Download Abstract

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Perturbation theory wikipedia, lookup

Two-body Dirac equations wikipedia, lookup

Plateau principle wikipedia, lookup

Relativistic quantum mechanics wikipedia, lookup

Lattice Boltzmann methods wikipedia, lookup

Routhian mechanics wikipedia, lookup

Navier–Stokes equations wikipedia, lookup

Mathematics of radio engineering wikipedia, lookup

Computational fluid dynamics wikipedia, lookup

Mathematical descriptions of the electromagnetic field wikipedia, lookup

Computational electromagnetics wikipedia, lookup

Finding real roots of Eisenstein series of weight
one by Painlevé VI equations
Chang-Shou Lin
Taida Institute for Mathematical Sciences
National Taiwan University
Taipei 10617, Taiwan
October 22, 2015
For any (r, s) ∈ C2 \ 21 Z2 , we define pr,s (τ ) ∈ C by
℘(pr,s (τ )|τ ) = ℘(r +sτ |τ )+
℘0 (r + sτ |τ )
2(ζ(r + sτ |τ ) − rη1 (τ ) − sη2 (τ ))
τ ∈H
The Hitchin theorem says that pr,s (τ ) is a solution of a Painlevé VI
equation. Remarkably, the denominator ζ(r + sτ |τ ) − rη1 (τ ) − sη2 (τ )
is the Eisenstein series of weight one if (r, s) is an N -torsion point.
Set Zr,s (τ ) = ζ(r + sτ |τ ) − rη1 (τ ) − sη2 (τ ). Then Zr,s (τ ) has
another link with geometric and PDE aspects. Any zero τ of Zr,s (τ )
for some real pair (r, s) ∈ R2 \ 12 Z2 , the PDE (mean field equation)
∆u + eu = 8πδ(0) in Eτ ,
has a solution, where Eτ is the torus C/∧τ , ∧τ is the lattice generated
by 1 and τ , and δ(0) is the Dirac measure at lattice points. In this talk,
I will give a survey about the story about the connection among such
subjects: mean field equations, Painlevé VI, and modular forms. The
center of these connections are Green functions and Lame equations.