School of Mathematics

We present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced with M. Goldman. Following De Giorgi’s philosophy for the regularity theory of minimal surfaces, it is...

The dichotomy between structure and randomness plays an important role in areas such as combinatorics and number theory. I will discuss a similar dichotomy in complexity theory, and illustrate it with three examples of my own work: (i) An...

There is growing interest in looking at operations on quantum cohomology that take into account symmetries in the holomorphic spheres (such as the quantum Steenrod powers, using a Z/p-symmetry). In order to prove relations between them, one needs to...

Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GL_n/F with p-torsion coefficients. These Galois representations are expected to satisfy local-global...

Flows of vector fields: classical and modern Consider a (possibly time-dependent) vector field v on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field v is Lipschitz in space...

There are significant gaps between legal and technical thinking around data privacy. Technical standards such as k-anonymity and differential privacy are described using mathematical language whereas legal standards are not rigorous from a...

The Kudla-Rapoport conjecture predicts a precise identity between the arithmetic intersection number of special cycles on unitary Rapoport-Zink spaces and the derivative of local representation densities of hermitian forms. It is a key local...

Meta-learning (or learning to learn) studies how to use machine learning to design machine learning methods themselves. We consider an optimization-based formulation of meta-learning that learns to design an optimization algorithm automatically...

A continuing mystery in understanding the empirical success of deep neural networks has been in their ability to achieve zero training error and yet generalize well, even when the training data is noisy and there are many more parameters than data...