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 based on the approximation of the displacement by a harmonic gradient, which leads to a One-Step Improvement Lemma, and feeds into a Campanato iteration on the C1,α-level for the displacement, capitalizing on affine invariance.
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 generalise this to include equivariant operations with more marked points, varying domains and different symmetry groups. We will look at the general method of construction of these operations, as well as two distinct examples of relations between them.
Member, School of Natural Sciences, Institute for Advanced Study
April 17, 2020
In this talk we will present details of quantum extremal surface computations in a simple setup, demonstrating the role of entanglement islands in resolving entropy paradoxes in gravity. The setup involves eternal AdS2 black holes in thermal equilibrium with auxiliary bath systems. We will also describe the extension of this setup to higher dimensions using Randall-Sundrum branes.
Despite the success of deep learning, much of its success has existed in settings where the goal is to learn one, single-purpose function from data. However, in many contexts, we hope to optimize neural networks for multiple, distinct tasks (i.e. multi-task learning), and optimize so that what is learned from these tasks is transferable to the acquisition of new tasks (e.g. as in meta-learning).
Physical processes in the world often have a modular structure, with complexity emerging through combinations of simpler subsystems. Machine learning seeks to uncover and use regularities in the physical world. Although these regularities manifest themselves as statistical dependencies, they are ultimately due to dynamic processes governed by physics. These processes are often independent and only interact sparsely..Despite this, most machine learning models employ the opposite inductive bias, i.e., that all processes interact.
Standard machine learning produces models that are highly accurate on average but that degrade dramatically when the test distribution deviates from the training distribution. While one can train robust models, this often comes at the expense of standard accuracy (on the training distribution). We study this tradeoff in two settings, adversarial examples and minority groups, creating simple examples which highlight generalization issues as a major source of this tradeoff.
There are several broad insights we can draw from computational models of human cognition in order to build more human-like forms of machine learning. (1) The brain has a great deal of built-in structure, yet still tremendous need and potential for learning. Instead of seeing built-in structure and learning as in tension, we should be thinking about how to learn effectively with more and richer forms of structure. (2) The most powerful forms of human knowledge are symbolic and often causal and probabilistic.
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 compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems. I will discuss joint work with J.
In this talk, I would like to share some of my reflections on the progress made in the field of interpretable machine learning. We will reflect on where we are going as a field, and what are the things that we need to be aware of to make progress. With that perspective, I will then discuss some of my work on 1) sanity checking popular methods and 2) developing more lay person-friendly interpretability methods. I will also share some open theoretical questions that may help us move forward.