Communications in Contemporary Mathematics

We define two conformal structures on S¹ which give rise to a different view of the affine curvature flow and a new curvature flow, the "Q-curvature flow". The steady states of these flows are studied. More specifically, we prove four sharp inequalities, which state the existence of the corresponding extremal metrics.

We extend recent results of Machihara and Pecher on low regularity well-posedness of the Dirac–Klein–Gordon (DKG) system in one dimension. Our proof, like that of Pecher, relies on the null structure of DKG, recently completed by D'Ancona, Foschi and Selberg, but we show that in 1d the argument can be simplified by modifying the choice of projections for the Dirac operator. We also show that the result is best possible up to endpoint cases, if one iterates in Bourgain–Klainerman–Machedon spaces.

We consider here the following biharmonic equations:

Given a compact surface (Σ,g), we prove the existence of a solution for the mean field equation on Σ. The problem consists of solving a second-order nonlinear elliptic equation with variational structure and exponential nonlinearity. Since the corresponding Euler functional is, in general, unbounded from above and from below, we employ topological methods and min-max schemes. Our result generalizes previous results by Lin [11], by Chen and Lin [6] and by Ding et al. [8]. The main point here is that, by taking values of the parameter greater than 8π, we make no assumption on the topology of the surface.

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an -operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from -operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

Let x be a complex random variable with mean zero and bounded variance σ². Let N_n be a random matrix of order n with entries being i.i.d. copies of x. Let λ₁, …, λ_n be the eigenvalues of . Define the empirical spectral distributionμ_n of N_n by the formula

The following well-known conjecture has been open since the 1950's:

Circular Law Conjecture: μ_n converges to the uniform distribution μ_∞ over the unit disk as n tends to infinity.

We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices.

The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.