Extremal combinatorics is a central research area in discrete
mathematics. The field can be traced back to the work of Turán and
it was established by Erdős through his fundamental contributions
and his uncounted guiding questions. Since then it has...
Suppose you have a set S of integers from {1 , 2 , … , N} that
contains at least N / C elements. Then for large enough N , must S
contain three equally spaced numbers (i.e., a 3-term arithmetic
progression)?
Max Meyerhof, born in 1874 as the only son of a German-Jewish
family in Hildesheim, spent half his life in Cairo, where he died
in 1945, shortly before the end of the war. He practiced as a
successful ophthalmologist, who was said to be able to cure...
In certain time-dependent backgrounds there are nonstandard
light strings that are created classically at an instant and cannot
be approximated by particles. We discuss the implications of these
strings to black holes and cosmology.
Through the random matrix analogy, Fyodorov, Hiary and Keating
conjectured very precisely the typical values of the Riemann zeta
function in short intervals of the critical line, in particular
their maximum. Their prediction relied on techniques...
In Euclidean geometry, bisectors are perpendicular lines. In
random plane geometry, the situation is more complicated. I will
describe bisectors in the directed landscape, the universal
geometry in the KPZ class. These help answer some open...
For a compact subset K of a closed symplectic manifold,
Entov-Polterovich introduced the notion of (super)heaviness, which
reveals surprising symplectic rigidity. When K
is a Lagrangian submanifold, there is a well-established
criterion for its...
Centers of galaxies are fertile environments for a variety of
dynamical processes, owing to the high density of stars and the
presence of a central supermassive black hole. Stars may
occasionally be deflected onto a deadly trajectory, passing
too...
The Mackey-Zimmer representation theorem is a key structural
result from ergodic theory: Every compact extension between ergodic
measure-preserving systems can be written as a skew-product by a
homogeneous space of a compact group. This is used, e.g...
Let X be a smooth projective variety over the field of complex
numbers. The classical Riemann-Hilbert correspondence supplies a
fully faithful embedding from the category of perverse sheaves on X
to the category of algebraic D_X-modules. In this...
