# Seminars and short courses

Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.

CAMGSD has recorded the calendar of its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.

For a full search interface see the Mathematics Department seminar page. Here you will be restricted to lists of forthcoming CAMGSD seminars for the next two weeks or to a given year.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

This will be the third session of a course on Microlocal Analysis.

### , Monday

#### , String Theory

Katrin Wendland, University of Freiburg.

Reflections on $K3$ theories.

This talk will give a lightning review on $K3$ theories, including some newer developments. In particular, we discuss a procedure recently devised in joint work with Anne Taormina in the context of Mathieu Moonshine. This procedure, which we call reflection, allows to transform certain superconformal field theories into super vertex operator algebras and their admissible modules, thus building a bridge between the two worlds.

### , Friday

#### , Partial Differential Equations

Thomas Johnson, Imperial College London.

The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised harmonic gauge.

The Schwarzschild solution in general relativity, discovered in 1916, describes a spacetime that contains a non-rotating black hole. A recent (2016) breakthrough paper of Dafermos, Holzegel and Rodnianski showed that the Schwarzschild exterior is linearly stable (in an appropriate sense) as a solution to the vacuum Einstein equations. Their method of proof involved an analysis of the (linearised) Bianchi equations for the Weyl curvature. In this talk, we shall present our proof that the Schwarzschild solution is in fact linearly stable in a generalised harmonic gauge. In particular, we focus on the (linearised) Einstein equations for the metric directly.

The result relies upon insights gained for the scalar wave equation by Dafermos and Rodnianski and a fortiori includes a decay statement for solutions to the Zerilli equation. Moreover, the issue of gauge plays a very important role in the problem and shall be discussed.

### , Tuesday

#### , Geometria em Lisboa

André Gama Oliveira, Centro de Matemática da Universidade do Porto.

Parabolic Higgs Bundles and Topological Mirror Symmetry.

In 2003, T. Hausel and M. Thaddeus proved that the Hitchin systems on the moduli spaces of $\operatorname{SL}(n,\mathbb{C})$- and $\operatorname{PGL}(n,\mathbb{C})$-Higgs bundles on a curve, verify the requirements to be considered SYZ-mirror partners, in the mirror symmetry setting proposed by Strominger-Yau-Zaslow (SYZ). These were the first non-trivial known examples of SYZ-mirror partners of dimension greater than $2$.

According to the expectations coming from physicists, the generalized Hodge numbers of these moduli spaces should thus agree — this is the so-called topological mirror symmetry. Hausel and Thaddeus proved that this is the case for $n=2,3$ and gave strong indications that the same holds for any $n$ prime (and degree coprime to $n$). In joint work in progress with P. Gothen, we perform a similar study but for parabolic Higgs bundles. We will roughly explain this setting, our study and some questions which naturally arise from it.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

This will be the second session of a course on Microlocal Analysis.

### , Wednesday

#### , Topological Quantum Field Theory

José Ricardo Oliveira, Univ. Nottingham.

EPRL/FK Asymptotics and the Flatness Problem: a concrete example.

Spin foam models are a "state-sum" approach to loop quantum gravity which aims to facilitate the description of its dynamics, an open problem of the parent framework. Since these models' relation to classical Einstein gravity is not explicit, it becomes necessary to study their asymptotics — the classical theory should be obtained in a limit where quantum effects are negligible, taken to be the limit of large triangle areas in a triangulated manifold with boundary.

In this talk we will briefly introduce the EPRL/FK spin foam model and known results about its asymptotics, proceeding then to describe a practical computation of spin foam and asymptotic geometric data for a simple triangulation, with only one interior triangle. The results are used to comment on the "flatness problem" — a hypothesis raised by Bonzom (2009) suggesting that EPRL/FK's classical limit only describes flat geometries in vacuum.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

This will be the first session of a course on Microlocal Analysis.

### , Tuesday

#### , Analysis, Geometry, and Dynamical Systems

Daniel Gonçalves, Universidade Federal de Santa Catarina.

Representations of graph algebras via branching systems and the Perron-Frobenius Operator.

In this talk we show how to obtain representations of graph algebras from branching systems and show how these representations connect to the Perron-Frobenius operator from Ergodic Theory. We will describe how every permutative representation of a graph algebra is unitary equivalent to a representation arising from a branching system. Time permitting we will give an application of the branching systems representations to the converse of the Cuntz-Krieger Uniqueness Theorem for graph algebras.

### , Friday

#### , Partial Differential Equations

Sari Ghanem, Albert Einstein Institute, Max-Planck Institute for Gravitational Physics.

The decay of spherically symmetric $SU(2)$ Yang-Mills fields on a black hole.

First, I will present the Yang-Mills fields on an arbitrary fixed curved space-time, valued in any Lie algebra, and then expose briefly the proof of the non-blow-up of the Yang-Mills curvature. Thereafter, I will present recent results obtained with Dietrich Häfner concerning the Yang-Mills fields on the Schwarzschild black hole. Unlike the free scalar wave equation, the Yang-Mills equations on a black hole space-time admit stationary solutions, which we eliminate by considering spherically symmetric initial data with small energy and satisfying a certain Ansatz. We then prove uniform decay estimates in the entire exterior region of the black hole, including the event horizon, for gauge invariant norms on the Yang-Mills curvature generated from such initial data, including the pointwise norm of the so-called middle components. This is done by proving in this setting, a Morawetz type estimate that is stronger than the one assumed in previous work, without decoupling the middle-components, using the Yang-Mills equations directly.

### , Friday

#### , Partial Differential Equations

Mahendra Pantee, Universidade Estadual de Campinas.

On well-posedness of some bi-dimensional dispersive models.

We consider an initial value problem (IVP) associated to a third order dispersive model posed in $T^2$. Using the techniques introduced by Ionescu and Kenig, we prove the local well-posedness result for given data in $H^s(T^2)$ whenever $s\gt 3/2$.

### , Friday

#### , Algebra

Ieke Moerdijk, University of Utrecht.

Shuffles and Trees.

The notion of "shuffle" of linear orders plays a central role in elementary topology. Motivated by tensor products of operads and of dendroidal sets, I will present a generalization to shuffles of trees. This combinatorial operation of shuffling trees can be understood by itself, and enjoys some intriguing properties. It raises several questions of a completely elementary nature which seem hard to answer.

### , Friday

#### , Topological Quantum Field Theory

Urs Schreiber, Czech Academy of Sciences, Prague.

Duality in String/M-theory from Cyclic cohomology of Super Lie $n$-algebras.

I discuss how, at the level of rational homotopy theory, all the pertinent dualities in string theory (M/IIA/IIB/F) are mathematically witnessed and systematically derivable from the cyclic cohomology of super Lie $n$-algebras. I close by commenting on how this may help with solving the open problem of identifying the correct generalized cohomology theory for M-flux fields, lifting the classification of the RR-fields in twisted K-theory.

This is based on joint work with D. Fiorenza and H. Sati arxiv:1611.06536

### , Tuesday

#### , Geometria em Lisboa

Rui Albuquerque, Universidade de Évora.

Riemannian $3$-manifolds and Conti-Salamon $\operatorname{SU}(2)$-structures.

We present an $\operatorname{SO}(2)$-structure and the associated global exterior differential system existing on the contact Riemannian manifold $\cal S$, which is the total space of the tangent sphere bundle, with the canonical metric, of any given $3$-dimensional oriented Riemannian manifold $M$. This is part of a wider theory which can be studied in any dimension. In this seminar we focus on the first interesting dimension and show several new $\operatorname{SU}(2)$-structures on $\cal S$, following the recent ideas introduced by D. Conti and S. Salamon for the study of $5$-manifolds with special metrics.