# 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.

### 27/02/2019, Wednesday

#### 15:00, Room P3.10, Mathematics Building, Mathematical Relativity

Artur Alho, Instituto Superior Técnico.

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.

### 26/02/2019, Tuesday

#### 15:00, Room P3.10, Mathematics Building, Geometria em Lisboa

Bruno Oliveira, University of Miami.

The presence of symmetric and more generally $k$-jet differentials on surfaces $X$ of general type play an important role in constraining the presence of entire curves (nonconstant holomorphic maps from $\mathbb{C}$ to $X$). Green-Griffiths-Lang conjecture and Kobayashi conjecture are the pillars of the theory of constraints on the existence of entire curves on varieties of general type.

When the surface as a low ratio $c_1^2/c_2$ a simple application of Riemann-Roch is unable to guarantee abundance of symmetric or $k$-jet differentials.

This talk gives an approach to show abundance on resolutions of hypersurfaces in $P^3$ with $A_n$ singularities and of low degree (low $c_1^2/c_2$). A new ingredient from a recent work with my student Michael Weiss gives that there are such hypersurfaces of degree $8$ (and potentially $7$). The best known result till date was degree $13$.

### 22/02/2019, Friday

#### 11:30, Room P3.10, Mathematics Building, Mathematical Relativity

Pedro Girão, Instituto Superior Técnico.

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_g\phi=0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A \in H^3(\Sigma)$ (where $\Sigma=\mathbb{R}^n$, $\mathbb{S}^n$ or $\mathbb{H}^n$ models the spatial hypersurfaces) there exists a unique solution $\phi$ of the wave equation converging to $A$ in $H^1(\Sigma)$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(\Sigma)$.

### 13/02/2019, Wednesday

#### 11:00, Seminar room (2.8.3), Physics Building, Mathematical Relativity

Juan Antonio Valiente Kroon, Queen Mary, University of London.
Construction of anti de Sitter-like spacetimes using the metric conformal field equations.

In this talk I with describe how to make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. I will consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3-dimensional Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions. I will also discuss extensions of this analysis to the case of the Einstein equations coupled with various tracefree matter models. This is work in collaboration with Diego Carranza.

### 12/02/2019, Tuesday

#### 15:00, Room P3.10, Mathematics Building, Analysis, Geometry, and Dynamical Systems

Federico Sau, Delft University.
Self-duality for conservative interacting particle systems.

In this talk, we will sketch some recent developments about the notion of duality for conservative interacting particle systems. In particular, we will show the simplification that arises in presence of self-duality when considering hydrodynamic limits in a dynamic disorder (joint work with F. Redig and E. Saada). We will find all particle systems which admit a special form of self-duality (joint work with F. Redig) and, in conclusion, we will use the spectral point of view of this notion to address some open questions.

### 07/02/2019, Thursday

#### 14:30, Room P3.10, Mathematics Building, Mathematical Relativity

Noa Zilberman, Technion.
Quantum effects near the inner horizon of a black hole.

The analytically extended Kerr and Reissner-Nordström metrics, describing respectively spinning or spherical charged black holes (BHs), reveal a traversable passage through an inner horizon (IH) to another external universe. Consider a traveler intending to access this other universe. What will she encounter along the way? Is her mission doomed to fail? Does this other external universe actually exist?

Answering these questions requires one to understand the manner in which quantum fields influence the internal geometry of BHs. In particular, this would include the computation of the renormalized stress-energy tensor (RSET) on BH backgrounds - primarily near the IH. Although a theoretical framework for such a computation does exist, this has been a serious challenge for decades (partially due to its inevitable numerical implementation). However, the recently developed pragmatic mode-sum regularization method has made the RSET computation more accessible.

In this talk, we will first consider the computation of the simpler quantity $\langle\phi^2\rangle_{ren}$, for a minimally-coupled massless scalar field inside a (4d) Reissner-Nordström BH. We shall then proceed with the long sought-after RSET, focusing on the computation of the semi-classical fluxes near the IH. Our novel results for the latter will be presented, with a closer look at the extremal limit. Finally - we will discuss possible implications to the fate of our traveler.

### 30/01/2019, Wednesday

#### 11:00, Room P3.10, Mathematics Building, Mathematical Relativity

Anne Franzen, Instituto Superior Técnico.
Flat FLRW and Kasner Big Bang singularities analyzed on the level of scalar waves.

We consider the wave equation, $\square_g\psi=0$, in fixed flat Friedmann-Lemaitre-Robertson-Walker and Kasner spacetimes with topology $\mathbb{R}_+\times\mathbb{T}^3$. We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface $\{t=0\}$. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate $L^2$-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the $L^2(\mathbb{T}^3)$ norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

### 16/01/2019, Wednesday

#### 15:00, Room P3.10, Mathematics Building, Algebra

I will explain how three integration techniques for producing cohomology classes — Chen integrals for loop spaces, Bott-Taubes integrals for knots and links, and Kontsevich integrals for configuration spaces — come together in the computation of the cohomology of spaces of braids. The relationship between various integrals is encoded by certain graph complexes. I will also talk about the generalizations to other spaces of maps into configuration spaces (of which braids are an example). This will lead to connections to spaces of link maps and, from there, to other topics such as rope length, manifold calculus of functors, and a conjecture of Koschorke, all of which I will touch upon briefly. This is joint work with Rafal Komendarczyk and Robin Koytcheff.

### 10/01/2019, Thursday

#### 15:00, Room P4.35, Mathematics Building, Topological Quantum Field Theory

I will first recall Lusztig's asymptotic Hecke algebra and its categorification, a fusion category obtained from the perverse homology of Soergel bimodules. For example, for finite dihedral Coxeter type this fusion category is a 2-colored version of the semisimplified quotient of the module category of quantum $\operatorname{sl}(2)$ at a root of unity, which Reshetikhin-Turaev and Turaev-Viro used for the construction of 3-dimensional Topological Quantum Field Theories.

In the second part of my talk, I will recall the basics of 2-representation theory and indicate how the fusion categories above can conjecturally be used to study the 2-representation theory of Soergel bimodules of finite Coxeter type.

This is joint work with Mazorchuk, Miemietz, Tubbenhauer and Zhang.

### 07/01/2019, Monday

#### 14:00, Room P3.10, Mathematics Building, Geometria em Lisboa

Hugues Auvray, Université Paris-Sud.
Complete extremal metrics and stability of pairs on Hirzebruch surfaces.

In this talk I will discuss the existence of complete extremal metrics on the complement of simple normal crossings divisors in compact Kähler manifolds, and stability of pairs, in the toric case.

Using constructions of Legendre and Apostolov-Calderbank-Gauduchon, we completely characterize when this holds for Hirzebruch surfaces. In particular, our results show that relative stability of a pair and the existence of extremal Poincaré type/cusp metrics do not coincide. However, stability is equivalent to the existence of a complete extremal metric on the complement of the divisor in our examples. It is the Poincaré type condition on the asymptotics of the extremal metric that fails in general.

This is joint work with Vestislav Apostolov and Lars Sektnan.