We show that the analogue of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.

The spectrum of a Schrödinger operator with a perfect honeycomb lattice potential has special points, called Dirac points, where the lowest two branches of the spectrum touch. Deformations can result in the merging and disappearance of the Dirac points, and the originally intersecting dispersion relation branches separate. Corresponding to these deformations, nonlinear envelope equations are derived and their dynamics are studied. In the region where Dirac points exist, a maximally balanced equation is derived which has limits to a nonlinear Schrödinger--Kadomtsev--Petviashvili (NLSKP)-type equation and its dispersionless reduction. When the Dirac points disappear and a gap opens, a different maximally balanced equation is derived which has the NLSKP equation and a one-dimensional nonlocal evolution equation as limits. When the gap is sufficiently wide, a nonlinear Dirac equation with nonzero mass and a nonlinear Schrödinger focusing-defocusing system are found. The latter two equations admit nonlinear localized modes. Typical dynamical behaviors of the effective envelope equations are presented.

We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [14] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \E \abs{h_{ij}}^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width $W\gg N^{1-\e_n}$ with some $\e_n>0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17,19,6].

We prove that a constrained Willmore immersion of a 2–torus into the conformal 4–sphere S4 is of “finite type”, that is, has a spectral curve of finite genus, or of “holomorphic type” which means that it is super conformal or Euclidean minimal with planar ends in R4 = S4\{1} for some point 1 2 S4 at infinity. This implies that all constrained Willmore tori in S4 can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin’s approach [19] to the study of harmonic tori in S3.

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