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Schwarzian equation, automorphic functions and functional transcendence Nagloo, Joel
Description
By a Schwarzian differential equation, we mean an equation of the form $S_{\frac{d}{dt}}(y) +(y')^2 R(y) =0,$ where $S_{\frac{d}{dt}}(y)$ denotes the Schwarzian derivative and $R$ is a rational function with complex coefficients. The equation naturally appears in the study of automorphic functions (such as the modular $j$-function): if $j_{\Gamma}$ is the uniformizing function of a genus zero Fuchsian group of the first kind, then $j_{\Gamma}$ is a solution of some Schwarzian equation. In this talk, we discuss recent work towards the proof of a conjecture/claim of P. Painlev\â e (1895) about the irreducibility of the Schwarzian equations. We also explain how, using the model theory of differentially closed fields, this work on irreducibility can be used to tackle questions related to the study of algebraic relations between the solutions of a Schwarzian equation. This includes, for example, obtaining the Ax-Lindemann-Weierstrass Theorem with derivatives for all Fuchsian automorphic functions.
Item Metadata
Title |
Schwarzian equation, automorphic functions and functional transcendence
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2020-11-10T10:00
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Description |
By a Schwarzian differential equation, we mean an equation of the form $S_{\frac{d}{dt}}(y) +(y')^2 R(y) =0,$ where $S_{\frac{d}{dt}}(y)$ denotes the Schwarzian derivative and $R$ is a rational function with complex coefficients. The equation naturally appears in the study of automorphic functions (such as the modular $j$-function): if $j_{\Gamma}$ is the uniformizing function of a genus zero Fuchsian group of the first kind, then $j_{\Gamma}$ is a solution of some Schwarzian equation.
In this talk, we discuss recent work towards the proof of a conjecture/claim of P. Painlev\â e (1895) about the irreducibility of the Schwarzian equations. We also explain how, using the model theory of differentially closed fields, this work on irreducibility can be used to tackle questions related to the study of algebraic relations between the solutions of a Schwarzian equation. This includes, for example, obtaining the Ax-Lindemann-Weierstrass Theorem with derivatives for all Fuchsian automorphic functions.
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Extent |
55.0 minutes
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Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: City University of New York
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Series | |
Date Available |
2021-05-10
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0397363
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Researcher
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International