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Research Interests

In preparation (april 2016)

The topological recursion was first discovered in random matrices in 2004 [AMM04, Eynard04].
It became a mathematical theory in 2007 [EO07].

The Topological recursion takes as
   input: a spectral curve S (an analytical plane curve with some structure)
and gives
   output: a sequence of invariants (the T.R. invariants) of the spectral curve, denoted Wg,n(S). The n=0 invariants are denoted Wg,0(S) = Fg(S).


- The Witten-Kontsevich intersection numbers
       (curve S: x=y^2 )

- The Mirzakhani relation among hyperbolic volumes of moduli spaces
       (curve S: y = sin( x^(1/2))  )

- The Gromov-Witten invariants of Calabi-Yau manifolds
      (curve S: = mirror curve)

- the large size expansion of random matrix laws
      (curve S: = equilibrium spectral density at large N)

- the enumeration of maps of all topologies
      (curve S: = counting function of rooted planar maps)

- the WKB asymptotic expansion of Hitchin systems
      ( curve S: = Hitchin's spectral curve)

- The Knot polynomials (Jones, Homfly,...)
     ( curve S: = A-polynomial of the knot)

The T.R. invariants have many beautiful mathematical properties:
- symplectic invariance
- modular properties, automorphicity
- form-cycle dualities for their deformations
- nice behavior under limits
- homogeneity

Study the statistical properties of eigenvalues of large random matrices.

of Harish-Chandra- Itzykson-Zuber type.
Let G be a semi-simple Lie group. Let X and Y be  matrices in the Lie Algebra of G.
For any polynomial function f(x,y) of 2 variables, I found in [E-PF05] [dFEPFZ] that:

where the product is over positive roots in a Cartan subalgebra, and the last integral is over the strictly positive Borel subalgebra, and is a Gaussian integral.
(for G=U(N), the integral is over strictly upper triangular matrices).
The case f=1, is Harish-Chandra formula.

Example of application:
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Enumerative geometry

Integrable systems