Consider a sparse system of \(n\) Laurent polynomials in \(n\) variables with complex coefficients and support in a finite lattice set \(A\). The maximal number of isolated roots in the torus of the system is known to be the normalized volume of the convex hull of \(A\) (the BKK bound). We explore the following question: if the cardinality of \(A\) equals \(n+m+1\), which is the maximum local intersection multiplicity at one point in the torus in terms of \(n\) and \(m\)? This study was initiated by Gabrielov in the multivariate case. In joint work with Frédéric Bihan and Jens Forsgård, we give an upper bound that is always sharp for circuits and, under a generic technical hypothesis, it is considerably smaller for any codimension \(m\). We also present a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.

Given a normal crossings degeneration \(f:(X,\Omega_X)\to\Delta\) of complex Kahler manifolds, in recent work together with T. Pelka, we have shown how to associate a smooth locally trivial fibration \(f_A:A\to \Delta_{log}\) over the real oriented blow up of the disc \(\Delta\). It is moreover endowed with a closed \(2\)-form \(\omega_A\) giving it the structure of a symplectic fibration. The restriction of \(\omega_A\) to every fibre of \(f_A\) «at positive radius» (that is over a point of \(\Delta\setminus\{0\}\) is the modification by a potential of the restriction of \(\omega_X\) to the same fibre. The construction is so that:

(1) We can produce symplectic representatives of the monodromy with very special dynamics, and based on this and on a spectral sequence due to McLean prpve the family version of Zariski’s multiplicity conjecture.

(2) If \(f\) is a maximal Calabi-Yau degeneration we can produce Lagrangian torus fibrations over a the complement of a codimension 2 set over the (expanded) essential skeleton of the degeneration, satisfying many of the properties conjectured by Kontsevich and Soibelman.

The analogies between the iteration of holomorphic maps and the action of Kleinian groups are numerous.
In this talk we will discuss how these two worlds can be combined together. More precisely, we will see how the dynamics of rational maps and kleinian groups can be united in a holomorphic correspondence.

In this talk, I will discuss some topological and geometric properties of diffeomorphisms groups that are hard to tackle via classical methods because of the lack of local compactness. In particular, I will elaborate on Gromov’s notion of distortion in this context. I will also draw the ideas of a result recently obtained in collaboration with Hélène Eynard-Bontemps (Ins. Fourier, Grenoble): The space of pairs of twice-differentiable commuting diffeomorphisms of \(1\)-manifolds is path-connected.

A basic question in algebraic geometry is whether there can be any non-constant maps between (smooth, projective) varieties of different types. I will explain some basic and some more sophisticated obstructions to the existence of such maps, the latter revolving around the notion of Kodaira dimension of an algebraic variety. I will also discuss some recent conjectures.

The volume of a line bundle on a smooth projective variety is a rough measure for the asymptotic growth of the dimension of the space of sections of its high tensor powers, and line bundles with positive volume are called big. The volume extends naturally to a continuous function on the real Neron-Severi group, which vanishes outside the big cone and is \(C^1\) differentiable inside of it, by work of Boucksom-Favre-Jonsson and Lazarsfeld-Mustata. An interesting question is then to understand the behavior of the volume near the boundary of the big cone. More precisely, given \(D\) a pseudoeffective \(R\)-divisor with volume zero and \(A\) an ample divisor, what is the behavior of the function \(vol(D+tA)\) as \(t\) decreases to zero? I will discuss joint work with Simion Filip and John Lesieutre where we construct examples where this volume function is \(C^1\) but not better, and answer negatively questions of Lazarsfeld, Fujino and others.

I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.