In part 1, I will survey the history of total positivity,
beginning in the 1930's with the introduction of totally positive
matrices, which turn out to have surprising linear-algebraic and
combinatorial properties. I will discuss some modern...
In this talk I will discuss various connections between the
dynamics of integrable Hamiltonian flows, gradient flows, and
combinatorial geometry. A key system is the Toda lattice
which describes the dynamics of interacting particles on the line.
I...
Consider a closed surface M of genus greater than or equal to 2.
For negatively curved metrics on M and their corresponding geodesic
flow, we can study the topological entropy, the Liouville entropy,
and the mean root curvature. In 2004, Manning...
The notion of singular support has its origin in the theory of
partial differential equations, and was introduced to the world of
constructible sheaves by M. Kashiwara and P. Schapira in the 1970s.
It is a basic invariant (like the support), and...
In 1962, Yudovich established the well-posedness of the
two-dimensional incompressible Euler equations for solutions with
bounded vorticity. However, uniqueness within the broader class of
solutions with L^p vorticity remains a key unresolved...
Although current large language models are complex, the most
basic specifications of the underlying language generation problem
itself are simple to state: given a finite set of training samples
from an unknown language, produce valid new strings...
In this talk, we will explore the relationship between the
geometry and topology of a complexity-one four-manifold and the
combinatorial data that encode it. We will use a
generators-and-relations description for the even part of the
equivariant...
Spatially resolved polarimetric images of black holes with the
Event Horizon Telescope (EHT) favor models with strong magnetic
fields, so-called Magnetically Arrested Disks or MAD models.
MADs can produce efficient jets via the Blandford-Znajek (BZ...
Configuration spaces of points in graphs are nonsmooth analogs
of braid arrangements, appearing in robotics applications and in
theory of moduli spaces of tropical curves. While their cohomology
is extremely difficult to understand, and depends on...
The Navier–Stokes (NS) equations describe fluid dynamics through
a high-dimensional, nonlinear partial differential equations (PDEs)
system. Despite their fundamental importance, their behavior in
turbulent regimes remains incompletely understood...
