Applied Maths Seminar: Takei -- WKB analysis via topological recursion for (confluent) hypergeometric differential equations

Dear all, 

Our upcoming AM seminar is held next Wednesday at 1pm in F11 Chemistry Lecture Theatre 4
(across from Carslaw; this will be the usual location for the seminars in Semester 1).  

Our speaker is Yumiko Takei, who is visiting us from the National Institute of
Technology (KOSEN) at Ibaraki College.  

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Title: WKB analysis via topological recursion for (confluent) hypergeometric
differential equations 

Abstract: The exact WKB analysis is a method to analyze differential equations with a
small parameter h.  The main ingredient of the exact WKB analysis is a formal solution
for h, called a WKB solution.  When we study differential equations by using the exact
WKB analysis, Voros coefficients provide important quantities for describing global
behavior of solutions of differential equations.  The Voros coefficient is defined as a
contour integral of the logarithmic derivative of WKB solutions.  

On the other hand, the topological recursion introduced by B.  Eynard and N.  Orantin is
a recursive algorithm to construct a formal solution to the loop equations that the
correlation functions of the matrix model satisfy.  

The quantization scheme connects WKB solutions with the topological recursion.  It is
found that WKB solutions can be constructed via the topological recursion.  

In this talk, we prove that the Voros coefficients for hypergeometric differential
equations are described by the generating functions of free energies defined in terms of
the topological recursion.  Furthermore, as its applications we show the following
objects can be explicitly computed for hypergeometric equations: (i) three-term
difference equations that the generating function of free energies satisfies, (ii)
explicit forms of the free energies, and (iii) explicit forms of Voros coefficients.  

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An ongoing list of AM seminars is posted here:
https://www.maths.usyd.edu.au/u/SemConf/Applied.html.  

See you there, 

 Ian