# Abstracts - Invited Addresses

**Wednesday Evening Public Lecture:** June 10th at 8:30 pm in the Guilhermina Suggia auditorium at the Casa da Música

Marcus du Sautoy, University of Oxford, UK

**Title:** The Secret Mathematicians.

**Abstract:** From composers to painters, writers to choreographers, the mathematician’s palette of shapes, patterns and numbers has proved a powerful inspiration. Artists can be subconsciously drawn to the same structures that fascinate mathematicians as they hunt for interesting new structures to frame their creative process.

Professor du Sautoy will explore the hidden mathematical ideas that underpin the creative output of well-known artists and reveal that the work of the mathematician is also driven by strong aesthetic values.

Rui Loja Fernandes, University of Illinois, USA

**Title:** Global Aspects of Poisson Geometry.

**Abstract:** A Poisson bracket on a manifold is the geometric structure relevant for Hamiltonian dynamics. A Poisson bracket induces a foliation of the manifold by symplectic leaves, which in general is singular: the dimension of the leaves can jump. The study of the global properties of a Poisson bracket combines the symplectic geometry of each leaf, the topology of the leaves and the singularities of the foliation at places where the dimension jumps. In this talk, I will survey some fascinating recent advances in the global geometry of manifolds equipped with a Poisson bracket.

Irene Fonseca, Carnegie Mellon University, USA

**Title:** Quantum Dots and Dislocations: Dynamics of Materials Defects.

**Abstract:** The formation and assembly patterns of quantum dots have a significant impact on the optoelectronic properties of semiconductors. We will discuss shapes of quantum dots and short time existence for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained three-dimensional films. Further, short time existence, uniqueness, and qualitative properties of solutions to an evolution law for systems of screw dislocations under the assumption of antiplane shear will be obtained.

Annette Huber, Albert-Ludwigs-Universität, Germany

**Title:** Differential forms in algebraic geometry – a new perspective in the singular case.

**Abstract:** Differential forms originally show up when integrating or differentiating on manifolds. However, the concept also makes perfect sense on algebraic varieties because the derivative of a polynomial is a polynomial.

The object has very many important uses, e.g., as a source of invariants needed in order to classify varieties. This approach was very successful for smooth varieties, but the singular case is less well-understood.

We explain how the use of the h-topology (introduced by Suslin and Voevodsky in order to study motives) gives a very good object also in the singular case, at least in characteristic zero. The approach unifies other ad-hoc notions and simplies many proofs. We also explain the necessary modifications in positive characteristic and the new problems that show up.

Mikhail Khovanov, Columbia University, USA

**Title:** Categorification at a prime root of unity.

**Abstract:** Rings important in quantum theory, such as quantum groups and Hecke algebras, exhibit different behaviours when the parameter q is generic and when a root of unity. Upon categorification, generic parameter q becomes a grading shift or, more generally, an automorphism of a triangulated category. How categorification works at a root of unity is a much more subtle and mostly unsolved problem, In this talk we’ll discuss categorifications when the degree of a root of unity is a prime number and categorification happens over a field of characteristic p, including a categorification of the small quantum sl(2).

André Neves, Imperial College London, UK

**Title:** Min-max methods in Geometry.

**Abstract:** Min-max methods were introduced in Geometry around 100 years ago by Birkhoff. Recently, with Fernando Marques, we used it to solve the Willmore Conjecture and other long standing open questions in Geometry. I will survey the method, its applications, and propose new directions in the area.

**EMS Distinguished Speaker**: Sylvia Serfaty, Université Pierre et Marie Curie Paris 6, France

**Title:** Questions of crystallization in systems with Coulomb and Riesz interactions.

**Abstract:** We are interested in systems of points with Coulomb, logarithmic or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. After rescaling we deal with a microscopic quantity, the associated empirical point process, for which we give a large deviation principle whose rate function is the sum of a relative entropy and of a renormalized energy that governs microscopic patterns of points. The former favors disorder, while the latter is expected to favor cristalline configurations. This is based on joint works with Etienne Sandier, Simona Rota Nodari, Nicolas Rougerie, Mircea Petrache, and Thomas Leblé.

Gigliola Staffilani, MIT, USA

**Title: **Recent developments on certain dispersive equations as infinite dimensional Hamiltonian systems.

**Abstract:** In this talk I will present some recent developments in the study of dispersive differential equations on compact manifolds that can also be viewed as infinite dimensional Hamiltonian systems. I will talk about Strichartz estimates, weak turbulence, Gibbs measures, symplectic structures and non-squeezing theorems. A list of open problems will conclude the talk.

Marcelo Viana, Instituto de Matemática Pura e Aplicada, Brazil

**Title:** Repellers of random walks.

**Abstract:** In his 2009 thesis at IMPA, Carlos Bocker proved that the Lyapunov exponents of random (iid) products of 2-by-2 matrices always depend continuously on the matrices' coefficients and their probability weights. The proof is based on a detailed analysis of the dynamics of the associated random walk in projective space.

Most recently, Avila, Eskin and the speaker announced that they are able to carry this analysis to arbitrary dimension, using a very different (cost functions) approach. Thus, continuity of Lyapunov exponents on the underlying data holds in full generality for iid random products of matrices.

This new approach has been extended in the thesis of Elaís Malheiro to prove that the 2-dimensional statement generalizes to Markov products of matrices. Moreover, it is in the basis of the work of Lucas Backes, another 2014 thesis at IMPA, which contains substantial progress towards proving that continuity of Lyapunov exponents holds for very general 2-dimensional Holder cocycles over hyperbolic systems.

These results are in stark contrast with observations of Ricardo Mañé in the 1980's, completed by Jairo Bochi and the speaker two decades later, according to which one can often annihilate the Lyapunov exponents of continuous linear cocycles, thus making continuity a very particular situation in that context.