Research Interests |

- The topological recursion

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 W

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^(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

- Random matrices

Study the statistical properties of eigenvalues of large random matrices.

- Group integrals

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:

(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.

- Random geometry

- Knots

Enumerative geometry

Integrable systems

others