La recherche

We survey recent results on soliton resolution for the energy critical nonlinear wave equation.

Let $X$ be a smooth algebraic variety over $\mathbb{R}$. We study the separation of the connected components of $X(\mathbb{R})$ by equivariant cohomology, unramified cohomology, and signatures of unramified quadratic forms. Our results highlight how the geometry of $X$ influences these questions. We apply them to cubic hypersurfaces, complete intersections of two quadrics, abelian varieties, and Enriques surfaces. We also obtain new bounds on the exponent of the cokernel of higher signature morphisms in $\mathbf{I}$-cohomology and investigate the optimality of these bounds for smooth projective varieties.

This PhD thesis presents contributions to the field of deep learning. From convolutional ResNets to Transformers, residual connections are ubiquitous in state-of-the-art deep learning models. The continuous depth analogues of residual networks, neural ODEs, have been widely adopted, but the connection between the discrete and continuous models still lacks a solid mathematical foundation. In this manuscript, we will show that for a formal correspondence between residual networks and neural ODEs to hold, the residual functions must be smooth with depth, and we will present an implicit regularization result of deep residual networks towards neural ODEs. We will then present two applications of this analogy to the design and study of new architectures. First, we will introduce a drop-in replacement for any residual network that can be trained with the same accuracy, but with much less memory. Second, by viewing the attention mechanism as an interacting particle system, where the particles are the tokens, we will study the impact of attention map normalization on the Transformer model. Finally, we will present some other contributions to Transformers: how Transformers perform in-context autoregressive learning and how to differentiably route tokens to experts in Sparse Mixture of Experts Transformers.