Abstract:I will discuss some new bounds on the spectra of Laplacian operators on hyperbolic 3-manifolds. One example of such a bound is that the spectral gap of the Laplace-Beltrami operator on a closed orientable hyperbolic 3-manifold must be less than 47.32, or less than 31.57 if the first Betti number is positive. The bounds are derived using two approaches, both of which employ linear prog...
AbstractWe study two related problems concerning the number of monochromatic cliques in two-colorings of the complete graph that go back to questions of Erdős. Most notably, we “significantly” improve the best known upper bounds on the Ramsey multiplicity of K_4 and K_5 and settle the minimum number of independent sets of size four in graphs with clique number at most four. Motivated by the ...