Consider a polynomial function f : ℂⁿ→ℂ with generic fiber F. Let B_f be the bifurcation set of f; hence f induces a smooth locally trivial fibration over ℂ\B_f. Then, for any integer q≥0 and any coefficient ring R, there is an associated monodromy representation

$\rho(f)_q : \pi_1(\mathbb {C}\backslash B_f,{\rm pt})\to {\rm Aut}(\tilde {H}_q(F,R))$

in (reduced) homology. Going around a circle in ℂ large enough to contain all of the bifurcation set gives rise to the monodromy operators at infinity, which we denote by M_∞(f)_q.

We show that these monodromy operators at infinity and a certain natural direct sum decomposition of the homology of F in terms of vanishing cycles determine the monodromy representation. The role played by this decomposition is crucial since there are examples of polynomials ℂ²→ℂ having distinct complex monodromy representations but whose monodromy operators at infinity have the same Jordan normal form.

In this paper we start a new approach to the uniformization problem of Riemann surfaces and algebraic curves by means of computational procedures. The following question is studied: Given a compact Riemann surface S described as the quotient of the Poincaré upper half-plane divided by the action of a Fuchsian group, find explicitly the polynomial describing S as an algebraic curve (in some normal form). The explicit computation given in this paper is based on the numerical computation of conformal capacities of hyperbolic domains. These capacities yield the period matrices of S in terms of the Fenchel-Nielsen coordinates, and from there one gets to the polynomial via theta-characteristics. The paper also contains a list of worked-out examples and a list of examples–new in the literature–where the polynomial for the curve, as a function of the corresponding Fuchsian group, is given in closed form.

The conserved polynomials of the Korteweg‒de Vries equation u_t=u_xxx−12uu_x are characterized by the vanishing of the residues of their associated differential polynomials evaluated on the formal power series of the kind u=x⁻²+u₀+ Σ_n≥2uⁿxⁿ.

We study the connection between the geometry and all time regularity of the interface in degenerated diffusion. Our model considers the porous medium equation u_t=Δu^m, m>1, with initial data u₀ nonnegative, integrable, and compactly supported. We show that if the initial pressure f₀=u₀^m− is smooth up to the interface and in addition it is root-concave and also satisfies the nondegeneracy condition |Df₀|≠0 at $\partial\overline {\rm supp}$f₀, then the pressure f^m−1 remains C^∞-smooth up to the interface and root-concave, for all time $0 < t < ∞$. In particular, the free boundary is C^∞-smooth for all time.

The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example, as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interval [(3/4)g, g−1). Otherwise, they still give a formula modulo third-order terms. Explicit generators and relations are also given unless d is in [(4/5)g−3/5, g−1). The virtual class on the space of stable maps plays a significant role. But the central ideas ultimately come from Brill-Noether theory: specifically, a formula of J. Harris and L. Tu for the Chern numbers of determinantal varieties. The case of d=g−1 is especially interesting: it resembles that of a Calabi-Yau 3-fold, and the Aspinwall-Morrison formula enters the calculations. A detailed analogy with A. Givental's work is also explained.

Singular Radon transforms are a type of operator combining characteristics of both singular integrals and Radon transforms. They are important in a number of settings in mathematics. In a theorem of M. Christ, A. Nagel, E. Stein, and S. Wainger [3], L^p boundedness of singular Radon transforms for 1<p<∞ is proven under a general finite-type condition using the method of lifting to nilpotent Lie groups. In this paper an alternate approach is presented. Geometric and analytic methods are developed which allow us to prove L^p-bounds in codimension 1 under a curvature condition equivalent to that of [3]. We restrict consideration to the important case where the hypersurfaces are graphs of C^∞-functions. Our methods do not involve the Fourier transform, lifting, or facts about Lie groups. This might prove useful in extending our work to related problems.