Formulating Szemerédi's theorem in terms of ultrafilters

Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that form an arithmetic progression. Szemerédi's theorem generalizes this statement and asserts that every subset of natural numbers with positive density contains arithmetic progressions of arbitrary length

Bulk-boundary correspondence for non-Hermitian Hamiltonians via Green functions

Genuinely non-Hermitian topological phases can be realized in open systems with sufficiently strong gain and loss; in such phases, the Hamiltonian cannot be deformed into a gapped Hermitian Hamiltonian without energy bands touching each other. Comparing Green functions for periodic and open boundary conditions we find that, in general, there is no correspondence between topological invariants computed for periodic boundary conditions, and boundary eigenstates observed for open boundary conditions. Instead, we find that the non-Hermitian winding number in one dimension signals a topological phase transition in the bulk: It implies spatial growth of the bulk Green function