MIT

Neural networks underpin both biological intelligence and modern AI systems, yet there is relatively little theory for how the observed behavior of these networks arises. Even the connectivity of neurons within the brain remains largely unknown, and popular deep learning algorithms lack theoretical justification or reliability guarantees.  In this talk, we consider paths towards a more rigorous understanding of neural networks. We characterize and, where possible, prove essential properties of neural algorithms: expressivity, learning, and robustness. We show how observed emergent behavior can arise from network dynamics, and we develop algorithms for learning more about the network structure of the brain.

Dedalus is a new open-source framework for solving general partial differential equations using spectral methods.  It is designed for maximum extensibility and incorporates features such as symbolic equation entry, custom domain construction, and automatic MPI parallelization.  I will briefly describe key algorithmic features of the code, including our sparse formulation and support for general tensor calculus in curvilinear domains.  I will then show examples of the code’s capabilities with various applications to astrophysical and geophysical fluid dynamics, including a compressible flow benchmark against a finite volume code, and direct numerical simulations of turbulent glacial melting

Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map).  Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants.  In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi.  Conjecturally, these should agree with the invariants as defined by stable maps.  I will also explain how to prove the conjectural correspondence for irreducible curves on local surfaces.  This is joint work with Yukinobu Toda.

Much of our understanding of the tropospheric dynamics relies on the concept of discrete internal modes. However, discrete modes are the signature of a finite system, while the atmosphere should be modeled as infinite and "is characterized by a single isolated eigenmode and a continuous spectrum" (Lindzen, JAS 2003). Is it then unphysical to use discrete modes? To resolve this issue we obtain an approximate radiation condition at the tropopause --- this yields an EBC. We then use this EBC to compute a new set of vertical modes: the leaky rigid lid modes. These modes decay, with decay time-scales for the first few modes ranging from an hour to a week. This suggests that the rate of energy loss through upwards propagating waves may be an important factor in setting the time scale for some atmospheric phenomena. The modes are not orthogonal, but they are complete, with a simple way to project initial conditions onto them.

The EBC formulation requires an extension of the dispersive wave theory. There it is shown that sinusoidal waves carry energy with the group speed c_g = d omega / dk, where both the frequency omega and wavenumber k are real. However, when there are losses, complex k's and omega's arise, and a more general theory is required. I will briefly comment on this theory, and on how the Laplace Transform can be used to implement generic EBC.

In his talk, Kerry will explore the pressing practical problem of how hurricane activity will respond to global warming, and how hurricanes could in turn be influencing the atmosphere and ocean

In his talk, Kerry will explore the pressing practical problem of how hurricane activity will respond to global warming, and how hurricanes could in turn be influencing the atmosphere and ocean.

There is a simple way to “glue together” a coupled pair of continuum random trees to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the “interface” between the trees). We present an explicit and canonical way to embed the sphere into the Riemann sphere. In this embedding, the measure is Liouville quantum gravity with parameter gamma in (0,2), and the curve is space-filling version of SLE with kappa=16/gamma^2. Based on joint work with Bertrand Duplantier and Scott Sheffield