# 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.

### 19/12/2018, Wednesday

#### 16:00, Room P4.35, Mathematics Building, Geometria em Lisboa

This is a report on joint work with Alex Waldron.

The Yang-Mills functional is the most studied functional on the space of connections on a vector bundle over an oriented Riemannian manifold. Its negative gradient flow leads to a semi-parabolic PDE known as the Yang-Mills flow.

I will introduce this flow and talk about its properties in the context of manifolds with special holonomy, particularly in Kahler, $G_2$, and Spin(7)-manifolds. I intend to explain a blow-up criteria and talk about relationships with certain minimal "submanifolds" known as calibrated.

### 18/12/2018, Tuesday

#### 11:00, Room P3.10, Mathematics Building, Analysis, Geometry, and Dynamical Systems

Benito Frazão Pires, Universidade de São Paulo.
Symbolic dynamics of piecewise contractions.

A map $f:[0,1]\to [0,1]$ is a piecewise contraction if locally $f$ contracts distance, i.e., if there exist $0<\lambda<1$ and a partition of $[0,1]$ into intervals $I_1,I_2,\ldots,I_n$ such that $\left\vert f(x)-f(y)\right\vert \le\lambda \vert x-y\vert$ for all $x,y\in I_i$ $(1\le i\le n)$. Piecewise contractions describe the dynamics of many systems such as traffic control systems, queueing systems, outer billiards and Cherry flows. Here I am interested in the symbolic dynamics of such maps. More precisely, we say that an infinite word $i_0 i_1 i_2\ldots$ over the alphabet $\mathcal{A}=\{1,2,\ldots,n\}$ is the natural coding of $x\in [0,1]$ if $f^k(x)\in I_{i_k}$ for all $k\ge 0$. The aim of this talk is to provide a complete classification of the words that appear as natural codings of injective piecewise contractions.

### 17/12/2018, Monday

#### 15:00, Room P4.35, Mathematics Building, Analysis, Geometry, and Dynamical Systems

Daniel Gonçalves, Universidade Federal de Santa Catarina, Brasil.
Inﬁnite alphabet ultragraph edge shift spaces: relations to $C^\ast$-algebras and chaos.

We explain the notion of ultragraphs, which generalize directed graphs, and use this combinatorial object to deﬁne a notion of (one-sided) edge shift spaces (which, in the ﬁnite case, coincides with the edge shift space of a graph). We then go on to show that these shift spaces have some nice properties, as for example metrizability and basis of compact open sets. We examine shift morphisms between these shift spaces: we give an idea how to show that if two (possibly inﬁnite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph $C^\ast$-algebras are isomorphic. Finally we describe Li-Yorke chaoticity associated to these shifts and remark that the results obtained mimic the results for shifts of finite type over finite alphabets (what is not the case for infinite alphabet shift spaces with the product topology).

### 10/12/2018, Monday

#### 14:00, Room P3.10, Mathematics Building, Mathematical Relativity

Moritz Reintjes, Instituto Superior Técnico.
IIntroduction to the Theory of Shock Waves.

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### Bibliography

• Joel Smoller, Shock waves and Reaction Diffusion Equations
• Constantine Dafermos, Hyperbolic Conservation Laws in Continuum Physics
• Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics
• Lecture notes of Blake Temple (on his webpage)

### 07/12/2018, Friday

#### 14:00, Room P4.35, Mathematics Building, Mathematical Relativity

Moritz Reintjes, Instituto Superior Técnico.
Introduction to the Theory of Shock Waves.

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### 05/12/2018, Wednesday

#### 14:00, Room P3.10, Mathematics Building, Topological Quantum Field Theory

Miguel Tierz, Grupo de Fisica Matematica, Universidade de Lisboa.
Random matrix theory in topological gauge theories.

We present an overview of analytical tools in random matrix theory and related areas, involving Toeplitz/ Hankel determinants and symmetric functions, with an emphasis on their relevance in the study of topological gauge theory and focussing on some specific Chern-Simons theories and 2d Yang-Mills theories. We will also explain how these methods and results are intertwined with localization results in supersymmetric gauge theories.

### 30/11/2018, Friday

#### 14:00, Room P3.10, Mathematics Building, Mathematical Relativity

Moritz Reintjes, Instituto Superior Técnico.
Introduction to the Theory of Shock Waves.

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### 29/11/2018, Thursday

#### 16:15, Room P4.35, Mathematics Building, Geometria em Lisboa

Dan Avritzer, Universidade Federal de Minas Gerais.
Classical Geometry and the Moduli Space of Higgs bundles.

One of the most beautiful objects of classical geometry is the Kummer Surface, that was studied by Kummer in the 19th century. In a celebrated paper of 1969 Narasimhan and Ramanan studied the moduli space of vector bundles of rank 2 and trivial determinant over a curve of genus 2, proving that this space is isomorphic to projective space of dimension 3. In this space the moduli space of non-stable bundles is parameterized by a Kummer Surface.

In this seminar, I will introduce the Kummer Surface in the classical setting and recall the main results of the paper of Narasimhan and Ramanan mentioned above. Then I will talk about joint work in progress with Peter Gothen, where we describe the moduli space of Higgs bundles over a curve of genus 2. We obtain a similar description as in the paper above of the moduli of Higgs bundles in the so called nilpotent cone. The aim is to study the geometry of this nilpotent cone as done in the Narasimhan-Ramanan paper.

### 26/11/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

Davide Polini, Instituto Superior Técnico.
Counting formulae for extremal black holes in an STU-model.

We present microstate counting formulae for BPS black holes in an $N=2$ STU-model.

### 21/11/2018, Wednesday

#### 14:00, Room P3.10, Mathematics Building, Topological Quantum Field Theory

Björn Gohla, CAMGSD, Universidade de Lisboa.
Internal Strictification.

It is well known that ordinary bicategories can always be replaced by bi-equivalent strict 2-categories. Special cases of this are the strictification of monoidal categories and categorical groups. We give an abstract strictification construction for pseudo-monoids in a monoidal 2-category. It is easy to see that bicategories internal to an appropriate category are such pseudo-monoids, and can hence be strictified. (Joint work with Nelson Martins-Ferreira)

### 16/11/2018, Friday

#### 14:00, Room P3.10, Mathematics Building, Mathematical Relativity

Moritz Reintjes, Instituto Superior Técnico.
Introduction to the Theory of Shock Waves.

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### 12/11/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

Alexandre Belin, University of Amsterdam.

I will discuss the application of Siegel modular forms for extracting the degeneracy of states of symmetric orbifold CFTs. These modular forms are closely related to the generating function for the elliptic genera of such CFTs and I will present an efficient technic for extracting their Fourier coefficients. I will then discuss to what extent symmetric orbifold CFTs can admit nice gravity duals and thus make an interesting connection between number theory and quantum gravity.

### 12/11/2018, Monday

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

Jeremie Szeftel, Laboratoire Jacques-Louis Lions de l'Université Pierre et Marie Curie.
The nonlinear stability of Schwarzschild.

I'll discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data subject to a certain symmetry class.

### 09/11/2018, Friday

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

Moritz Reintjes, Instituto Superior Técnico.
Introduction to the Theory of Shock Waves.

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### 05/11/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

Zhihao Duan, École Normale Supérieure Paris.
Instantons in the Hofstadter butterfly: resurgence and quantum mirror curves.

Recently an interesting connection between topological string theory and lattice models in condensed matter physics was discussed by several authors. In this talk, we will focus on the Harper-Hofstadter Hamiltonian. For special values of the magnetic flux, its energy spectrum can be exactly solved and its graph has a beautiful shape known as Hofstadter's butterfly. We are interested in the non-perturbative information inside the spectrum. First we consider the weak magnetic field limit and write down a trans-series ansatz for the energies. We then discuss fluctuations around instanton sectors as well as resurgence relations. For the second half of the talk, our goal is to present another powerful way to compute those fluctuations using the topological string formalism, after reviewing all the necessary background. The talk will be based on arXiv: 1806.11092.

### 31/10/2018, Wednesday

#### 11:30, Room P3.10, Mathematics Building, Topological Quantum Field Theory

Leonardo Santilli, Grupo de Fisica Matematica, Univ. Lisboa.
A Chern-Simons view on noncommutative scalar field theory.

We present a recent result establishing a bridge between noncommutative scalar field theory in $2$ dimensions and topological field theory in $3$ dimensions.

The content of the seminar is split in two main parts, according to the twofold aspect of the result. In the first half, we show that a version of Abelian gauge theory on $\mathbb{R}^3 _{\lambda}$, when restricted to a single fuzzy sphere, reduces in the large $N$ limit to the Langmann-Szabo-Zarembo (LSZ) matrix model, which originally emerges in the study of scalar field theory on the Moyal plane. Then, throughout the second part, we prove that the LSZ matrix model is actually equivalent to the matrix model of $U(N)$ Chern-Simons theory on the three-sphere. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices of the LSZ model, the Chern-Simons matrix model either describes the Chern-Simons partition function, the unknot invariant, given by quantum dimensions, or the Hopf link invariant. Equivalently, the partition function of the LSZ model can be written in terms of the $S$ and $T$ modular matrices of the WZW model.

Based on: arXiv:1805.10543 [hep-th].

### 24/10/2018, Wednesday

#### 14:00, Room P3.10, Mathematics Building, Partial Differential Equations

Rolf Stenberg, Aalto University, Finland.
Stabilised Finite Element Methods for Variational Inequalities.

We survey our recent and ongoing work [1,2] on finite element methods for contact problems. Our approach is to first write the problem in mixed form, in which the contact pressure act as a Lagrange multiplier. In order to avoid the problems related to a direct mixed finite element discretisation, we use a stabilised formulation, in which appropriately weighted residual terms are added to the discrete variational forms. We prove that the formulation is uniformly stable, which implies an optimal a priori error estimate. Using the stability of the continuous problem, we also prove a posteriori estimates, the optimality of which is ensured by local lower bounds. In the implementation of the methods, the discrete Lagrange multiplier is locally eliminated, leading to a Nitsche-type method [3].

For the problems of a membrane and plate subject to solid obstacles, we present numerical results.

Joint work with Tom Gustafsson (Aalto) and Juha Videman (Lisbon).

### References

1. T. Gustafsson, R. Stenberg, J. Videman. Mixed and stabilized finite element methods for the obstacle problem. SIAM Journal of Numerical Analysis 55 (2017) 2718–2744
2. T. Gustafsson, R. Stenberg, J. Videman. Stabilized methods for the plate obstacle problem. BIT– Numerical Mathematics (2018) DOI: 10.1007/s10543-018-0728-7
3. E. Burman, P. Hansbo, M.G. Larson, R. Stenberg. Galerkin least squares finite element method for the obstacle problem. Computer Methods in Applied Mechanics and Engineering 313 (2017) 362–374

### 19/10/2018, Friday

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

Moritz Reintjes, Instituto Superior Técnico.
Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory.

I am going to present Blake Temple's and my recent breakthrough regarding optimal metric regularity: We recently derived a set of nonlinear elliptic equations, with differential forms as unknowns, (the "Regularity Transformation equations" or "RT-equations"), and proved existence of solutions. The RT-equations determine whether optimal metric regularity can be achieved in General Relativity. Our existence result applies to connections in and Riemann curvature in $W^{m,p}$, $m\geq1$, $p>n$, and thus yields that such connections can always be smoothed to optimal regularity (one derivative above their curvature) by coordinate transformation. Extending this existence theory to the case of GR shock waves, when the connection is in $L^{\infty}$, is subject of our ongoing research. Our current existence result demonstrates that the method of determining optimal metric regularity by the RT-equations works.

### 17/10/2018, Wednesday

#### 11:30, Room P3.10, Mathematics Building, Topological Quantum Field Theory

Björn Gohla, GFM Univ. Lisboa.

We will present some classical facts about the relationship between monoids and monads. We will use ordinal sums of categories and the join product of topological spaces to define the abstract and topological simplices. Along the way we show how the simplicial identities can be obtained. Time permitting we will indicate a 2-categorical generalization of this circle of ideas.

### 09/10/2018, Tuesday

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

Franco Severo, Institut des Hautes Études Scientifiques.
Existence of phase transition for percolation on general graphs.

The first step in the study of percolation on a graph $G$ is proving that its critical point $p_c(G)$ for the emergence of an infinite cluster is nontrivial, that is, $p_c(G)\lt 1$. In this talk we prove that, if the isoperimetric dimension of a graph $G$ (with bounded degree) is strictly larger than $4$, then $p_c(G)\lt 1$. This settles a conjecture of Benjamini and Schramm saying that $p_c(G)\lt 1$ for any transitive graph with super-linear growth.

The proof proceeds by first proving the existence of an infinite cluster for percolation with certain random edge-parameters induced by the Gaussian Free Field (GFF). Then we integrate out the randomness in the environment by using a multi-scale decomposition of the GFF.

Joint work with Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi and Ariel Yadin.

### 03/10/2018, Wednesday

#### 14:30, Room P3.10, Mathematics Building, String Theory

Abhiram Kidambi, Technical University of Vienna.
BPS algebras and Moonshine.

We give a brief introduction to BPS algebras and Moonshine in this informal seminar.

### 01/10/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

Abhiram Kidambi, Tecnical University of Vienna.
$\Gamma_0(N)$, quantum black holes and wall crossing.

The degeneracies of supersymmetric dyonic black holes are known to be encoded in the Fourier coefficients of certain modular objects. For the case of $N = 4$, $d=4$ theory which I shall discuss, the spectrum of quarter BPS dyons is prone to wall crossing phenomena. The number theory machinery behind wall crossing in $4d$ $N = 4$ theories was described systematically in a comprehensive paper by Atish Dabholkar, Sameer Murthy and Don Zagier. There have also been supergravity localisation calculations thereafter which confirm some of the results that were shown by DMZ.

In this talk, I shall provide some of the number theoretic background for BPS state counting and review some of the key results known so far from both the microscopic and macroscopic side. I shall comment on black hole metamorphosis studied by Sen (and collaborators) and Nampuri et.al from a number theoretic framework. The remainder of the talk will be devoted to the generalisation of the number theory machinery of DMZ to congruence subgroups of $\operatorname{SL}(2,\mathbb{Z})$ i.e. for orbifolded CHL black holes and the supergravity approach for the CHL case.

This talk summarises some of the ongoing work with Sameer Murthy, Valentin Reys, Abhishek Chowdhury and Timm Wrase.

### 28/09/2018, Friday

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

Maxime Van de Moortel, University of Cambridge.
Stability and instability in spherical symmetry of Reissner-Nordström black holes for the Einstein-Maxwell-Klein-Gordon model.

Penrose’s Strong Cosmic Censorship Conjecture is one of the central problems of Mathematical General Relativity. Its proof for the Einstein-Maxwell-Uncharged-Scalar-Field (EMSF) model in spherical symmetry relies on the formation of a Cauchy horizon that is $C^0$ regular but $C^2$ singular for generic Cauchy data. EMSF model however only admits two-ended black holes, unlike its charged analogue that allows for one-ended black holes, relevant to the study of charged gravitational collapse in spherical symmetry. In this talk, I will present my work about spherically symmetric charged and massive scalar fields on black holes. This includes a study of the black hole interior, that relates the behaviour of fields on the event horizon to the formation of a $C^0$ regular and $C^2$ singular Cauchy horizon. I will also mention my more recent work on the black hole exterior stability, for weakly charged massless scalar fields.

### 26/09/2018, Wednesday

#### 13:30, Room P3.10, Mathematics Building, Partial Differential Equations

This talk is intend to review the theory of global well-posedness for the initial value problem associated to the generalized Korteweg-de Vries equation. In the supercritical case we will present a new result describing the assymptotic behavior of some global solutions.

### 25/09/2018, Tuesday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Levi Lima, Universidade Federal do Ceará.
The mass of asymptotically hyperbolic manifolds with a noncompact boundary.

We discuss a positive mass inequality (and its consequences) for the class of manifolds in the title, under the spin assumption. This is a natural extension to this setting of a previous result by P. Chrusciel and M. Herzlich, who treated the boundaryless case. Joint work with S. Almaraz.

### 19/09/2018, Wednesday

#### 13:30, Room P3.10, Mathematics Building, Mathematical Relativity

Volker Schlue, Sorbonne Université.
On some stability and instability problems for hard stars in spherical symmetry.

I will review Christodoulou's two phase model of relativistic fluids in the context of gravitational collapse, and describe the properties of static solutions to the hard phase with vacuum boundary (stars). We expect small stars to be (orbitally) stable, but limiting configurations with large central density to be unstable, and I will explain some of the underlying heuristics, and related results for these scenarios.

### 06/09/2018, Thursday

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

Arick Shao, Queen Mary University of London.
Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes.

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an aAdS solution of the Einstein equations uniquely determined by its data on its conformal boundary?

In these talks, we report on recent progress in this direction, and we highlight the connections between correspondence conjectures in physics, unique continuation theory for wave equations, and the geometry of aAdS spacetimes. We discuss recent unique continuation theorems for waves on aAdS spacetimes that form the key step toward correspondence results, as well as novel geometric obstructions to these results. As an application, we provide an answer to the following symmetry extension question: when can a symmetry on the conformal boundary be extended into the interior?

This is joint work with Gustav Holzegel (Imperial College London).

### 05/09/2018, Wednesday

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

Arick Shao, Queen Mary University of London.
Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes.

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an aAdS solution of the Einstein equations uniquely determined by its data on its conformal boundary?

In these talks, we report on recent progress in this direction, and we highlight the connections between correspondence conjectures in physics, unique continuation theory for wave equations, and the geometry of aAdS spacetimes. We discuss recent unique continuation theorems for waves on aAdS spacetimes that form the key step toward correspondence results, as well as novel geometric obstructions to these results. As an application, we provide an answer to the following symmetry extension question: when can a symmetry on the conformal boundary be extended into the interior?

This is joint work with Gustav Holzegel (Imperial College London).

### 28/08/2018, Tuesday

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

Marco Morandotti, TUM, Munique.
Spatially inhomogeneous evolutionary games.

We study an interaction model of a large population of players based on an evolutionary game, which describes the dynamical process of how the distribution of strategies changes in time according to their individual success.

Differently from spatially homogeneous dynamical games, we assume that the population of players is distributed over a state space and that they are each endowed with probability distributions of pure strategies, which they draw at random to evolve their states. Simultaneously, the mixed strategies evolve according to a replicator dynamics, modeling the success of pure strategies according to a payoff functional.

We establish existence, uniqueness, and stability of Lagrangian and Eulerian solutions of this dynamical game by using methods of ODE and optimal transport on Banach spaces.

### 26/07/2018, Thursday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Bruno Oliveira, University of Miami.
Hyperbolicity of projective manifolds.

We continue to discuss several ideas and methods used in studying the Kobayshi hyperbolicity of projective manifolds. A manifold $X$ is said to be hyperbolic if there are no nonconstant holomorphic maps from the complex line to $X$. This is a subject that brings together methods of algebraic geometry, complex analysis and differential geometry.

We will discuss the key and well understood case of dimension $1$. We will have several distinct characterizations of hyperbolicity and see how that extend for projective manifolds of higher dimension. We will also discuss the related Green-Griffiths-Lang conjecture.

### 24/07/2018, Tuesday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Martin Pinsonnault, University of Western Ontario.
Stability of Symplectomorphism Groups of Small Rational Surfaces.

Let $(X_k,\omega_k)$ be the symplectic blow-up of the projective plane at $k$ balls, $1\leq k\leq 9$, of capacities $c_1,\ldots, c_k$. After reviewing some facts on Kahler cones and curve cones of tamed almost complex structures, we will give sufficient conditions on two sets of capacities $\{c_i\}$ and $\{c_i’\}$ for the associated symplectomorphism groups to be homotopy equivalent. In particular, we will explain when those groups are homotopy equivalent to stabilisers of points in $(X_{k-1},\omega_{k-1})$. We will discuss some corollaries for the spaces of symplectic balls.

### 18/07/2018, Wednesday

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

Moritz Reintjes, Instituto Superior Técnico.
The quantised Dirac field and the fermionic signature operator.

In this mini course I am going to introduce the Dirac equation (describing fermions) in Minkowski spacetime and explain how to extend the equation to curved spacetimes (Lecture 1). In Lecture 2, I will introduce the canonical quantisation of the Dirac field in Minkowski spacetime and describe the problem of time-dependent external fields in the canonical quantisation formalism. In Lecture 3, I will present a proposal of myself and Felix Finster addressing the problem of time-dependent external fields and spacetimes, based on the fermionic signature operator.

### 17/07/2018, Tuesday

#### 11:00, Room P3.10, Mathematics Building, Geometria em Lisboa

Jun Li, University of Minnesota, Minneapolis.
The symplectomorphism groups of rational surfaces.

This talk is on the topology of $\operatorname{Symp}(M, \omega)$, where $\operatorname{Symp}(M, \omega)$ is the symplectomorphism group of a symplectic rational surface $(M, \omega)$. We will illustrate our approach with the 5 point blowup of the projective plane. For an arbitrary symplectic form on this rational surface, we are able to determine the symplectic mapping class group (SMC) and describe the answer in terms of the Dynkin diagram of Lagrangian sphere classes. In particular, when deforming the symplectic form, the SMC of a rational surface behaves in the way of forgetting strand map of braid groups. We are also able to compute the fundamental group of $\operatorname{Symp}(M, \omega)$ for an open region of the symplectic cone. This is a joint work with Tian-Jun Li and Weiwei Wu.

### 13/07/2018, Friday

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

Moritz Reintjes, Instituto Superior Técnico.
The quantised Dirac field and the fermionic signature operator.

In this mini course I am going to introduce the Dirac equation (describing fermions) in Minkowski spacetime and explain how to extend the equation to curved spacetimes (Lecture 1). In Lecture 2, I will introduce the canonical quantisation of the Dirac field in Minkowski spacetime and describe the problem of time-dependent external fields in the canonical quantisation formalism. In Lecture 3, I will present a proposal of myself and Felix Finster addressing the problem of time-dependent external fields and spacetimes, based on the fermionic signature operator.

### 11/07/2018, Wednesday

#### 16:00, Room P3.10, Mathematics Building, Geometria em Lisboa

Bruno Oliveira, University of Miami.
Hyperbolicity of projective manifolds.

In this talk we will discuss several ideas and methods used in studying the Kobayshi hyperbolicity of projective manifolds. A manifold $X$ is said to be hyperbolic if there are no nonconstant holomorphic maps from the complex line to $X$. This is a subject that brings together methods of algebraic geometry, complex analysis and differential geometry.

We will discuss the key and well understood case of dimension $1$. We will have several distinct characterizations of hyperbolicity and see how that extend for projective manifolds of higher dimension. We will also discuss the related Green-Griffiths-Lang conjecture.

### 11/07/2018, Wednesday

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

Moritz Reintjes, Instituto Superior Técnico.
The quantised Dirac field and the fermionic signature operator.

In this mini course I am going to introduce the Dirac equation (describing fermions) in Minkowski spacetime and explain how to extend the equation to curved spacetimes (Lecture 1). In Lecture 2, I will introduce the canonical quantisation of the Dirac field in Minkowski spacetime and describe the problem of time-dependent external fields in the canonical quantisation formalism. In Lecture 3, I will present a proposal of myself and Felix Finster addressing the problem of time-dependent external fields and spacetimes, based on the fermionic signature operator.

### 09/07/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

$L_{\infty}$ bootstrap approach to non-commutative gauge theories.

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying $L_{\infty}$ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. In this talk I will give a brief introduction to $L_{\infty}$ algebras and discuss in more details the $L_{\infty}$ bootstrap program: the existence of the solution, uniqueness and particular examples. The talk is mainly based on: arXiv:1803.00732 and 1806.10314.

### 06/07/2018, Friday

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

Jesus Oliver, California State University, East Bay.
Boundedness of energy for the Wake Klein-Gordon model.

We consider the global-in-time existence theory for the Wave-Klein-Gordon model for the Einstein-Klein-Gordon equations introduced by LeFloch and Ma. By using the hyperboloidal foliation method, we prove that a hierarchy of weighted energies of the solutions remain (essentially) bounded for all times.

### 04/07/2018, Wednesday

#### 15:15, Room P4.35, Mathematics Building, Analysis, Geometry, and Dynamical Systems

Rajesh Kumar, BITS Pilani, India.
Convergence analysis of finite volume scheme for solving coagulation-fragmentation equations.

### 04/07/2018, Wednesday

#### 14:00, Room P4.35, Mathematics Building, Analysis, Geometry, and Dynamical Systems

Ankik K. Giri, IIT Roorkee, India.
Recent developments in the theory of coagulation-fragmentation models.

### 04/07/2018, Wednesday

#### 11:30, Room P3.10, Mathematics Building, Topological Quantum Field Theory

Björn Gohla, GFM Univ. Lisboa.
A Categorical Model for the Hopf Fibration.

We give a description up to homeomorphism of $S^3$ and $S^2$ as classifying spaces of small categories, such that the Hopf map $S^3\longrightarrow{}S^2$ is the realization of a functor.

### 28/06/2018, Thursday

#### 16:30, Room P4.35, Mathematics Building, Geometria em Lisboa

José Mourão, CAMGSD, Instituto Superior Técnico, Universidade de Lisboa.
Imaginary time Hamiltonian flows and applications to Kahler geometry, Kahler reduction and representation theory.

The formalism to complexify time in the flow of a nonholomorphic vector field on a complex manifold is reviewed. The complexified flow, besides acting on $M$, changes also the complex structure. We will describe the following applications:

1. For a compact Kahler manifold the imaginary time Hamiltonian flows correspond to Mabuchi geodesics in the infinite dimensional space of Kahler metrics on $M$. These geodesics play a very important role in the study of stability of Kahler manifolds. A nontrivial nontoric example on the two-dimensional sphere will be described.
2. Let the compact connected Lie group $G$ act in an Hamiltonian and Kahler way on a Kahler manifold $M$ and assume that its action extends to $G_C$. Then, by taking geodesics of Kahler structures generated by convex functions of the $G$-momentum to infinite geodesic time, one gets (conjecturally always, proved on several important examples) a concentration of holomorphic sections of holomorphic line bundles on inverse images of coadjoint orbits under the $G$-momentum map. A nontrivial toric example and the case of $M=G_C$ will be described.

On work with T Baier, J Hilgert, O Kaya, JP Nunes, M Pereira, P Silva.

### 28/06/2018, Thursday

#### 14:30, Seminar room (2.8.3), Physics Building, Mathematical Relativity

Edgar Gasperin, CENTRA, Instituto Superior Técnico.
Perturbations of the asymptotic region of the Schwarzschild-de Sitter spacetime.

Although the study of the Cauchy problem in General Relativity started in the decade of 1950 with the work of Foures-Bruhat, addressing the problem of global non-linear stability of solutions to the Einstein field equations is in general a hard problem. The first non-linear global non-linear stability result in General Relativity was obtained for the de Sitter spacetime by H. Friedrich in the decade of 1980. In this talk the main tool used in the above result is introduced: a conformal (regular) representation of the Einstein field equations — the so-called conformal Einstein field equations (CEFE). Then, the conformal structure of the Schwarzschild-de Sitter spacetime is analysed using the extended conformal Einstein field equations (XCEFE). To this end, initial data for an asymptotic initial value problem for the Schwarzschild-de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild-de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild-de Sitter spacetime in the asymptotic region. Finally, we'll show that small enough perturbations of asymptotic initial data for the Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild-de Sitter spacetime.

### 20/06/2018, Wednesday

#### 11:30, Room P3.10, Mathematics Building, Topological Quantum Field Theory

Ricardo Schiappa, Instituto Superior Técnico.
Co-equational (i.e. Parametric) Resurgence and Topological Strings.

I will briefly review the uses and applications of resurgence applied to topological string theory, with emphasis on nonperturbative completions and the large-order behaviour of enumerative invariants. Due to the nature of the holomorphic anomaly equations, there is a clear need to develop methods of co-equational (i.e. parametric) resurgence in order to achieve a complete description of the topological string transseries.

### 19/06/2018, Tuesday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Lino Amorim, Kansas State University.
Closed mirror symmetry for orbifold spheres.

In this talk I will describe a closed mirror symmetry theorem for a sphere with three orbifold points. More precisely I will construct an isomorphism between the quantum cohomology ring of the orbifold and the Jacobian ring of a certain power series built from the Lagrangian Floer theory of an immersed circle. This is joint work with Cho, Hong and Lau.

### 19/06/2018, Tuesday

#### 11:00, Room P4.35, Mathematics Building, Analysis, Geometry, and Dynamical Systems

Marcel de Jeu, Leiden University.
Banach lattice algebra representations in harmonic analysis.

If $G$ is a locally compact group, then natural spaces such as $L^1(G)$ or $M(G)$ carry more structure than just that of a Banach algebra. They are also vector lattices, so that they are, in fact, Banach lattice algebras. Therefore, if they act by convolution on, say, $L^p(G)$, it is a meaningful question to ask if the corresponding map into the Banach lattice algebra $L_r(L^p(G))$ of regular operators on $L^p(G)$ is not only an algebra homomorphism, but also a lattice homomorphism. Analogous questions can be asked in similar situations, such as the left regular representation of $M(G)$.

In this lecture, we shall give an overview of what is known in this direction, and which approaches are available. The rule of thumb, based on an underlying general principle, seems to be that the answer is affirmative whenever the question is meaningful.

This is joint work with Garth Dales and David Kok.

### 12/06/2018, Tuesday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Nitu Kitchloo, Johns Hopkins University.
The Stable Symplectic Category and a Conjecture of Kontsevich.

Motivated by his work on deformation quantization and his computations of Feynman integrals, Kontsevich conjectured that a certain group (related to the Grothendieck Teichmuller group) acts on the moduli space of quantum field theories. Even though this moduli space is not well-defined in general, we will show that a stable version of this space makes sense and can be identified as a space that represents an interesting cohomology theory. In addition, we will show that a solvable quotient of the Grothendieck-Teichmuller group acts on the stable moduli space, and as such, it can be identified with an algebraic functor of the underlying cohomology theory.

### 08/06/2018, Friday

#### 16:00, Room P3.10, Mathematics Building, Partial Differential Equations

Bifurcation theory for non autonomous systems.

We consider bifurcation problems arising in mathematical biology, specifically in pattern formation on nonplanar, growing domains. This setting leads to the study of reaction-diffusion equations with variable coefficients. We present both analytical and numerical results and discuss the Turing-Hopf bifurcation for a Fitzhugh-Nagumo system. This is joint work with J. Castillo and F. Sánchez.

### 06/06/2018, Wednesday

#### 11:30, Room P3.10, Mathematics Building, Topological Quantum Field Theory

2-representation theory.

I will give an overview of 2-representation theory, following Mazorchuk and Miemietz' approach. After explaining the general setup, I will sketch the 2-representation theory of dihedral Soergel bimodules as an example.

After the seminar, for those interested we will continue with a discussion of approaches to 2-representation theory.

### 04/06/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

Frank Ferrari, Université Libre de Bruxelles.
On Melonic Matrix Models and SYK-like Black Holes.

I will illustrate three aspects of the new large $D$ limit of matrix models and their applications to black hole physics:

1. Graph theory aspect: I will review the basic properties of the new large $D$ limit of matrix models and provide a simple graph-theoretic argument for its existence, independent of standard tensor model techniques, using the concepts of Tait graphs and Petrie duals.
2. Phase diagrams: I will outline the interesting phenomena found in the phase diagrams of simple fermionic matrix quantum mechanics/tensor/SYK models at strong coupling, including first and second order phase transitions and quantum critical points. Some of these phase transitions can be argued to provide a quantum mechanical description of the phenomenon of gravitational collapse.
3. Probe analysis: I will briefly describe how the matrix point of view allows to naturally define models of D-particles probing an SYK-like black hole and discuss the qualitative properties of this class of models, emphasizing the difference between models based on fermionic and on bosonic strings. This approach provides an interesting strategy to study the emerging geometry of melonic/SYK black holes. In particular, it will be explained how a sharply defined notion of horizon emerges naturally.

### 29/05/2018, Tuesday

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

Elvira Zappale, Università degli Studi di Salerno.
Optimal design problems for energies with nonstandard growth.

Some recent results dealing with optimal design problems for energies which describe composite materials, mixed materials and Ogden ones will be presented.

### 28/05/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

Salvatore Baldino, Instituto Superior Técnico.
Introduction to resurgence.

This is the third in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.

### 23/05/2018, Wednesday

#### 11:30, Room P3.10, Mathematics Building, Topological Quantum Field Theory

Ana Bela Cruzeiro, Department of Mathematics, Instituto Superior Técnico.
Stochastic Clebsch variational principles.

We derive the equations of motion associated with stochastic Clebsch action principles for mechanical systems whose configuration space is a manifold on which a Lie algebra acts transitively. These are stochastic differential equations (spde's in infinite dimensions).

We give the Hamiltonian version of the equations, as well as the corresponding Kolmogorov equations.

This is a joint work with D. D. Holm and T. S. Ratiu.

### 18/05/2018, Friday

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

Rodrigo Vicente, Instituto Superior Tecnico.
Test fields cannot destroy extremal black holes.

We prove that (possibly charged) test fields satisfying the null energy condition at the event horizon cannot overspin/overcharge extremal Kerr-Newman or Kerr-Newman-anti de Sitter black holes, that is, the weak cosmic censorship conjecture cannot be violated in the test field approximation. The argument relies on black hole thermodynamics (without assuming cosmic censorship), and does not depend on the precise nature of the fields. We also discuss generalizations of this result to other extremal black holes.

### 08/05/2018, Tuesday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Alessandro Ghigi, Università di Pavia.
Compactifying automorphism groups of Kaehler manifolds.

Around 1978 Akira Fujiki and David Lieberman independently introduced a natural compactification of the connected component of the identity in the automorphism group of a compact Kaehler manifold. In the talk I will recall the construction of this compactification using Barlet cycle space. Then I will describe some recent results obtained jointly with Leonardo Biliotti. The main result is the interpretation of boundary points in terms of non-dominant meromorphic inmaps of the manifold in itself.

### 07/05/2018, Monday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Cyril Lecuire, Centre National de la recherche scientifique.
Geometry in groups.

The intent of Geometric Group Theory is to deduce algebraic properties of groups from their actions on metric spaces. A natural way to obtain such an action is to equip a group with an invariant distance.

First, to motivate the study of Geometric Group Theory, I will expose some of its achievements (solvability of the word problem, Tits alternative). Then I will define the word metric on a finitely generated group and explain the difficulties raised by the definition. Finally, as an example of geometric properties of interest, I will introduce hyperbolic groups.

### 02/05/2018, Wednesday

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

Melanie Graf, University of Vienna.
The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics.

The classical singularity theorems of General Relativity show that a Lorentzian manifold with a $C^2$-metric that satisfies physically reasonable conditions cannot be geodesically complete. One of the questions left unanswered by the classical singularity theorems is whether one could extend such a spacetime with a lower regularity Lorentzian metric such that the extension still satisfies these physically reasonable conditions and does no longer contain any incomplete causal geodesics. In other words, the question is if a lower differentiability of the metric is sufficient for the theorems to hold. The natural differentiability class to consider here is $C^{1,1}$. This regularity corresponds, via the field equations, to a finite jump in the matter variables, a situation that is not a priori regarded as singular from the viewpoint of physics and it is the minimal condition that ensures unique solvability of the geodesic equations. Recent progress in low-regularity Lorentzian geometry has allowed one to tackle this question and show that, in fact, the classical singularity theorems of Penrose, Hawking, and Hawking-Penrose remain valid for $C^{1,1}$-metrics. In this talk I will focus on the Hawking-Penrose theorem, being the most recent and in a sense most general of the aforementioned results, and some of the methods from low regularity causality and comparison geometry that were employed in its proof. This is joint work with J. D. E. Grant, M. Kunzinger and R. Steinbauer.

### 02/05/2018, Wednesday

#### 11:30, Room P3.10, Mathematics Building, Topological Quantum Field Theory

Marko Stošić, Instituto Superior Técnico.
Knots-quivers correspondence and applications.

In this talk I shall present the knots-quivers correspondence, as well as some surprising implications in combinatorics involving counting of lattice paths and number theory. The knots-quivers correspondence relates the colored HOMFLY-PT invariants of a knot with the motivic Donaldson-Thomas invariants of the corresponding quiver. This correspondence is made completely explicit at the level of generating series. The motivation for this relationship comes from topological string theory, BPS (LMOV) invariants, as well as categorification of HOMFLY-PT polynomial and A-polynomials. We compute quivers for various classes of knots, including twist knots, rational knots and torus knots.

One of the surprising outcomes of this correspondence is that from the information of the colored HOMFLY-PT polynomials of certain knots we get new expressions for the classical combinatorial problem of counting lattice paths, as well as new integrality/divisibility properties.

The main goal of this talk is to present basic ideas and to present numerous open questions and ramifications coming from knots-quivers correspondence.

(based on joint works with P. Sulkowski, M. Reineke, P. Kucharski, M. Panfil and P. Wedrich).

### 30/04/2018, Monday

#### 16:00, Room P3.10, Mathematics Building, String Theory

Maximilian Schwick, Instituto Superior Técnico.
Introduction to resurgence.

This is the second in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.

### 27/04/2018, Friday

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

Pedro Freitas, Instituto Superior Técnico.
The spectral determinant of the quantum harmonic oscillator in arbitrary dimensions.

We show that the spectral determinant of the isotropic quantum harmonic oscillator converges exponentially to one as the space dimension grows to infinity. We determine the precise asymptotic behaviour for large dimension and obtain estimates valid for all cases with the same asymptotic behaviour in the large.

As a consequence, we provide an alternative proof of a conjecture posed by Bar and Schopka concerning the convergence of the determinant of the Dirac operator on $S^{n}$, determining the exact asymptotic behaviour for this case and thus improving the estimate on the rate of convergence given in the proof by Moller.

### 24/04/2018, Tuesday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Matias Del Hoyo, Universidade Federal Fluminense.
Discrete dynamics and differentiable stacks.

In a joint work with A. Cabrera (UFRJ) and E. Pujals (IMPA) we study actions of discrete groups over connected manifolds by means of their orbit stacks. Stacks are categorified spaces, they generalize manifolds and orbifolds, and they remember the isotropies of the actions that give rise to them. I will review the basics, show that for simply connected spaces the stacks recover the dynamics up to conjugacy, and discuss the general case. I will also discuss several examples, involving irrational rotations of the circle, hyperbolic toral automorphisms, and the lens spaces.

### 24/04/2018, Tuesday

#### 11:00, Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa, String Theory

Panagiotis Betzios, University of Crete.
Matrix Quantum Mechanics and the $S^1/\mathbb{Z}_2$ orbifold.

We revisit $c=1$ non-critical string theory and its formulation via Matrix Quantum Mechanics (MQM). In particular we study the theory on an $S^1/\mathbb{Z}_2$ orbifold of Euclidean time and try to compute its partition function in the grand canonical ensemble that allows one to study the double scaling limit of the matrix model and connect the result to string theory (Liouville theory). The result is expressed as the Fredholm Pfaffian of a Kernel which we describe in several bases. En route we encounter interesting mathematics related to Jacobi elliptic functions and the Hilbert transform. We are able to extract the contribution of the twisted states at the orbifold fixed points using a formula by Dyson for the determinant of the sine kernel. Finally, we will make some comments regarding the possibility of using this model as a toy model of a two dimensional big-bang big-crunch universe.

### 23/04/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

FZZT branes and non-singlets of Matrix Quantum Mechanics.

We will discuss the non-singlet sectors of the matrix model associated with two dimensional non-critical string theory. These sectors of the matrix model contain rich physics and are expected to describe non-trivial states such as black holes. I will present how one can turn on the non-singlets by adding $N_f \times N$ fundamental and anti-fundamental fields in the gauge matrix quantum mechanics model as well as a Chern-Simons term. Then, I will show how one can rewrite our model as a spin-Calogero model in an external magnetic field. By introducing chiral variables we can define spin-currents that in the large $N$ limit satisfy an $SU(2N_f )_k$ Kac-Moody algebra. Moreover, we can write down the canonical partition function and study different limits of the parameters and possible phase transitions. In the grand canonical ensemble the partition function is a $\tau$ - function obeying discrete soliton equations. Also, in a certain limit we recover the matrix model of Kazakov-Kostov-Kutasov conjectured to describe the two dimensional black hole. Finally, I will discuss several implications that our model has for the understanding of the thermodynamics and the physics of such string theory states.​

### 19/04/2018, Thursday

#### 11:30, Room P3.10, Mathematics Building, Algebra

During this talk, I would like to give an overview of the (relative) delooping theorems as well as applications to spaces of long embeddings. In particular, we show that the space of long embeddings and the space of ($k$)-immersions from $\mathbb{R}^d$ to $\mathbb{R}^m$ are weakly equivalent to an explicit ($d+1$)-iterated loop space and an explicit ($d+1$)-iterated relative loop space, respectively. Both of them can be expressed in term of derived mapping spaces of coloured operads. Such a pair is a typical example of Swiss-Cheese algebra.

### 18/04/2018, Wednesday

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

Oliver Lindblad Petersen, University of Potsdam.
Wave equations with initial data on compact Cauchy horizons.

I will present a new energy estimate for wave equations close to compact non-degenerate Cauchy horizons. The estimate allows one to conclude several existence and uniqueness results for wave equations with initial data on the Cauchy horizon. This generalizes classical results that were proven under the assumption of either analyticity or symmetry of the spacetime or closedness of the generators. In particular, the results are useful in understanding the geometry of vacuum spacetimes with a compact non-degenerate Cauchy horizon (without making any extra assumptions). This problem is closely related to the strong cosmic censorship conjecture.

### 16/04/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

Masazumi Honda, Weizmann Institute of Science.
Resurgent transseries and Lefschetz thimble in 3d $\mathcal{N}=2$ supersymmetric Chern-Simons matter theories.

We show that a certain class of supersymmetric (SUSY) observables in 3d $\mathcal{N}=2$ SUSY Chern-Simons (CS) matter theories has nontrivial resurgent structures with respect to coupling constants given by inverse CS levels, and that their exact results are expressed as appropriate resummations of weak coupling expansions given by transseries. With a real mass parameter varied, we encounter Stokes phenomena infinitely many times, where the perturbative series gets non-Borel-summable along positive real axis of the Borel plane. We also decompose integral representations of the exact results in terms of Lefschetz thimbles and study how they are related to the resurgent transseries. We further discuss connections between the non-perturbative effects appearing in the transseries and complexified SUSY solutions which formally satisfy SUSY conditions but are not on original path integral contour. We explicitly demonstrate the above for partition functions of rank-1 3d $\mathcal{N}=2$ CS matter theories on sphere. This talk is based on arXiv:1604.08653, 1710.05010, and an on-going collaboration with Toshiaki Fujimori, Syo Kamata, Tatsuhiro Misumi and Norisuke Sakai.​

### 12/04/2018, Thursday

#### 11:30, Room P3.10, Mathematics Building, Analysis, Geometry, and Dynamical Systems

Anastasiia Panchuk, Academia Nacional das Ciências de Kiev.
A piecewise linear map with two discontinuities: bifurcation structures in the chaotic domain.

In the current work we consider a one-dimensional piecewise linear map with two discontinuity points and describe different bifurcation structures observed in its parameter space. The structures associated with periodic orbits have been extensively studied before (see, e.g., Sushko et al., 2015 or Tramontana et al., 2012, 2015). By contrast, here we mainly focus on the regions associated with robust multiband chaotic attractors. It is shown that besides the standard bandcount adding and bandcount incrementing bifurcation structures, occurring in maps with only one discontinuity, there also exist peculiar bandcount adding and bandcount incrementing structures involving all three partitions. Moreover, the map's three partitions may generate intriguing bistability phenomena.

1. Sushko I., Tramontana F., Westerhoff, F. and Avrutin V. (2015): Symmetry breaking in a bull and bear financial market model. Chaos, Solitons and Fractals, 79, 57-72.
2. Tramontana, F., Gardini L., Avrutin V. and Schanz M. (2012): Period Adding in Piecewise Linear Maps with Two Discontinuities. International Journal of Bifurcation & Chaos, 22(3) (2012) 1250068 (1-30).
3. Tramontana, F., Westerhoff, F. and Gardini, L. (2015): A simple financial market model with chartists and fundamentalists: market entry levels and discontinuities. Mathematics and Computers in Simulation, Vol. 108, 16-40.

### 09/04/2018, Monday

#### 15:00, Room P3.10, Mathematics Building, String Theory

Roberto Vega, Instituto Superior Técnico.
Introduction to resurgence.

This is the first in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.

### 20/03/2018, Tuesday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Pedro Boavida, CAMGSD, Instituto Superior Técnico, Universidade de Lisboa.
Spaces of smooth embeddings and the little disks operad.

I will describe a homotopy theoretic approach, based on a method due to Goodwillie and Weiss, to study spaces of smooth embeddings of a manifold into another. This approach opened up interesting relations to operad theory and as such to fundamental objects in topology (e.g. configuration spaces) and algebra (e.g. graph complexes). I will survey some of these developments, focusing on the case of long knots and higher-dimensional variants, for which these relations are the sharpest.

### 09/03/2018, Friday

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

Katharina Radermacher, KTH Royal Institute of Technology.
On the Cosmic No-Hair Conjecture in $\mathbb{T}^2$-symmetric non-linear scalar field spacetimes.

At late times, cosmological spacetimes solving Einstein's field equations, at least when assuming a positive cosmological constant, are conjectured to isotropise and appear like the de Sitter spacetime to late time observers. This is the statement of the Cosmic No-Hair conjecture. In this talk, I consider Einstein's non-linear scalar field equations and spacetimes with $\mathbb{T}^2$-symmetry. I present results on future global existence of such solutions and discuss the conjecture in the setting of a constant potential.

This talk is based on arXiv:1712.01801.

### 07/03/2018, Wednesday

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

Masashi Kimura, Instituto Superior Técnico.

We introduce a novel type of ladder operators, which map a solution to the massive Klein-Gordon equation into another solution with a different mass. It is shown that such operators are constructed from closed conformal Killing vector fields in arbitrary dimensions if the vector fields are eigenvectors of the Ricci tensor. As an example, we explicitly construct the ladder operators in AdS spacetime. It is shown that the ladder operators exist for masses above the Breitenlohner-Freedman bound. We also discuss their applications, ladder operator for spherical harmonics, the relation between supersymmetric quantum mechanics, and some phenomenon around extremal black holes whose near horizon geometry is AdS2.

### 28/02/2018, Wednesday

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

Pedro Oliveira, Instituto Superior Técnico.
Cosmic no-hair in spherically symmetric black hole spacetimes.

We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordström-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon ("near" $i^+$), in such a way that $r=+\infty$ is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the Cosmic No-Hair Conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.

### 07/02/2018, Wednesday

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

David Hilditch, Instituto Superior Tecnico.
Free-evolution formulations of GR for numerical relativity.

In this talk I will give an overview of the formulations of GR used in numerical relativity. I will summarize what is known about their mathematical properties and explain how local well-posedness of the IVP is achieved. Subsequently I will discuss a dual-foliation formulation of the field equations. The new formalism allows a larger class of coordinates to be employed in applications. These include choices popular in mathematical relativity.

### 30/01/2018, Tuesday

#### 14:00, Room P3.10, Mathematics Building, Mathematical Relativity

Aspects of cosmological perturbation theory.

Cosmological perturbation theory has been around for over 70 years and is the underlying theory for the interpretation of observations that have resulted in several Nobel prizes. Is there really anything new one can say about this field from a mathematical physics perspective? In this talk, which consists of two parts, I will try to convince you that the answer is yes. The first part deals with second order perturbations, which is a field that gives rise to notoriously messy equations. However, I will show that by using underlying physically motivated mathematical structures significant simplifications can be achieved, which give rise to new conserved quantities and simple explicit solutions in the so-called long wavelength limit, and for the currently dominating cosmological paradigm, the $\Lambda$CDM models. The second part is about a new research program where first order cosmological perturbation equations are reformulated as dynamical systems, which allows one to use dynamical systems methods and approximations, complementing previous investigations. Throughout the talk I will focus on ideas rather than technical details.

### 23/01/2018, Tuesday

#### 16:30, Room P3.10, Mathematics Building, Geometria em Lisboa

Zuoqin Wang, University of Science and Technology of China Heifei.
Equivariant Eigenvalues on Manifolds with Large Symmetry.

Let $M$ be a compact Riemannian manifold on which a compact Lie group acts by isometries. In this talk I will explain how the symmetry induces extra structures in the spectrum of Laplace-type operators, and how to apply symplectic techniques to study the induced equivariant spectrum. In particular, I will discuss a) my joint works with V. Guillemin on inverse spectral results for Schrodinger operators on toric manifolds; b) my joint work with Y. Qin on the first equivariant eigenvalues of toric Kahler manifolds.

### 17/01/2018, Wednesday

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

Moritz Reintjes, Instituto Superior Técnico.
The Question of Essential Metric Regularity at General Relativistic Shock Waves.

It is an open question whether shock wave solutions of the Einstein Euler equations contain "regularity singularities'', i.e., points where the spacetime metric would be Lipschitz continuous ($C^{0,1}$), but no smoother, in any coordinate system. In 1966, Israel showed that a metric $C^{0,1}$ across a single shock surface can be smoothed to the $C^{1,1}$ regularity sufficient for spacetime to be non-singular and for locally inertial frames to exist. In 2015, B. Temple and I gave the first (and only) extension of Israel's result to shock wave interactions in spherical symmetry by a new constructive proof involving non-local PDE's. In 2016, to address most general shock wave solutions (generated by Glimm's random choice method), we introduced the "Riemann flat condition" on $L^\infty$ connections and proved our condition necessary and sufficient for the essential metric regularity to be smooth (i.e. $C^{1,1}$). In our work in progress, we took the Riemann flat condition to derive an elliptic system which determines the essential metric regularity at shock waves (and beyond). Our preliminary results suggest that our elliptic system is well-posed and we believe this system to provide a systematic way for resolving the problem of regularity singularities completely.