I will talk about a tropical analogue of the Monge-Ampère equation and its connection to the SYZ conjecture. First, I introduce a differential calculus on tropical varieties that allows to properly formulate the tropical Monge-Ampère equation. This uses tropical Hodge theory developed in our previous work. Then, I discuss our current understanding of this equation. I explain how a recent work by Yang Li and some new results about tropicalizations of algebraic varieties allow to deduce the metric SYZ conjecture from the existence of a solution to the tropical Monge-Ampère equation.  Based on joint work with Matthieu Piquerez.

In the setting of Arakelov geometry, one can define a notion of height of a polarized arithmetic variety endowed with a hermitian metric. For a metrized arithmetic Fano variety, the anticanonical polarization thus gives us a natural notion of height. The height in this case is an arithmetic analogue of the degree of a Fano variety over C. Among K-semistable Fano manifolds, K. Fujita proved that the degree is maximal for projective space. Inspired by his result, we conjecture that, (after an appropriate normalization) the height of a metrized arithmetic Fano variety whose complexification is K-semistable is maximal for the projective space over the integers, endowed with the Fubini-Study metric. I will present some results in favour of the conjecture in the toric case, and for diagonal hypersurfaces, highlighting connections to Kähler-Einstein metrics and K-stability along the way. This is based on joint work with Robert Berman. 

In a work in collaboration with Marion Jeannin, we establish a Harder-Narasimhan theory for slope functions defined on a bounded lattice and taking values in a totally ordered set. In this talk, I will describe this theory and explain how it leads to a unified interpretation for several existence and uniqueness results in very different contexts.

Low-energy expansion of scattering amplitudes of elementary particles arising in string theory are often linked to an inhomogeneous Laplace equation. For example, for the IIB string theory in 10+1 dimensions and scattering of 4-gravitons, the equation is the inhomogeneous Laplace equation on the modular surface, where the inhomogeneous part contains products of Eisenstein series. In this talk, we discuss how to solve this partial differential equation. Additionally, from asymptotic behavior of its solutions we obtain exact identities for infinite convolution sums of even divisor functions weighted by Laurent polynomials with logarithms. 

BCOV invariants of Calabi-Yau manifolds are certain combination of analytic torsions and conjecturally correspond to genus-1 Gromov-invariants under mirror symmetry. I will report on a joint work with Yeping Zhang, proving the birational invariance of BCOV invariants, predicted by mirror symmetry. Our methods, inspired by motivic integration, also allow us to extend the definition of BCOV invariants to mildly singular Calabi-Yau varieties. 

Singularities appear naturally in many situations, e.g. through quotients or through contractions, and often package interesting information. The music playing in the background of this talk is joint upcoming work with D. Eriksson and J. Nicaise, where we investigate how one can use certain singular models of a curve, in order to compute the base change conductor of its Jacobian. I will give a brief account of this approach, and mention some natural questions concerning singularities that arise from this.

Let k be a complete, algebraically closed non-trivial non-archimedean field. We discuss faithful tropicalizations by linear systems of the skeletons of abelian varieties defined over k and faithful embeddings by linear systems of tropical abelian varieties. Our results can be seen as non-archimedean and tropical analogues of the classical theorem of Lefschetz on abelian varieties. We also discuss lifting of tropical theta functions on tropical abelian varieties to theta functions on abelian varieties, which leads us to a converse result of tropicalizations of theta functions by Foster-Rabinoff-Shokrieh-Soto. This is a joint work with Kazuhiko Yamaki.

Given a family of projective hypersurfaces converging to a tropical hypersurface, I would like to understand what this convergence tells us about the variation of the moduli of the family. During the talk, I intend to overview how the latter convergence, expressed in terms of Hausdorff limit of amoebas, can be understood in terms of convergence of certain period matrices. The long term goal is to generalize the tropical compactification of the moduli space of stable curves to higher dimensions.

To any variation of Hodge structures on a complex projective curve C is attached its Griffiths canonical line bundle over C, and its Griffiths height defined as the degree of this line bundle. It is the geometric counterpart of the height of motives over number fields, recently defined by Kato. In my talk I will present the computation of Griffiths heights of (the middle-dimensional cohomology of pencils of complex projective hypersurfaces, using Steenbrink’s theory and the Grothendieck-Riemann-Roch theorem. I will also relate these Griffiths heights to more naive heights defined by means of geometric invariant theory. 

Jet differentials on complex varieties are geometric objects that can be used to formalize algebraic differential equations. For example, they are useful in the theory of hyperbolicity in the sense of Kobayashi, when we seek to show the algebraic degeneracy of entire curves on projective varieties of general type. In the talk, based on work in progress with Pierre-Emmanuel Chaput and Lionel Darondeau, I will present a construction of jet differentials and some applications. 

Genus one mirror symmetry describes a duality between a spectral quantity -- BCOV invariant, and enumerative-geometric quantities -- genus one Gromov-Witten invariants. It has been previously proven for hypersurfaces, and we have recently extended it to a new case. Despite this, I will provide an overview of the phenomenon rather than focusing solely on this specific new case.

Many important recent results in birational and complex geometry take on the following form: given a complex (projective) variety, can one find a way to degenerate it to a variety with exceptional geometric properties? In this mostly expository talk, I will give an illustrative explanation as to how to proofs of such results broadly rely on the same structural idea: that they can, through non-Archimedean analytic tools, be formulated as a minimisation problem that is then solved through a variational approach - so that one can find purely geometric and algebraic objects via methods inherited from PDE theory.

Have you heard the question, "Can one hear the shape of a drum?" Do you know the answer? In 1966, M. Kac's article of the same title popularized the inverse isospectral problem for planar domains. Twenty six years later, Gordon, Webb, and Wolpert demonstrated the answer, but many naturally related problems remain open today. We will discuss results and open problems inspired by "hearing the shape of a drum'" and involving the fields of geometry, mathematical physics, (microlocal) analysis, and number theory. 

This talk is about "p-adic riemann surfaces", or more precisely p-adic analytic curves: that is, curves which locally look like the solution set of a system of converging p-adic power series in a p-adic polydisc. Often, results from complex analytic geometry have analogies in such a p-adic setting, but due to the issues of wild ramification many new subtleties arise. I will illustrate this principle by explaining a p-adic analogon of the Riemann-Hurwitz formula and some applications to ramification of p-adic analytic curves. 

I will discuss a generalization to the case of parabolic Higgs bundles of the conformal limit correspondence between the Bialinicki-Birula stratification of the Dolbeault moduli space and Simpson's partial Oper stratification of the deRham moduli space.

I would like to report on a recent progress in the asymptotic behavior of small eigenvalues of the Laplacian for degenerations of compact Riemann surfaces, i.e., the eigenvalues converging to zero, where the Laplacian is considered with respect to the metric induced from the ambient space.