It is an open question as to whether the prime numbers contain
the sum A+B of two infinite sets of natural numbers A, B (although
results of this type are known assuming the Hardy-Littlewood prime
tuples conjecture). Using the Maynard sieve and the...
The four dimensional ellipsoid embedding function of a toric
symplectic manifold M measures when a symplectic ellipsoid embeds
into M. It generalizes the Gromov width and ball packing numbers.
This function can have a property called an infinite...
The Lefschetz property is central in the theory of projective
varieties, detailing a fundamental property of their Chow rings,
essentially saying that the multiplication with a geometrically
motivated class is of full rank.
et K be a convex body in Rn. In some cases (say when K is a
cube), we can tile Rn using translates of K. However, in general
(say when K is a ball) this is impossible. Nevertheless, we show
that one can always cover space "smoothly" using translates...
Following Birkhoff's proof of the Pointwise Ergodic Theorem, it
has been studied whether convergence still holds along various
subsequences. In 2020, Bergelson and Richter showed that under the
additional assumption of unique ergodicity, pointwise...
Can you hear the shape of LQG? We obtain a Weyl law for the
eigenvalues of Liouville Brownian motion: the n-th eigenvalue grows
linearly with n, with the proportionality constant given by the
Liouville area of the domain (times a certain...
Graph Crossing Number is a fundamental and extensively studied
problem with wide ranging applications. In this problem, the goal
is to draw an input graph G in the plane so as to minimize the
number of crossings between the images of its edges. The...
Many cohomology theories in algebraic geometry, such as
crystalline and syntomic cohomology, are not homotopy invariant.
This is a shame, because it means that the stable motivic homotopy
theory of Morel--Voevodsky cannot be employed in studying