Research Interests

In preparation (mid 2017)

• The topological recursion

The topological recursion was first discovered in random matrices in 2004 [Eynard04, AMM04].
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).
Fg(S) is a complex number, and Wg,n(S) is a differential n-form.

Some examples:

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

- The Mirzakhani relation among hyperbolic volumes of moduli spaces       (curve S: y = sin( √x)  )

- The Gromov-Witten invariants of toric Calabi-Yau 3-manifolds      (curve S: = mirror curve)

where GWg,n(X,L;ß,µ) is the open Gromov-Witten invariant of the Calabi-Yau 3-fold X, i.e. the "number" of holomorphic immersions in X of surfaces of genus g, with n boundaries, so that boundaries are immersed on a special Lagrangian submanifold L, with winding number µi, and of relative homology class ß.

- 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

• Random Matrices
A main question in random matrix theory is the large size asymptotic behavior of the eigenvalues distribution.
Below: plots of histograms for eigenvalues af a randdom Hermitian matrix, in a Gaussian model (semi-circle law), and non Gaussian. In the large size (N) limit, the eigenvalue density tends to an algebraic curve. The deviation, of order 1/N2, is also an algebraic function, and more generally, all the coefficients in the 1/N2 expandion are algebraic functions.
• See here : my GOE online generator
• See here : spectral curve of a quartic matrix model.

• Group integrals
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 dual 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:

• Angular integrals.

I found the following results

• Random geometry

• Knots

Enumerative geometry

Integrable systems

others