Flatness is a subtle algebraic notion that expresses continuity of the fibres of a mapping, but it has remained geometrically elusive. The aim of this talk is to understand the notion of flatness in explicit geometric terms. We show that non-flatness of a morphism of schemes of finite type with a regular target of dimension n manifests in the existence of the so-called vertical components in the n-fold fibred power of the morphism (i.e., components with a nowhere dense image). This leads to an effective algorithm for flatness over regular affine algebras, by means of Grobner bases.

This work was done with E. Bierstone and P. D. Milman.

Can we find the smallest class of singularities S with the following properties:

(1) S includes all normal-crossings singularities;
(2) every reduced variety X admits a birational morphism s: X¢® X such that X¢ has only singularities in S, and s is an isomorphism over the locus of points of X having only singularities in S?

Coherent systems are the analogous for higher classical linear systems. That is, a coherent system of type (n,d,k) is a pair (E,V) where E is a holomorphic bundle of rank n and degree d and V is a linear subspace of its space of holomorphic sections of dimension k. There is a stability notion for a pair (E,V), distinct from the stability of the bundle E. The natural definition of such stability depends on a real parameter a and leads to a finite family of moduli spaces of a-stable coherent systems. In this talk we will describe such moduli spaces for certain values of (n,d,k).

I will present joint work with Angelo Vistoli and Zinovy Reichstein on essential dimension of the moduli stacks of curves and of abelian varieties. The main technical result used to compute this essential dimension is a "genericity theorem" which reduces the computation of the essential dimension of a smooth Deligne-Mumford stack to the essential dimension of its generic gerbe. I will sketch the proof of this result.

There are two primary theories of "curve counting" on a Calabi-Yau threefold, Donaldson-Thomas theory and Gromov-Witten theory. Both theories extend to Calabi-Yau orbifolds. A three dimensional Calabi-Yau orbifold X has a canonical Calabi-Yau resolution Y. The four curve counting theories DT(X), DT(Y), GW(X), and GW(Y) are expected to be equivalent. We give an overview of these theories and conjectural equivalences.

In this talk we present a generalized algebraic index formula for certain class of real-analytic vector fields with non-algebraically isolated singularities. In the algebraically isolated case, we compute the index with the library PHindex writing for the software Singular, using the Eisenbud-Levine-Khimshiashvili's algebraic index formula.

Let C be an smooth complex algebraic curve of genus g ³ 8. Denote by K_C the canonical line bundle on C. Let |L| = g¹_g-2 a pencil on C free of base points such that the residual g²_g of the g¹_g-2 determines a birational map onto a plane curve of degree g and geometric genus g with d = [((g-1)(g-2))/2] nodes as singularities. Consider the Petri map m_L : H⁰(C,L) ÄH⁰(C,K_C ÄL^-1) ® H⁰(C,K_C). We show that m_L is not injective if and only there exists a curve F of degree g-5 containing d-1 nodes of G.

Now consider the Severi Variety V^g,g,d of reduced and irreducible plane curves of degree g and genus g having d nodes as singularities. Let V_g : = {G Î V^g,g,d : d-1 nodes lie on a curve of degree g-5}. Let f: V^d_g,g ® M_g the natural morphism to the moduli space of curves M_g. We show that the image f(V_g) is a divisor in M_g. We discuss the irreducibility of f(V_g) in some cases.

Given a holomorphic map-germ f : (Cⁿ,0) ® (C,0) with a critical value at 0 Î C, there are two equivalent ways of defining its Milnor fibration. The first is:

We show that there is a canonical decomposition of the whole ball B_e into real analytic hypersurfaces X_q that spin around their "axis" V_e = f^-1(0) ÇB_e forming a kind of "open-book" with singular binding. Using this decomposition we improve, or refine, Milnor's fibration theorem in several directions.

This is joint work with J. Seade and J. Snoussi.

We consider a particular family of singular Calabi-Yau 3-folds parametrized by U = P/1 - {q₁,...,q₆}. The singular locus of the fiber over u consists of exactly 100 nodes for every u Î U.

We study the monodromy action and in particular the variation of Hodge structure associated to the smooth family of the desingularizations.

Holomorphic Poisson manifolds are highly constrained compared to their relatively flexible smooth counterparts. I shall describe some new results concerning the structure and classification of holomorphic Poisson manifolds.

The moduli spaces of instantons on the four-sphere and of calorons (instantons on the circle times R³) turn out to be describable in terms of holomorphic maps from the Riemann sphere into a flag manifold of a loop group. These in turn, like for their finite dimensional cousins, admit a poles and principal parts description that allows one to describe the moduli and prove, for example, a topological stability theorem for the moduli.

Joint work with Michael Murray.

Let X be a algebraic variety equipped with an action of a reductive algebraic group G. Also let L be a finite dimensional subspace of rational functions invariant under G. In this talk we discuss various convex bodies that one can associate to (X,L) which encode information about number of solutions of generic systems of equations from L plus information on multiplicities of irreducible representations appearing in powers L^k. Similarly one associates convex bodies to a projective G-variety X and a G-linearized line bundle L on X. These far generalize the notion of Newton convex polytope in toric geometry as well as Gelfand-Cetlin polytopes associated to irreducible representations of GL(n) (i.e., flag variety of GL(n)).

This is a joint work with A. G. Khovanskii.

In this talk, I will examine the geometry of moduli spaces of stable bundles on Kodaira surfaces, which are non-Kaehler compact surfaces that can be realised as torus fibrations over elliptic curves. These moduli spaces are interesting examples of holomorphic symplectic manifolds whose geometry is similar to the geometry of Mukai's moduli spaces on K3 and abelian surfaces.