Under convenient geometric assumptions, the saddle-point method for multidimensional Laplace integrals is extended to the case where the contours of integration have boundaries. The asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points. The incidence of the Stokes phenomenon is related to the monodromy of the homology via generalized Picard-Lefschetz formulae and is quantified in terms of geometric indices of intersection. Exact remainder terms and the hyperasymptotics are then derived. A direct consequence is a numerical algorithm to determine the Stokes constants and indices of intersections. Examples are provided.

We prove that the filtration on the pure braid group on g strands, induced by the lower central series of the Torelli group of a genus g surface with one boundary component, coincides with its lower central series, shifted by one.

In particular, the cubic Jacobi relations in the pure braid group are quadratic relations in the Torelli group.

We establish borderline regularity for solutions of the Beltrami equation $f\sb z-\mu f\sb {\overline {z}}=0$ on the plane, where $\mu$ is a bounded measurable function, $\parallel\mu\parallel\sb \infty=k<1$. What is the minimal requirement of the type $f\in W \sp {1,q}\sb {{\rm loc}}$ which guarantees that any solution of the Beltrami equation with any $\parallel\mu\parallel\sb \infty=k<1$ is a continuous function? A deep result of K. Astala says that $f\in W \sp {1,1+k+\varepsilon}\sb {{\rm loc}}$ suffices if $\varepsilon>0$. On the other hand, O. Lehto and T. Iwaniec showed that $q<1+k$ is not sufficient. In [2], the following question was asked: What happens for the borderline case $q=1+k$? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia's extrapolation technique and two-weight estimates for the [26].

The evaluation homomorphisms from the Yangian ${\rm Y}(\mathfrak {gl}\sb n)$ to the universal enveloping algebra ${\rm U}(\mathfrak {gl}\sb n)$ allow one to regard the irreducible finite-dimensional representations of $\mathfrak {gl}\sb n$ as Yangian modules. We give necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.

We study the tensor product $W$ of any number of irreducible finite-dimensional modules $V\sb 1,\ldots V\sb k$ over the Yangian ${\rm Y}(\mathfrak {gl}\sb N)$ of the general linear Lie algebra $\mathfrak {gl}\sb N$. For any indices $i,j=1,\ldots k$, there is a canonical nonzero intertwining operator $J\sb {ij} : V\sb i\otimes V\sb j\to V\sb j\otimes V\sb i$. It has been conjectured that the tensor product $W$ is irreducible if and only if all operators $J\sb {ij}$ with $i<j$ are invertible. We prove this conjecture for a wide class of irreducible ${\rm Y}(\mathfrak {gl}\sb N)$-modules $V\sb 1,\ldots V\sb k$. Each of these modules is determined by a skew Young diagram and a complex parameter. We also introduce the notion of a Durfee rank of a skew Young diagram. For an ordinary Young diagram, this is the length of its main diagonal.

If $r\geq 6,r\neq 9$, we show that the minimal resolution conjecture (MRC) fails for a general set of $\gamma$ points in $\mathbb {P}\sp r$ for almost $(1/2)\sqrt {r}$ values of $\gamma$. This strengthens the result of D. Eisenbud and S. Popescu [EP1], who found a unique such $\gamma$ for each $r$ in the given range. Our proof begins like a variation of that of Eisenbud and Popescu, but uses exterior algebra methods as explained by Eisenbud, G. Fløystad, and F.- O. Schreyer [EFS] to avoid the degeneration arguments that were the most difficult part of the Eisenbud-Popescu proof. Analogous techniques show that the MRC fails for linearly normal curves of degree $d$ and genus $g$ when $d\geq 3g-2,g\geq 4$, re-proving results of Schreyer, M. Green, and R. Lazarsfeld.