Inspired by the Erdos--Renyi model for random graphs, Linial and
Meshulam devised in 2006 a model for random 2-dimensional
simplicial complexes. The goal of this talk (and the next) is to
present some nice results about the behavior of these random...
This talk is concerned with solutions of the 3D incompressible
Navier-Stokes equations that are bounded in a critical space. From
small initial data, these solutions are known to be globally
well-posed due to classical work of Fujita-Kato and others...
The compressible Euler equation can lead to the emergence of
shock discontinuities in finite time, notably observed behind
supersonic planes. A very natural way to justify these
singularities involves studying solutions as inviscid limits of
Navier...
Riemannian metrics are the simplest generalizations of Euclidean
geometry to smooth manifolds. The Ricci curvature of a metric
measures, in an averaged sense, how the geometry deviates from
being flat. The tensor −2Ric can be viewed as a Laplacian...
Large language models (LLMs) sometimes generate statements that
are plausible but factually incorrect—a phenomenon commonly called
"hallucination." We argue that these errors are not mysterious
failures of architecture or reasoning, but rather...
The Birkhoff attractor is a closed invariant subset associated
with any dissipative twist map of the annulus (of dimension 2),
which was introduced by Birkhoff in 1932. We will see that it can
be generalized to higher dimensions using tools from...
The goal of these lectures is to present the fundamentals of
Simpson’s correspondence, generalizing classical Hodge theory,
between complex local systems and semistable Higgs bundles with
vanishing Chern classes on smooth projective varieties.
The fields of computer science and game theory both trace their
roots to the first half of the 20th century, with the work of
Turing, von Neumann, Nash, and others. The 21st century has seen
many fruitful points of contact between these two fields...
The goal of these lectures is to present the fundamentals of
Simpson’s correspondence, generalizing classical Hodge theory,
between complex local systems and semistable Higgs bundles with
vanishing Chern classes on smooth projective varieties.
