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Mathematical theory
Ahlfors theory is a mathematical theory invented by Lars Ahlfors as a geometric counterpart of the Nevanlinna theory. Ahlfors was awarded one of the two
Ahlfors_theory
Finnish mathematician (1907–1996)
Mathematics. Ahlfors was born in Helsinki, Finland. His mother, Sievä Helander, died at his birth. His father, Karl Axel Mauritz Ahlfors, was a professor
Lars_Ahlfors
asymptotic value, but is of order 0. Lars Ahlfors later generalized the theorem, in his work on Ahlfors theory, which he published in 1935. The generalized
Denjoy–Carleman–Ahlfors theorem
Denjoy–Carleman–Ahlfors_theorem
Mathematics award
"Lars Valerian Ahlfors (1907–1996)" (PDF). Ams.org. Archived (PDF) from the original on 3 March 2022. Retrieved 31 March 2017. "Lars Ahlfors (1907–1996)"
Fields_Medal
transformations. Ahlfors theory A part of complex analysis being the geometric counterpart of Nevanlinna theory. It was invented by Lars Ahlfors. Algebra One
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Area of mathematics
called the Ahlfors–Shimizu characteristic. The bounded term O(1) is not important in most questions. The geometric meaning of the Ahlfors—Shimizu characteristic
Nevanlinna_theory
Schwarzian derivative Analytic capacity Disk algebra Univalent function Ahlfors theory Bieberbach conjecture Borel–Carathéodory theorem Corona theorem Hadamard
List of complex analysis topics
List_of_complex_analysis_topics
Study of space and shapes locally given by a convergent power series
a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes
Geometric_function_theory
Mathematical theorem
Ahlfors, who proved it in the case that the Kleinian group has a fundamental domain with a finite number of sides. Richard Canary proved the Ahlfors conjecture
Ahlfors_measure_conjecture
Mathematical theory
Kleinian group. The theorem was proved by Lars Ahlfors, apart from a gap that was filled by Greenberg. The Ahlfors finiteness theorem states that if Γ is a
Ahlfors_finiteness_theorem
Parametrizes complex structures on a surface
1988, p. 40. Thurston 1988, p. 42. Ahlfors 2006, p. 69. Ahlfors 2006, p. 71. Ahlfors 2006, Chapter VI.C. Ahlfors 2006, p. 96. Thurston, William (1998)
Teichmüller_space
Finnish mathematician (1895–1980)
famous doctoral student was Lars Ahlfors, one of the first two Fields Medal recipients. The research for which Ahlfors was awarded the prize (proving the
Rolf_Nevanlinna
discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Branch of mathematics studying functions of a complex variable
Fokas, Complex Variables: Introduction and Applications (Cambridge, 2003). Ahlfors, L., Complex Analysis (McGraw-Hill, 1953). Cartan, H., Théorie élémentaire
Complex_analysis
Analysis, 3 ed. (McGraw-Hill, 1986). Ahlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979). Kato, T., Perturbation theory for linear operators, 2 ed. (Springer-Verlag
Right_half-plane
Finnish mathematician (1870–1946)
meromorphic functions, created by Rolf Nevanlinna in 1922–1925. The young Lars Ahlfors rose at the end of the decade to become one of the world's leading researchers
Ernst_Leonard_Lindelöf
Matrix used in complex analysis
= |eg|2. Ahlfors 1966 Schiffer 1959, p. 261 Schiffer & Hawley 1962, p. 246 Schiffer & Hawley 1962, pp. 245–246 Takhtajan & Teo 2006 Ahlfors, Lars V. (1952)
Grunsky_matrix
Mathematical term in complex analysis
test Munkres. Topology, Theorem 46.8. See for example (Ahlfors 1953), (Ahlfors 1966), (Ahlfors 1978), (Conway 1978) and (Beardon 1979). Gamelin. Complex
Normal_family
Limit of roots of sequence of functions
Rouché's theorem Ahlfors 1966, p. 176, Ahlfors 1978, p. 178 Gamelin, Theodore (2001). Complex Analysis. Springer. ISBN 978-0387950693. Ahlfors, Lars V. (1966)
Hurwitz's theorem (complex analysis)
Hurwitz's_theorem_(complex_analysis)
theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct
Measurable Riemann mapping theorem
Measurable_Riemann_mapping_theorem
Japanese mathematician
theory, in particular the theory of meromorphic functions. A new form of the Nevanlinna characteristic generalised by him (and separately by Ahlfors)
Tatsujiro_Shimizu
Number with a real and an imaginary part
Company. p. 112. ISBN 978-1-78326-547-3. Extract of page 112 Ahlfors 1979, p. 15. Ahlfors 1979, p. 16. Bourbaki 1998, §VIII.1 Kline, Morris. A history
Complex_number
geometry) Rashevsky–Chow theorem (control theory) Rauch comparison theorem (Riemannian geometry) Schwarz–Ahlfors–Pick theorem (differential geometry) Soul
List_of_theorems
Undergraduate math course at Harvard University
Mathematical Analysis, Ahlfors' Complex Analysis, Spivak's Calculus on Manifolds, Axler's Linear Algebra Done Right, Halmos's Naive Set Theory and Finite-Dimensional
Math_55
Mathematical metric for Riemann surfaces
Riemannian metric. Weil (1958) stated, and Ahlfors (1961) proved, that the Weil–Petersson metric is a Kähler metric. Ahlfors (1961b) proved that it has negative
Weil–Petersson_metric
Number, approximately 3.14
Rudin, Walter (1986). Real and complex analysis. McGraw-Hill. p. 2. Ahlfors, Lars (1966). Complex analysis. McGraw-Hill. p. 46. Arndt & Haenel 2006
Pi
German mathematician (1913–1943)
mappings. He was already an expert in the Nevanlinna theory and evidently greatly influenced by Ahlfors' contributions to it. Teichmüller's habilitation thesis:
Oswald_Teichmüller
Concept in complex analysis
bounded analytic functions outside K. It was first introduced by Lars Ahlfors in the 1940s while studying the removability of singularities of bounded
Analytic_capacity
American mathematician and computer scientist (1924–2019)
Problem for Univalent Mappings of the Exterior of the Unit Circle under Lars Ahlfors. From 1949 to 1951 Springer was a C.L.E. Moore Instructor at Massachusetts
George Springer (mathematician)
George_Springer_(mathematician)
American mathematician (1927–2010)
D., also from Harvard University, in 1948 under the direction of Lars Ahlfors. It was at Brown University that he met his longtime colleague and collaborator
Paul_Garabedian
Austrian-American mathematician (1927–2026)
latter on the thesis Some Estimates for Extremal Distance advised by Lars Ahlfors. Pollak then joined Bell Labs (1951), where he later became director of
Henry_O._Pollak
Homeomorphism between plane domains
importance in the theory of quasiconformal mappings in two dimensions is the measurable Riemann mapping theorem, proved by Lars Ahlfors and Lipman Bers
Quasiconformal_mapping
American mathematician
Canonical Conformal Maps of Multiply Connected Regions was advised by Lars Ahlfors and Joseph Leonard Walsh. Landau later became a Distinguished Member of
Henry_Landau
Branch of mathematics
Texts in Mathematics. New York: Springer-Verlag. ISBN 978-0387950600. Ahlfors, Lars Valerian (1979). Complex Analysis (3rd ed.). New York: McGraw-Hill
Mathematical_analysis
Series of four mathematics textbooks
complex analysis covered in Lars Ahlfors's textbook but noted that Stein and Shakarchi also treat some topics Ahlfors skips. Stein, Elias M.; Shakarchi
Princeton Lectures in Analysis
Princeton_Lectures_in_Analysis
Proposition 3.9.3. Bers 1958; Chern 1955; Ahlfors 2006, p. 90. Morrey 1938. Imayoshi & Taniguchi 1992, pp. 20–21 Ahlfors 2006, pp. 85–115 Imayoshi & Taniguchi
Isothermal_coordinates
American mathematician
Contributions to the Problem of Type (on Riemann surfaces) supervised by Lars Ahlfors. He joined Stanford University in 1955. He joined the Mathematical Sciences
Robert_Osserman
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
Ahlfors, Lars V.; Sario, Leo (1960), Riemann surfaces, Princeton Mathematical Series, vol. 26, Princeton University Press Beltrami's equation Ahlfors
Uniformization_theorem
Branch of mathematics
March 2023. Retrieved 9 September 2022. Ahlfors, Lars V. (1979). Complex analysis: an introduction to the theory of analytic functions of one complex variable
Geometry
Mathematical concept
equal to R2. This relation has been used classically in Vekua (1962) and Ahlfors (1966) to establish the continuity properties of T on Lp spaces. The results
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
Perception of meaningful patterns or images in random or vague stimuli
ISBN 978-0-394-53512-8. Hadjikhani, Nouchine; Kveraga, Kestutis; Naik, Paulami; Ahlfors, Seppo P. (2009). "Early (M170) activation of face-specific cortex by face-like
Pareidolia
Property determining comparison and ordering
Beginning Calculus. McGraw-Hill Professional. p. 2. ISBN 978-0-07-148754-2. Ahlfors, Lars V. (1953). Complex Analysis. Tokyo: McGraw Hill Kogakusha. Nykamp
Magnitude_(mathematics)
Connected open subset of a topological space
1978, §1.3 pp. 21–22). For instance (Churchill 1948, §1.9 pp. 16–17); (Ahlfors 1953, §2.2 p. 58); (Rudin 1974, §10.1 p. 213) reserves the term domain
Domain (mathematical analysis)
Domain_(mathematical_analysis)
Mathematical approximation of a function
pp. 116–117. Stein & Shakarchi 2003, pp. 98–100. Ahlfors 1979, p. 38. Lang 1999, p. 166. Ahlfors 1979, pp. 184–186. Bliss 1933, Chapter II. Hörmander
Taylor_series
Quasiconformal complex image of a circle
admits a quasiconformal reflection. Ahlfors (1963) proved that this property characterizes quasicircles. Ahlfors noted that this result can be applied
Quasicircle
Concept of complex analysis
is denoted by B n {\displaystyle B_{n}} in Whittaker & Watson's book. Ahlfors, Lars (1979). Complex Analysis. McGraw Hill. ISBN 0-07-085008-9. Lindelöf
Residue_theorem
American mathematician
added to Colorado mountain lexicon". Denver Post. Retrieved 2011-07-23. Ahlfors, Lars V. (1955). "Review of Functionals of finite Riemann surfaces. By
Donald_C._Spencer
American mathematician (1928-1993)
his Ph.D. in 1951 at Harvard University under the supervision of Lars Ahlfors with thesis Harmonic functions on open Riemann surfaces. At Stanford University
Halsey_Royden
24 mathematical problems stated in 1982
tame? Solved independently by Agol and by Calegari-Gabai. 2004 10th The Ahlfors measure zero problem: group obtained as a limit set of finitely-generated
Thurston's_24_questions
Complex numbers with non-negative imaginary part
Kleinian group Modular group Moduli stack of elliptic curves Riemann surface Schwarz–Ahlfors–Pick theorem Weisstein, Eric W. "Upper Half-Plane". MathWorld.
Upper_half-plane
1214/EJP.v7-101. ISSN 1083-6489. Ahlfors, Lars V. (1973). Conformal invariants: topics in geometric function theory. Series in Higher Mathematics. McGraw-Hill
Conformal_radius
Chinese-American mathematician (born 1949)
generalize the classical Schwarz−Pick lemma of complex analysis. Lars Ahlfors, among others, had previously generalized the lemma to the setting of Riemann
Shing-Tung_Yau
Function family in complex analysis
Encyclopedia of Mathematics, ISBN 1402006098. Ahlfors, Lars (1953). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable
Antiholomorphic_function
American mathematician
Ph.D. at Harvard University in 1959. His doctoral supervisor was Lars Ahlfors. His dissertation topic was Mappings, Automorphisms and Coverings of Riemann
Dale_Husemoller
Every polynomial has a real or complex root
MR 4308140 Axler, Sheldon (2026), Linear Algebra Done Right, p. 125 Ahlfors, Lars, Complex Analysis (2nd ed.), McGraw-Hill Book Company, p. 122 A proof
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Mathematical formula in complex analysis
|f(z)|} . Paley–Wiener theorem Ahlfors, Lars V. (1979). "5.3.1, Jensen's formula". Complex analysis : an introduction to the theory of analytic functions of
Jensen's_formula
Characteristic property of holomorphic functions
analysis (3rd ed.). McGraw Hill (published 1987). ISBN 0-07-054234-1. Ahlfors, Lars (1953). Complex analysis (3rd ed.). McGraw Hill (published 1979)
Cauchy–Riemann_equations
analytic functions. Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw Hill, ISBN 978-0-07-000657-7 Markushevich, A. I. (1977). Theory of Functions of
Global_analytic_function
Partial differential equation
1992, p. 88 Ahlfors 1966, p. 98 See Ahlfors 1966, p. 99 Bers, John & Schechter 1979, p. 277 See: Douady & Buff 2000 Glutsyuk 2008 Ahlfors 1966 Astala
Beltrami_equation
Polygon associated with a compact Riemann surface
47–57 Shastri 2011 Farb & Margalit 2012 Ahlfors 2006, pp. 67–68 Farb & Margalit 2012, pp. 230–236 Ahlfors, Lars V. (2006), Lectures on quasiconformal
Fundamental_polygon
Wayback Machine Ahlfors, Lars V. (1979), Complex Analysis, Third Edition, McGraw-Hill, ISBN 0-07-085008-9 Remmert, Reinhold (1990), Theory of Complex Functions
Schwarz_integral_formula
Mathematical theorem
Green’s function. Ahlfors, Lars (1953), L. Ahlfors; E. Calabi; M. Morse; L. Sario; D. Spencer (eds.), "Developments of the Theory of Conformal Mapping
Riemann_mapping_theorem
Mathematical function that preserves angles
Masters of theory: Cambridge and the rise of mathematical physics. University of Chicago Press. pp. 404–424. ISBN 978-0-226-87375-6. Ahlfors, Lars V. (1973)
Conformal_map
Theorem in complex analysis
Scandinavian Mathematicians, pp. 59–90 Ahlfors, Lars V. (2010), Conformal invariants: topics in geometric function theory, AMS Chelsea Publishing, ISBN 978-0-8218-5270-5
Carathéodory's theorem (conformal mapping)
Carathéodory's_theorem_(conformal_mapping)
Power series theorem in mathematics
sequences Ahlfors, Lars Valerian (September 1, 1980). Complex Analysis (Third ed.). McGraw Hill Higher Education. pp. 41–42. ISBN 0-07-085008-9. - Ahlfors called
Abel's_theorem
and Geometry: Contributions from the Programme Ergodic Theory, Geometric Rigidity and Number Theory, Isaac Newton Institute for the Mathematical Sciences
List_of_conjectures
Discrete group of Möbius transformations
called the domain of discontinuity or the ordinary set or the regular set. Ahlfors' finiteness theorem implies that if the group is finitely generated then
Kleinian_group
46 (4): 438. He and Eveline accepted an invitation from Lars Ahlfors to visit the Ahlfors family in Finland. A little later he was jailed in Finland as
List of mathematicians who did research in prison
List_of_mathematicians_who_did_research_in_prison
American mathematician
received his PhD in 1962 from Harvard University with thesis advisor Lars Ahlfors. Marden has been a professor at the University of Minnesota since the 1970s
Albert_Marden
Awarded every year by the American Mathematical Society
Mathematical Society. ISBN 9780821836446. Ahlfors, Lars (2010) [1973]. Conformal Invariants. Topics in Geometric Function Theory (2nd ed.). Providence: American
Leroy_P._Steele_Prize
British mathematician (1900–1998)
theorem she used a new approach, applying a technique introduced by Lars Ahlfors for conformal mappings. While at Cambridge Cartwright attended the mathematical
Mary_Cartwright
Pseudometric of complex manifolds
constant, then X is Kobayashi hyperbolic. In dimension 1, this is called the Ahlfors–Schwarz lemma. The results above give a complete description of which complex
Kobayashi_metric
One-dimensional complex manifold
of Maryland. Ann. Math. Studies. Vol. 79. pp. 207–226. ISBN 0691081387. Ahlfors, Lars; Sario, Leo (1960), Riemann Surfaces (1st ed.), Princeton, New Jersey:
Riemann_surface
Statement in complex analysis
{\displaystyle z_{2}} tend to z 1 {\displaystyle z_{1}} . The Schwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds. De
Schwarz_lemma
Theorem in complex analysis
Analysis, Cambridge Stud. Adv. Math., 107, CUP, ISBN 978-0-521-80937-5 Ahlfors, Lars (2000), Complex Analysis, McGraw-Hill series in Mathematics, McGraw-Hill
Cauchy's_integral_theorem
Angle of complex number about real axis
the standard orientation. For example, in Lars Ahlfors' Complex Analysis: An introduction to the theory of analytic functions of one complex variable (1979)
Argument_(complex_analysis)
Extension of the domain of an analytic function (mathematics)
See the example given on the MathWorld page for natural boundary. Lars Ahlfors (1979). Complex Analysis (3 ed.). McGraw-Hill. pp. 172, 284. Ludwig Bieberbach
Analytic_continuation
Nonlinear differential operator used to study conformal mappings
Lehto 1987, p. 90 Nehari 1952 von Koppenfels & Stallmann 1959 Klein 1922 Ahlfors 1966 Lehto 1987 Imayoshi & Taniguchi 1992 Ovsienko & Tabachnikov 2005,
Schwarzian_derivative
American mathematician (1892–1977)
607–612. doi:10.1090/s0002-9904-1936-06362-7. Ahlfors, L. (1948). "Review: Topological methods in the theory of functions of a complex variable, by Marston
Marston_Morse
Mathematical function
Texts in Mathematics. Vol. 89 (2nd ed.). Springer. ISBN 978-0-387-90710-9. Ahlfors, Lars (1979). Complex Analysis. McGraw Hill. van der Waerden, B.L. (1931)
Algebraic_function
Manifold with Riemannian, complex and symplectic structure
the underlying complex manifold. It is an elementary consequence of the Ahlfors Schwarz lemma that if ( X , ω ) {\displaystyle (X,\omega )} is a Hermitian
Kähler_manifold
Complex-differentiable (mathematical) function
Encyclopedia of Mathematics, EMS Press, 2001 [1994], retrieved February 26, 2021 Ahlfors, L. (1979), Complex Analysis (3rd ed.), McGraw-Hill Henrici, P. (1986)
Holomorphic_function
Point of interest for complex multi-valued functions
213–269, doi:10.1007/978-3-642-20545-3_5, ISBN 978-3-642-20544-6 (page 6) Ahlfors 1979 Solomentsev 2001; Markushevich 1965 "Logarithmic branch point - Encyclopedia
Branch_point
Location around which a function displays irregular behavior
Methods and Function Theory. 20 (3): 389–401. arXiv:2007.14661. doi:10.1007/s40315-020-00316-x. hdl:2324/4483207. ISSN 2195-3724. Ahlfors, Lars V. (1979),
Essential_singularity
Aspect of group theory in mathematics
mapping class groups from the geometrical viewpoint. In the tradition of Ahlfors-Bers. IV, pp. 119–141, Contemporary Mathematics series, 432, American Mathematical
Ping-pong_lemma
Mapping brain activity by recording magnetic fields produced by currents in the brain
Plenum Press. pp. 399–408. ISBN 978-1-4757-1785-3. Cohen D, Schläpfer U, Ahlfors S, Hämäläinen M, Halgren E (August 2002). "New Six-Layer Magnetically-Shielded
Magnetoencephalography
Russian-American mathematician (1899–1986)
the Steele Prize in 1981, and the Wolf Prize in Mathematics with Lars Ahlfors in 1981. During 1969–1970 he was President of the American Mathematical
Oscar_Zariski
Theorem in complex analysis
Mathematics). McGraw-Hill. ISBN 978-0-07-054234-1. Ahlfors, Lars (1979). Complex analysis: an introduction to the theory of analytic functions of one complex variable
Argument_principle
Integral criterion for holomorphy
contour integration Residue (complex analysis) Mittag-Leffler's theorem Ahlfors, Lars (January 1, 1979), Complex Analysis, International Series in Pure
Morera's_theorem
Pseudonym of a group of mathematicians
the summer of 1939 in Finland with his wife Eveline, as guests of Lars Ahlfors. Due to their travel near the border, the couple were suspected as Soviet
Nicolas_Bourbaki
(1998). Postmodern Analysis. Springer. Ahlfors, Lars V. (1978), Complex analysis. An introduction to the theory of analytic functions of one complex variable
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
Schiffer 1981 Shapiro 1992, pp. 66–67 See also: Schiffer 1957 Schiffer 2011 Ahlfors 1952 Khavinson, Putinar & Shapiro 2007 Burbea 1986 Partyka 1997 Krzyż &
Neumann–Poincaré_operator
Mathematics award
Executive Committee and Vice President for Programs. 1960: Lars Valerian Ahlfors 1961: Mark Kac 1962: Jean Leray 1963: Stanislaw M. Ulam 1964: Solomon Lefschetz
John_von_Neumann_Prize
operators, Hardy spaces, a Kerzman–Stein formula and a Π, or Beurling–Ahlfors, transform. These have all found applications in solving boundary value
Clifford_analysis
Board game
Archived from the original on 27 October 2014. Retrieved 7 May 2024. Rainer Ahlfors [@GalaGalaxia] (12 December 2021). "La Granja: Deluxe Master Set: Obligatory-
Ticket_to_Ride_(board_game)
was able to predict or conjecture several important results such as the Ahlfors's Five Islands theorem, Cartan's theorem on holomorphic curves omitting
Bloch's_principle
Degree of differentiability of a function or map
Mathematical Monthly. 107 (3): 264–266. doi:10.2307/2589322. JSTOR 2589322. Ahlfors, Lars V. (1979). Complex Analysis (3rd ed.). McGraw-Hill. ISBN 978-0-07-000657-7
Smoothness
Functions of an angle
Reinhold (1991), Theory of complex functions, Springer, p. 327, ISBN 978-0-387-97195-7. Whittaker & Watson 1927, p. 137. Ahlfors 1966, p. 197. Bourbaki
Trigonometric_functions
363–410. ISBN 3-540-20488-1. Mumford, David (1974), "Prym varieties. I", in Ahlfors, Lars V.; Kra, Irwin; Nirenberg, Louis; et al. (eds.), Contributions to
Prym_variety
American mathematician
Research Award for his solution to the Marden Tameness Conjecture and the Ahlfors Measure Conjecture. In 2011 he was awarded a Royal Society Wolfson Research
Danny_Calegari
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