Based on work of Rasmussen [Ras03], we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsv ́ath-Szab ́o τ invariant.

It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satises the circular law. We consider the product $T X$, where $T$ is a deterministic $N\times M$ matrix and $X$ is a random $M\times  N$ matrix with independent entries having zero mean and variance $(N \wedge M) ^{-1}$. We prove a general local circular law for the empirical spectral distribution (ESD) of $TX$ at any point $z$ away from the unit circle under the assumptions that$ N \sim M$, and the matrix entries $X_{ij} have suciently high moments. More precisely, if $z$ satisfes $| |z|
- 1| \ge \tau$  for arbitrarily small $\tau > 0, the ESD of $TX$ converges to $\tilde{X}_D (z) d A(z)$, where $\tilde{X}_D$ is a rotation-invariant function determined by the singular valuau of $T$ and $dA$ denotes the Lebesgue measure on $\mathbb{C}$. The local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/4+\epsilon}$ for any  $\epsilon >0$. Moreover, if $|z| >1$ or the matrix entries of $X$ have vanishing third moments, the local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/2+\epsilon}$ for any  $\epsilon >0$.

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in F. Almgren, and J. Pitts, corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$of positive Ricci curvature with 2 ≤ n ≤ 6. We characterize the Morse index, volume and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.

In these notes we collect some results about finite-dimensional representations of $U_{q}(\mathfrak {gl}(1\mid1))$ and related invariants of framed tangles, which are well-known to experts but difficult to find in the literature. In particular, we give an explicit description of the ribbon structure on the category of finite-dimensional $U_{q}(\mathfrak {gl}(1\mid1))$ -representations and we use it to construct the corresponding quantum invariant of framed tangles. We explain in detail why this invariant vanishes on closed links and how one can modify the construction to get a non-zero invariant of framed closed links. Finally we show how to obtain the Alexander polynomial by considering the vector representation of $U_{q}(\mathfrak {gl}(1\mid1))$ .

Previous work has shown that in a two-dimensional periodic medium under focusing or defocusing cubic nonlinearities, gap solitons in the form of low-amplitude and slowly modulated single-Bloch-wave packets can bifurcate out from the edges of Bloch bands. In this paper, linear stability properties of these gap solitons near band edges are determined both analytically and numerically. Through asymptotic analysis, it is shown that these gap solitons are linearly unstable if the slope of their power curve at the band edge has the opposite sign of nonlinearity
here focusing nonlinearity is said to have a positive sign, and defocusing nonlinearity to have a negative sign. An equivalent condition for linear instability is that the power of the gap solitons near the band edge is lower than the limit power value on the band edge. Through numerical computations of the power curves, it is found that this condition is always satisfied, thus two-dimensional gap solitons near band edges are linearly unstable. The analytical formula for the unstable eigenvalue of gap solitons near band edges is also asymptotically derived. It is shown that this unstable eigenvalue is proportional to the cubic power of the soliton’s amplitude, and it induces width instabilities of gap solitons. A comparison between this analytical eigenvalue formula and numerically computed eigenvalues shows excellent agreement.