This essay explains how the maximum bisection value of sparse Erdos-Renyi graphs with asymptotically large average degree is related to the Sherrington-Kirkpatrick model of spin glasses.
This essay explains how the matrix-tree theorem can be combined with the theory of local convergence of graphs to enumerate, up to leading exponential order, the number of spanning trees in various sequences of graphs. It is possible to compute, for example, the number of spanning trees in typical regular graphs of large size.
The Hausdorff-Young inequality states that for 1 < p < 2, the Fourier transform of an L^p function lies in L^q where q is the Holder dual of p. How small is the image of L^p in L^q under the Fourier transform? The Stein-Tomas restriction theorem states that for p in a certain range, the Fourier transform of an L^p function in R^n lies in L^2 of the (n-1)-sphere. This essay contains a proof of the theorem along with an application to Schrodinger's equation and a discussion on the connection between restriction problems and the Kakeya conjecture.
In the 1950s Roth proved that any sufficiently large subset of the the first N natural numbers contains a three term arithmetic progression. Since then arithmetic problems of this nature have seen remarkable progress with important tools developed through combinatorics, ergodic theory, harmonic analysis and number theory. This essay provides Roth's Fourier analytic proof of the theorem.