubomir Iliev, Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria.
    Consultant: A. A. Goldberg, Lvov State University, Ukraine, USSR.

    M.S. (Complex Analysis)
    Thesis: On the boundary behavior of conformal maps of parabolically exhaustible symmetric strips, Sofia University, Bulgaria, 1981.
    Advisor: V. Alexandrov, Bulgarian Academy of Sciences, Institute of Mathematics, Sofia, Bulgaria.

    Ed.M. (Mathematics Education) Harvard University, Cambridge, MA, 1993.

  • Academic Positions

    • Visiting Research Scholar, CUNY Graduate Center, NY, August 2022 – August 2023 (on leave from Fordham.)
    • Professor, Fordham University, Bronx, NY, September 2020 – Current.
    • Associate Professor, Fordham University, Bronx, NY, September 2011 – August 2020.
    • Visiting Scientist, Max Planck Institute for Mathematics, Bonn, Germany, September 1–September 30 and December 7– December 22, 2017.
    • Senior Fellow, Institute for Pure and Applied Mathematics, UCLA, program on Interactions between Analysis and Geometry, March 11, 2013 – June 14, 2013.
    • Assistant Professor, Fordham University, Bronx, NY, September 2004 – June 2011.
    • Teacher of Mathematics, Hotchkiss School, Lakeville, CT, September 1993 – 2004, on leave (Sabbatical) 2002 – 2003.
    • Visiting Professor, Institute for Mathematical Sciences and Department of Mathematics, SUNY, Stony Brook, NY, September 2002 – June 2003.
    • Analytics Instructor, KSG Summer Program, Harvard University, Cambridge, MA, August 1995.
    • Visiting Lecturer, Tufts University, Medford, MA, Spring 1993.
    • Assistant Professor, American University in Bulgaria, Blagoevgrad, BG, September 1991 – June 1992.
    • Researcher, Bulgarian Academy of Sciences, Institute of Mathematics, Sofia, Bulgaria, 1986 – 1991.
    • Visiting Assistant Professor, Washington University, St. Louis, MO, September 1990 – June 1991.
    • Visiting Lecturer, University of Minnesota, Minneapolis, MN, September 1989 – June 1990.
    • Wissenschaftliche Hilfskraft, University of Regensburg, Regensburg, West Germany, Fall 1988 – Spring 1989.
    • Teaching Assistant, Sofia University, Sofia, Bulgaria, Fall 1986.
    • Teaching Assistant, High Institute of Engineering and Architecture, Sofia, Bulgaria, Fall 1984.

    Administrative Positions (Fordham University, NY)

    • Department Chair, June 2020 – July 2021.
    • Associate Chair, Rose Hill campus, Bronx, July 2019 – July 2020;
    • Associate Chair, Lincoln Center campus, Manhattan, July 2018 – July 2020.
    • Department Chair, July 2011 – July 2014 (on Faculty fellowship leave, Spring 2013)

  • My work falls mostly under the umbrella of geometric function theory, in particular, quasiconformal mappings and Teichmüller theory. I am also interested in the theory of minimal surfaces, harmonic mappings, as well as dynamical systems. I am currently working on smoothness and geometric properties of the p–integrable subspaces of the universal Teichmüller space.

  • M Brakalova-Trevithick, Properties of quasisymmetric homeomorphisms in the Weil-Petersson class, Teichm ̈uller Theory and Grothendieck-Teichm ̈uller Theory, Editors: Lizhen Ji, Athanase Papadopoulos, Weixu Su, Higher Education Press, Beijing, 2022, p. 51—65.

    V. Alberge and M. Brakalova, On Smoothness of the elements of some integrable Teichm ̈uller spaces. Math. Rep. 23(73), 95–105, 2021. MR4275907

    V. Alberge, M. Brakalova and A. Papadopoulos, Teichm ̈uller’s work on the type problem. in Handbook of Teichm ̈uller Theory (A. Papadopoulos, ed.), Volume VII, EMS Publishing House, Z ̈urich, 2020, arXiv:1912.11291. MR4321192

    V. Alberge, M. Brakalova and A. Papadopoulos, A commentary on Teichm ̈uller’s paper ”Untersuchungen ̈uber konforme and quasikonforme Abbildung”, (Investigations on conformal and quasiconformal mappings) in Handbook of Teichm ̈uller Theory (A. Papadopoulos, ed.), Volume VII, EMS Publishing House, Z ̈urich, 2020, arXiv:1912.11290. MR4321191

    M. Brakalova and M. Weiss, translation from the German of a paper by O. Teichm ̈uller, Investigation on conformal and quasiconformal mappings. (UntersuĚungen §ber konforme und quasikonforme Abbildung, Deutsche Math. 3 (1938), 621–678) in Handbook of Teichm ̈uller Theory (A. Pa- padopoulos, ed.), Volume VII, EMS Publishing House, Z ̈urich, 2020.

    M. Brakalova, translation from the German of a paper by H. Gr ̈otzsch On the distortion of univalent non-conformal mappings and a related extension of the Picard theorem. (Uber die Verz- ̈ errung bei schlichten nichtkonformen Abbildungen and ̈uber eine damit zusammenh ̈angende Erweiterung des Picardschen Satzes. Berichte ̈uber die Verhandlungen der Sachsischen Akad. der Wissenschaften zu Leipzig Math.-Physische Klasse. Bd. 80 (1928), 503–507 von Herbert Gr ̈otzsch in Leipzig Vorgelegt von Herrn Koebe) in Handbook of Teichm ̈uller Theory (A. Pa- padopoulos, ed.), Volume VII, EMS Publishing House, Z ̈urich, 2020.

    M. Brakalova, translation from the German of a paper by H. Gr ̈otzsch On closest-to-conformal mappings of schlicht domains II. (Uber m ̈oglist konformen Abbildungen von schlichten Bere- ̈ ichen. Berichte ber die Verhandlungen der Sachsischen Akad. der Wissenschaften zu Leipzig ̈ Math.-Physische Klasse. Bd. 84 (1932), 114–120. von Herbert Gr ̈otzsch in Leipzig Vorgelegt von Herrn Koebe) in Handbook of Teichm ̈uller Theory (A. Papadopoulos, ed.), Volume VII, EMS Publishing House, Z ̈urich, 2020.

    M. Brakalova, Translation of a paper by H. Gr ̈otzsch On Some Extremal problems of the conformal mapping II. (from German: Uber einige Extremalprobleme der konformen Abbildung ̈ II Berichte ̈uber die Verhandlungen der Sachsischen Akad. der Wissenschaften zu Leipzig Math.-Physische Klasse. Bd. 80 (1928), 497–502 von Herbert Gr ̈otzsch in Leipzig, Vorgelegt von Herrn Koebe) in Handbook of Teichm ̈uller Theory, (A. Papadopoulos, ed.), Volume VII, EMS Publishing House, Z ̈urich, 2020.

    M. Brakalova, Translation from the German of a paper by O. Teichm ̈uller, An application of quasiconformal mappings to the type problem. (Eine Anwendung quasikonformer Abbildungen auf das Typenproblem Deutsche Math. 2, 321–327 (1937)) in Handbook of Teichm ̈uller Theory (A. Papadopoulos, ed.), Volume VII, EMS Publishing House, Z ̈urich, 2020.

    M. Brakalova, Symmetric properties of pintegrable Teichm ̈uller spaces. Analysis and Mathematical Physics, 8 (2018) 541–549, Springer Verlag. MR3881012

    M. Brakalova, I. Markina and A. Vasil’ev, Extremal functions for modules of systems of measures. J. d’Analyse Math. 133, 1 (2017) 335–359. (arXiv:1409.1626) MR3736495

    M. Brakalova, I. Markina and A. Vasil’ev, Modules of systems of measures on polarizable Carnot groups. Arkiv f ̈or Mat. 54 (2016) 2, 371–401. (originally submitted to arXiv:1409.1626) MR3546358

    M. Brakalova, On local stability of solutions to the Beltrami equation with degeneration. Proceedings of International Conference in Complex Analysis and Applications’13, Sofia (2013) 67–77. MR3312225

    M. Brakalova. Sufficient and necessary conditions for conformality. Part II. Analytic viewpoint. Ann. Acad. Sci. Fenn. 35 (2010) 235–254. MR2643407

    M. Brakalova. Sufficient and necessary conditions for conformality. Part I. Geometric view- point. Complex Variables and Elliptic Equations, issue dedicated to Cabiria Andreian - Cazacu) 57 (2010) 137–155. MR2599616

    M. Brakalova, Boundary extension of μ - homeomorphisms. Complex and Harmonic Anal- ysis, Edited by: A. Carberry, P. Duren, D. Khavinson and A. G. Siskakis (2007) 231–247. MR2387293

    M. Brakalova and J. A. Jenkins, On solutions of the Beltrami equation II. Publ. l’Inst. Math., 75 (2004) 3–8. MR2107992

    M. Brakalova and J. A. Jenkins, On a paper of Carleson. Ann. Acad. Sci. Fenn, 27 (2002) 485–490. MR1922202

    M. Brakalova and D. Coughlin, Teaching fractals and dynamical systems at the Hotchkiss school, in Fractals, Computer Graphics, and Mathematics Education, edited by B. Mandelbrot and M. Frame, MAA Notes No. 58 (2002) 35–48.

    M. Brakalova and J. A. Jenkins, On solutions of the Beltrami Equations with ∥μ∥∞ = 1. J. Anal. Math., 76 (1998) 67–92. MR1676936

    M. Brakalova and J. A. Jenkins, On the local behavior of certain homeomorphisms. II. Zap. Nauchn. Sem. S. Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), Anal. Teor. Chisel i Teor. Funkts. 14, 11 – 20, 227; translation in J. Math. Sci. (New York) 95 (1999) no. 3, 2178–2184 MR1691279

    M. Brakalova and J. A. Jenkins, On the local behavior of certain homeomorphisms. Kodai Math. J. 17 (1994) 201–213. MR1282210

    M. Bracalova, A Generalization of the Teichm ̈uller theorem. Complex analysis, Joensuu 1987, Lecture Notes in Math., 1351, Springer Verlag, Berlin (1988) 69–77. MR0982073

    M Brakalova, On the asymptotic representation of some quasiconformal mappings. Complex analysis, Publ. House Bulgar. Acad. Sci., Sofia (1989) 49–83. MR1127618

    M. Brakalova, On the asymptotic behavior of some quasiconformal and strip conformal map- pings. C. R. Acad. Bull. Sci., (I) 41 (1988) 11–14. MR0937105

    M. Brakalova, On asymptotic behavior of some conformal strip mappings. Constructive theory of functions, (Varna, 1987), Publ. House Bulgar. Acad. Sci., Sofia (1988) 65–71. MR0914511 (89a:30004)

    M. Brakalova, Asymptotic behavior of strip mappings of some symmetric strips. Complex analysis and applications, Sofia (1986) 127–134. MR1127618

  • M. Brakalova, Properties of quasisymmetric homeomorphisms in the Weil–Petersson class. Te- ichm ̈uller theory and its impact, Advanced Lectures in Mathematics (ALM) book series, Inter- national Press and Higher Education Press of China, based on the Workshop on Grothendieck– Teichm ̈uller Theories, Chern Institute, Nankai University, China, 2016. (in print)

    V. Alberge and M. Brakalova, On Smoothness of the elements of some integrable Teichm ̈uller spaces. Math. Rep. 23(73), 95–105, 2021. MR4275907

    M. Brakalova, Symmetric properties of p-integrable Teichm ̈uller spaces. Analysis and Mathematical Physics, 8 (2018) 541–549, Springer Verlag. MR3881012

    M. Brakalova, I. Markina and A. Vasil’ev, Extremal functions for modules of systems of measures. J. d’Analyse Math. 133, 1 (2017) 335–359. (arXiv:1409.1626) MR3736495

    M. Brakalova, I. Markina and A. Vasil’ev, Modules of systems of measures on polarizable Carnot groups. Arkiv f ̈or Mat. 54 (2016) 2, 371–401. (originally submitted to arXiv:1409.1626) MR3546358

    M. Brakalova, On local stability of solutions to the Beltrami equation with degeneration. Proceedings of International Conference in Complex Analysis and Applications’13, Sofia (2013) 67–77. MR3312225

    M. Brakalova. Sufficient and necessary conditions for conformality. Part II. Analytic viewpoint. Ann. Acad. Sci. Fenn. 35 (2010) 235–254. MR2643407

    M. Brakalova. Sufficient and necessary conditions for conformality. Part I. Geometric viewpoint. Complex Variables and Elliptic Equations, issue dedicated to Cabiria Andreian - Cazacu) 57 (2010) 137–155. MR2599616

    M. Brakalova, Boundary extension of μ - homeomorphisms. Complex and Harmonic Analysis, Edited by: A. Carberry, P. Duren, D. Khavinson and A. G. Siskakis (2007) 231–247. MR2387293

    M. Brakalova and J. A. Jenkins, On solutions of the Beltrami equation II. Publ. l’Inst. Math., 75 (2004) 3–8. MR2107992

    M. Brakalova and J. A. Jenkins, On a paper of Carleson. Ann. Acad. Sci. Fenn, 27 (2002) 485–490. MR1922202

    M. Brakalova and D. Coughlin, Teaching fractals and dynamical systems at the Hotchkiss school, in Fractals, Computer Graphics, and Mathematics Education, edited by B. Mandelbrot and M. Frame, MAA Notes No. 58 (2002) 35–48. 1

    M. Brakalova and J. A. Jenkins, On solutions of the Beltrami Equations with ‖μ‖∞ = 1. J. Anal. Math., 76 (1998) 67–92. MR1676936

    M. Brakalova and J. A. Jenkins, On the local behavior of certain homeomorphisms. II. Zap. Nauchn. Sem. S. Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), Anal. Teor. Chisel i Teor. Funkts. 14, 11 – 20, 227; translation in J. Math. Sci. (New York) 95 (1999) no. 3, 2178–2184 MR1691279

    M. Brakalova and J. A. Jenkins, On the local behavior of certain homeomorphisms. Kodai Math. J. 17 (1994) 201–213. MR1282210

    M. Bracalova, A Generalization of the Teichm ̈uller theorem. Complex analysis, Joensuu 1987, Lecture Notes in Math., 1351, Springer Verlag, Berlin (1988) 69–77. MR0982073

    M Brakalova, On the asymptotic representation of some quasiconformal mappings. Complex analysis, Publ. House Bulgar. Acad. Sci., Sofia (1989) 49–83. MR1127618

    M. Brakalova, On the asymptotic behavior of some quasiconformal and strip conformal mappings. C. R. Acad. Bull. Sci., (I) 41 (1988) 11–14. MR0937105

    M. Brakalova, On asymptotic behavior of some conformal strip mappings. Constructive theory of functions, (Varna, 1987), Publ. House Bulgar. Acad. Sci., Sofia (1988) 65–71. MR0914511 (89a:30004)

    M. Brakalova, Asymptotic behavior of strip mappings of some symmetric strips. Complex analysis and applications, Sofia (1986) 127–134. MR1127618 2

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