Given a one-parameter degenerating family of rational maps on
the projective line, it is possible to construct a non-archimedean
limit which captures how this family degenerates. Recently,
Luo used ultrafilters to construct limits for an arbitrary...
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...
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 talk about the well known relation between cocycles and
covers. Then, I will discuss the relation of cocycle
expansion to approximate covers, and finally how these show up in
low-soundness PCP agreement tests.
What is the cost of a task if we have to perform it many times?
Can we achieve economies of scale? Such "direct-sum problems" play
a central role in many areas of mathematics, physics and computer
science. Protagonistic problems of this kind are...
