### Research assistant

### Ecole Polytechnique Fédérale de Lausanne

**E-mail**: franck.gabriel@normalesup.org

### Research Interest:

Holonomy fields and quantum field theory,

Random matrices and combinatorics,

Singular EDPS and regularity structures,

Neural networks,

CFT and Ising model

## New preprints:

### Neural Tangent Kernel: Convergence and Generalization in Neural Networks (A. Jacot, F. Gabriel, C. Hongler) to appear in Proceedings of Neural Information Processing Systems 2018 (8-page version).

We study the asymptotic dynamics of neural networks when the widths of the layers tend to infinity. We achieve this by using the "kernel methods" framework. This provides the first asymptotic guarantee for the training of neural networks.

### Large permutation invariant random matrices are asymptotically free over the diagonal (B. Au, G. Cébron, A. Dahlqvist, F. Gabriel, C. Male)

We show that independent permutation invariant matrices are asymptotically free over the diagonal. This provides a connection between C. Male's traffic theory and Voiculescu's freeness theory.

**Conference :** I organized the conference **"Distributional symmetries and independences"****,***Bordeaux, November, 16 - 17, 2017*. You will find some information on the conference website. Funded by a *"Projet Exploratoire Premier Soutien Jeunes chercheur-e-s"* obtained by the organizers as well as by the the *GDR MEGA* (Matrices and Random Graphs).

For an introduction to my early research work, you can refer to the thesis introduction available on this website. My thesis, * "Holonomy fields and random matrices: symmetries under braiding and permutation"*, conducted under the direction of Pr. Thierry Lévy, in Paris 6 at the LPMA laboratory, deals with

*two-dimensional Markovian holonomy fields*and the study of

*random matrices, invariant in law by conjugation by the symmetric group,*via the combinatorics of partitions.

## Articles :

### Pre-publications

** [1] - F. Gabriel:** Planar Markovian Holonomy Fields. arXiv:1501.05077,* 2015. *

**[2] - F. Gabriel:**** **Geodesic order on partitions: structures and convergence. arXiv:1503.02792, *2015*.

** [3] - F. Gabriel: **Random matrices in the light of A-tracial algebras and Schur-Weyl-Jones dualities. arXiv:1507.02465, *2015. *

** [4] - F. Gabriel: **Two dimensional S(N)-Yang-Mills theory and random ramified N-coverings of the disk in the large N-limit. (The large N-limit of random walks on S(N)), arXiv:1510.01046,* 2015. *

** [5] - B. Au, G. Cébron, A. Dahlqvist, F. Gabriel, C. Male : **Large permutation invariant random matrices are asymptotically free over the diagonal. arXiv:1805.07045,* 2018.*

### Publications

** [6] - G. Cébron, A. Dahlqvist, F. Gabriel : **The generalized master fields, arXiv:1601.00214, Journal of Geometry and Physics*, *Volume 119, Septembre 2017, pp 34-53.

** [7] - B. K. Driver, F. Gabriel, B. C. Hall, T. Kemp : **The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces, Communications in Mathematical Physics, June 2017, Volume 352, Issue 3, pp 967–978.

** [8] - A. Jacot, F. Gabriel, C. Hongler : **Neural Tangent Kernel: Convergence and Generalization in Neural Networks, arXiv:1806.07572,* 2018.*

### Educational articles

** [a] - F. Gabriel:** Generalized characteristic polynomials, graphical computations and Caley-Hamilton's theorem.

### Applied mathematics

** [b] - P. Bochard, S. Carré, R. Catellier, F. Gabriel, V. Letizia, T. Tran: **Optimal planning of energy production under technological constraints - **(**Planification optimale de production d'énergie sous contraintes technologiques - Semaine d'étude mathématiques et entreprises 4. )

### Presentations

** [c] - Presentations: **Cambridge’s probability seminar, Warwick's probability seminar & Montréal PIMS probability summer school.

** [d] - Working groups:** Basic quantum field theory.

## Collaborators :

G. Cébron, A. Dahlqvist, B. K. Driver, B. C. Hall, T. Kemp, C. Male, B. Au, M. Hairer, L. Zambotti, Y. Bruned, C. Hongler, A. Jacot, J. Fageot, S. Carré, P. Collin-Dufresne