# Franck Gabriel

### 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, Economics

## Latest news :

- New video and New Poster about the article Neural Tangent Kernel: Convergence and Generalization in Neural Networks (A. Jacot, F. Gabriel, C. Hongler), which will be explained during a Spotlight Talk at NIPs2018
- New preprints :
- Insider Trading with Penalties (S. Carré, P. Collin-Dufresne, F. Gabriel)
- Neural Tangent Kernel: Convergence and Generalization in Neural Networks. (A. Jacot, F. Gabriel, C. Hongler)
- Large Permutation Invariant Random Matrices are Asymptotically Free over the Diagonal. (B. Au, G. Cébron, A. Dahlqvist, F. Gabriel, C. Male)

## More about the new preprints:

### Insider trading with penalties (S. Carré, P. Collin-Dufresne, F. Gabriel), Septembre 2018

We consider a one-period Kyle (1985) framework where the insider can be subject to a penalty if she trades. We establish existence and uniqueness of equilibrium for virtually any penalty function when noise is uniform. In equilibrium, the demand of the insider and the price functions are in general non-linear and remain analytically tractable because the expected price function is linear. We use this result to investigate the trade off between price efficiency and 'fairness': we consider a regulator that wants to minimise post-trade standard deviation for a given level of uninformed traders' losse and the minimisation is over the function space of penalties. Optimal penalties are characterized in closed-form. They must increase quickly with the magnitude of the insider's order for small orders and become flat for large orders: high trades-if they occur-are costly for liquidity traders but they signal extreme events and therefore incorporate a lot of information into prices. We generalize this result by imposing a budget constraint on the regulator. Under this constraint, if the penalties are non-pecuniary, optimal ones are a subset of the previously optimal penalties: the patterns of equilibrium trade volumes and prices is unchanged. If the penalties are pecuniary, we show that new patterns emerge in the demand schedules of the insider trader and the associated price functions.

### 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.*

** [6] - S. Carré, P. Collin-Dufresne, F. Gabriel :** Insider Trading with Penalties. hal:1874923, *2018*.

### Publications

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

** [8] - 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.

** [9] - A. Jacot, F. Gabriel, C. Hongler : **Neural Tangent Kernel: Convergence and Generalization in Neural Networks, to appear in Proceedings of Neural Information Processing Systems 2018 , 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