## Notes of Journee francilienne d’accueil des post-doctorants en mathematiques

Journee francilienne d’accueil des post-doctorants en mathematiques

1. Eleonora di Nezza: Special metrics in Kähler geometry

I work in Kähler geometry. It has to do with complex manifolds.

Let us start with the uniformization theorem. It implies that compact Riemann surfaces (complex 1-manifolds) admits metrics of constant curvature, the sign of which is determined by the sign of Euler characteristic. This is tremendously useful in the study of the space of Riemann surfaces.

In higher dimensions, do complex manifolds admit special metrics? Kähler metrics are Hermitian metrics which satisfy an extra first order intergability condition: they osculate a flat metric one order more that a general hermitian metric. This is expressed via the imaginary part of the Hermitian metric, which is a differential 2-form ${\omega}$: ${\omega}$ has to be closed. Alternatively, two connections are associated to a Hermitian metrics, the Riemannian Levi-Civita connection, and the Chern connection. The metric is Kähler iff these two connections coincide.

Nevertheless, a given complex manifold admits many Kähler metrics. To single a preferred one, we require that its Ricci curvature be constant. We call them Kähler-Einstein metrics, since constancy of Ricci curvature is the content of Einstein’s equations for vacuum with a cosmological consant.

Not every complex manifold admits KE metrics. Indeed, Ricci curvature, viewed a differential 2-form, is a representative of the first Chern class of the (complex) tangent bundle. In the KE case, Ricci curvature is proportional to the Kähler form. For a complex manifold, the fact that the first Chern class can be represented by a closed ${(1,1)}$-form which is positive definite (resp. zero, resp. negative definite) is rather restrictive. Let us assume that this condition is satisfied. Then Calabi showed that KE equations are equivalent to a scalar second order nonlinear PDE of complex Monge-Ampère type. The problem splits into 3 different cases. The negative and zero cases were solved in the late 1970’s. The positive case is very recent (Chen-Donaldson-Sun, Tian), after decades of co siderable work by a large community.

The present trend is to extend the theory to singular varieties. Indeed, the classification program (“Minimal Model Program”) requires to handle singular varieties. The unknown in the complex Monge-Ampère equation is then a real function defined on the complement of a fixed divisor (i.e. complex mildly singular hypersurface).

Guedj-Zeriahi 2007: existence and uniqueness (up to an additive constant) of a weak solution. With Lu (then in Orsay), I proved in 2014 that their solution is smooth in the complement of the divisor. This is analysis, dealing with a degenerate equation. The french school (Boucksom, Essidieux, Guedj, Zeriahi) has introduced a pluripotential theory which is very efficient for this problem.

2. Claudio Llosa Isenrich : Kähler groups from branched covers of elliptic curves

A complex manifold is a real manifold with a preferred atlas ro open subsets of ${{\mathbb C}^n}$ where changes of charts are holomorphic diffeomorphisms. A Kähler manifold is a complex mnifol with a Hermitian metric whose imaginary part, a differential 2-form, is closed.

The most important examples are smooth complex submanifolds of complex projective space (the space of lines in ${{\mathbb C}^{n+1}}$). They are defined as zero sets of homogeneous polynomials in ${n+1}$ variables.

The fundamental group of a topological space is the set of homotopy classes of based loops. Loops can be concatenated, which gives rise to a group structure. This is the most ancient, and still the most important, algebraic invariant of topology.

Say a group is Kähler if it is isomorphic to the fundamental group of a compact Kähler manifold. In the 1950’s, Serre raised the question of which groups are Ka\”hler. Notes that every (finitely presented) groups is isomorphic to the fundamental group of a compact complex manifold.

Here are examples. Even rank free abelian groups ${{\mathbb Z}^{2n}}$ are Kähler. Surface groups (fundamental groups of compact surfaces) are Kähler. Serre showed that all finite groups are Kähler. The class of Kähler groups is able under taking direct products and finite index subgroups.

Here are restrictions. The first Betti number of a Kähler is even (this rules out even rank free abelian groups, free groups).

My contribution is a construction of new examples of Kähler groups, insped by Dimca-Papadima-Suciu 2009. Let ${E={\mathbb C}/{\mathbb Z}^2}$ be a complex torus (also called a complex elliptic curve). For ${r\geq 3}$, consider branched covers of ${E}$, ${f_i:S_i\rightarrow E}$, ${1\leq i\leq r}$, where ${S_i}$ is a Riemann surface. For instance, cut ${E}$ open along several slits, take several copies and glue them together along slits in a suitable combinatorial pattern. Since ${E}$ has a commutative group structure, one can take the sum of these maps, and get a map

$\displaystyle \begin{array}{rcl} h=\sum_{i=1}^r f_i:\prod_{i=1}^r S_i \rightarrow E. \end{array}$

This is a holomorphic map. A generic fiber ${f^{-1}(p)}$ is smooth, its fundamental group is Kähler, by construction. It is the kernel of the homomorphism induced by ${h}$ on fundamental groups. I show that if ${h}$ is surjective on fundamental groups, the group enjoys an interesting finiteness property (which I will not define today): it is ${\mathcal{F}_{r-1}}$ but not ${\mathcal{F}_{r}}$. No previous example was known.

Peng: higher dimensions? Unclear.

3. Cyril Marzouk: Scaling limits of large random combinatorial structures: Brownian objects

Motivation from quantum gravity: how to sample a random metric on the 2-sphere? Possibly easier: how to sample a random function?

Approximate answer: look at random walks on ${{\mathbb Z}}$. The endpoints obeys a binomial law, which converges to a Gaussian law. In fact (Donsker), the whole path converges in law to Brownian motion, a probability measure on real functions on ${{\mathbb R}_+}$.

Closer to random metrics: random quadrangulations, also known as planar maps. The graph distance is thought of as a metric on the sphere. The Gromov-Hausdorff distance makes such planar maps into a Polish topological space.

Le Gall and Miermont (independantly): a uniformly random quadrangulation with ${n}$ faces, renormalized by ${n^{1/4}}$, converges to a random metric space in Gromov-Hausdorff distance. It is homeomorphic to the 2-sphere, but it has Hausdorff dimension 4.

Enrich the model by admitting faces with variable (even) number of edges, and specifiy the number ${n_i}$ of faces of degree ${2i}$. Assume ${n_i/n}$ converges to ${p(i)}$, ${\sum_i in_i/n}$ converges to ${\sum_i ip(i)}$, and ${\sum_i i^2 n_i/n}$ converges to ${\sum_i i^2p(i)}$. Then I show that a uniformly random such map converges to the same random metric space (slightly rescaled).

Le Gall’s method relies on Schaeffer’s encoding of quadrangulations with rooted trees carrying integers at vertices. It was known earlier (Aldous) that the uniformly random tree with ${n+1}$ vertices, rescaled by ${n^{1/2}}$, converges to the Brownian tree. I merely explain where the constants come from. A finite tree can be viewed as a walk on ${{\mathbb Z}}$. The depth of the tree is the maximal distance reached by the walk. Since random walks are well understood, depths of random trees are known.

4. Jie Lin: Special values of ${L}$-functions and a conjecture of Deligne

Considering sums of inverse squares, cubes,… leads to Riemann’s zeta function. Euler showed that ${\zeta(2)=\pi^2/6}$. More generally, ${\zeta(2m)}$ is commensurable to ${\pi^{2m}}$. To show this, one extends ${\zeta}$ meromorphically to ${{\mathbb C}}$, and gets a functional equation, which relates ${\zeta(2m)}$ to ${\zeta(1-2m)}$, which is a rational number (one says that ${2m}$ is critical, since ${\zeta}$ is holomorphic both at ${2m}$ and ${1-2m}$).

For odd integers, one must compute residues, which is much harder.

A Dirichlet ${L}$-function is a weighted inverse power sums, where weights are characters mod ${N}$. The above theorem extends to ${L}$-functions. Example: ${N}$

Adèles. Euler’s product formula for ${\zeta}$ suggests formulae involving all prime numbers, i.e. all absolute values on ${{\mathbb Q}}$. The ring of adeles is the (restricted) product of all ${p}$-adic fields (completion of ${{\mathbb Q}}$ for the ${p}$-adic absolute value), and the reals. The idèles are the units of adele ring. The idea of a ${L}$-function extends to Hecker characters, i.e. continuous characters of ${{\mathbb Q}^\times \setminus\mathbb{A}^\times}$.

A Hecke character is a one-dimensional representation of ${Gl_1(\mathbb{A})}$, we must consider more generaly automorphic representations of ${Gl_N(\mathbb{A})}$. Motives are an even wider generalization introduced by Grothendieck. These are geometric objects, like elliptic curves. Analytic continuation of ${L}$-functions in this generality is a conjecture by Hasse-Weil. In 1979, Deligne conjectured the values of ${L}$-functions for critical integers.

Langlands’ program relates these 3 types of objects, motives, automorphic representations and Galois representations. This contains the Taniyama-Shimura-Weil conjecture whose solution is used in the solution of Fermat’s last problem.

With coauthors, I have translated Deligne conjecture in automorphic representation terms.

5. Dena Kazerani : Symmetry, from hyperbolic systems to Green-Naghdi models

I work in fluid mechanics of incompressible flows. When viscous forces are low, Navier-Stokes equations simplify to Euler equations. A difficulty arises since the domain occupied by the fluid evolves.

I focus today on shallow water, this is the Green-Naghdi model (1976): one assumes that the fluid is irrotational, that the ground is planar and horizontal, vertical velocity depends linearly on the vertical variable, horizontal velocity does not depend on vertical velocity. Then the unknown of the equation, defined on a fixed planar domain, are the horizontal velocity and the water height. One gets a hyperbolic Saint-Venant system with extra dispersive terms.

Hyperbolic systems often have symmetry: Lax-entropy pairs, Godunov structures. It is the case for the Saint-Venant system. The symmetry is expressed by changes of variables involving matrices

$\displaystyle \begin{array}{rcl} \partial_t U+\partial_x F(U)=0. \end{array}$

We need to generalize matrices to operators on Banach spaces. We give a general definition of Godunov structure, extend the classical results (equivalence with existence of Lax-entropy pairs). We show that Green-Naghdi’s model has this property.

We use this symmetry to establish well-posedness of the equations: global existence, asymptotic stability.

Di Nezza: the definitions make sense only for smooth solutions, is this a problem? Yes, singular solutions behave differently (shocks).