The mathematical core of deep learning is function approximation
by neural networks trained on data using stochastic gradient
descent. I will present a collection of sharp results on training
dynamics for the deep linear network (DLN), a...
Higher-dimensional Heegaard Floer homology (HDHF) is defined by
extending Lipshitz's cylindrical reformulation of Heegaard Floer
homology from surfaces to arbitrary Liouville domains. The HDHF
also serves as a model for Lagrangian Floer homology of...
In this talk, we will discuss new algorithms for solving
instances of random and semirandom planted constraint satisfaction
problems (CSPs). Random CSP are generated by first choosing a
solution x and then sampling constraints so that x satisfies...
I will discuss the path integral of pure 3D gravity on a finite
region of spacetime, with boundary conditions that fix dihedral
angles or geodesic lengths. This amplitude calculates corrections
to the Gaussian statistics of OPE coefficients in the...
A natural problem in the study of local systems on complex
varieties is to characterize those that arise in a family of
varieties. We refer to such local systems as motivic. Simpson
conjectured that for a reductive group G, rigid G-local
systems...
Consider a point mass traveling in a polygon. It travels in a
straight line, with constant speed, until it hits a side, at which
point it obeys the rules of elastic collision. What can we say
about this? When all the angles of the polygon are...
In this work, we develop a new method for proving lower bounds
for static data structures in the classical cellprobe model of Yao.
Our methods give the strongest known lower bounds for any explicit
problem in this model (quadratically stronger for...
I will explain how the irreversibility of the renormalization
group together with anomalies, including anomalies in the space of
coupling constants, can be used to constrain the IR phases of
defects in familiar quantum field theories. As an example...
In the era of high-dimensional data and simulation-based
science, machine learning is transforming the different stages of
the scientific method in astrophysics. I will present a summary of
my research connected to generative models in two main...
