# 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

Tobias Weth, Goethe-Universität Frankfurt.

On the unique continuation property for sublinear elliptic equations.

In the framework of linear elliptic equations of second order, the unique continuation principle states that if a solution vanishes on an open subset of a domain, then it vanishes identically in the domain. The principle applies under very general assumptions on the data and has various applications — in particular it implies the strict monotonicity property of eigenvalues with respect to domain inclusion. The unique continuation principle admits a straightforward extension to semilinear equations with Lipschitz nonlinearities, but it fails in general in the case of sublinear equations. In the talk, I will discuss very recent positive results for a rather large class of sublinear equations and also for some problems with discontinuous nonlinearities.

This is joint work with Nicola Soave (Politecnico di Milano).

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Wednesday

#### , Topological Quantum Field Theory

Paul Wedrich, Imperial College, London.

On colored link homologies.

Link homology theories are powerful generalizations of classical (and quantum) link polynomials, which are being studied from a variety of mathematical and physical viewpoints. Besides providing stronger invariants, these theories are often functorial under link cobordisms and carry additional topological information. The focus of this talk is on the Khovanov-Rozansky homologies, which categorify the Chern-Simons/Reshetikhin-Turaev $\mathfrak{sl}(N)$ link invariants and their large $N$ limits. I will survey recent results about their behaviour under deformations as well as their stability at large $N$, which together lead to a rigorous proof of a package of conjectures originating in string theory.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Monday

#### , Geometria em Lisboa

José Natário, Instituto Superior Técnico.

A Minkowski-type inequality for convex surfaces in the hyperbolic 3-space.

In this talk we derive a new Minkowski-type inequality for closed convex surfaces in the hyperbolic 3-space. The inequality is obtained by explicitly computing the area of the family of surfaces arising from the normal flow and then applying the isoperimetric inequality. Using the same method, we also we give elementary proofs of the classical Minkowski inequalities for closed convex surfaces in the Euclidean 3-space and in the 3-sphere.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Wednesday

#### , Topological Quantum Field Theory

Marco Mackaay, Universidade do Algarve.

A diagrammatic categorification of the higher level Heisenberg algebras.

Khovanov defined a diagrammatic 2-category and conjectured (and partially proved) that it categorifies the level-one Heisenberg algebra. Since then, several interesting generalizations and applications have been found, e.g. Cautis and Licata's generalization involving Hilbert schemes and their construction of categorical vertex operators. However, these are all for level one. In my talk, I will explain Alistair Savage and my results on a generalization of Khovanov's original results for higher level Heisenberg algebras. This is work in progress.

### , Monday

#### , String Theory

Valentin Reys, University of Milano-Bicocca.

Exact entropy of $1/4$-BPS black holes in $N=4$ supergravity and the mixed Rademacher expansion.

In this talk, I will present some recent developments in computing the exact entropy of dyonic $1/4$-BPS black holes in four-dimensional $N=4$ supergravity theories originating from Type IIB string theory compactified on $K3 \times T_2$. The exact entropy is obtained in the Quantum Entropy Function formalism by means of supersymmetric localization techniques. The result can then be compared to the degeneracy of the brane/momentum system making up the black hole in the string theory picture. Such degeneracies are given by the Fourier coefficients of so-called mock Jacobi forms, a concept I will review. An exact formula for the coefficients can be obtained via a suitable generalization of the Hardy-Ramanujan-Rademacher circle method which takes into account the mock character of the counting functions. After presenting these results, I will outline some discrepancies (at sub-leading order in the charges) between the supergravity result for the exact entropy and the degeneracies of the brane/momentum system, and point to some aspects of the supergravity calculations which should be examined in more detail if one hopes to get a complete matching.

### , Monday

#### , Geometria em Lisboa

Thomas Baier, Instituto Superior Tecnico.

Higher rank Prym varieties and Hitchin's connection.

Prym varieties are abelian varieties similarly associated to a double covers of algebraic curves as Jacobians are to a curve. In this talk, we define a higher rank analogue of Prym varieties and investigate some of their geometric properties. In particular we are interested in deformation theoretic aspects that permit the construction of a generalized Hitchin's connection in this setting.

This talk is based on joint work in progress with Michele Bolognesi, Johan Martens and Christian Pauly.

### , Friday

#### , Partial Differential Equations

Claude Warnick, Imperial College London.

Linear fields on anti-de Sitter spacetimes.

Spacetimes with negative cosmological constant are of interest both from a mathematical point of view, but also from a physical perspective in view of the conjectured AdS/CFT correspondence. A crucial feature of these spacetimes is their timelike null infinity, on which boundary conditions must be imposed. I will discuss several results in the theory of linear fields on anti-de Sitter backgrounds, including renormalisation and well-posedness, quasinormal modes and black hole stability.

### , Wednesday

#### , Partial Differential Equations

Adam Levi, Technion, Haifa.

Regularization of the stress-energy tensor, and semi-classical effects in black holes.

Regularization of the stress-energy tensor was the main obstacle to study semi-classical effects in non-trivial backgrounds. I'll talk about a new method, "pragmatic mode-sum regularization", that overcomes this obstacle. And show results calculated using the new method, including recent results in Kerr background.

### , Wednesday

#### , Topological Quantum Field Theory

Lucile Vandembroucq, Universidade do Minho.

Topological Complexity of the Klein Bottle.

The notion of topological complexity of a space has been introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem in robotics. Surprisingly, the determination of this invariant for non-orientable surfaces has turned out to be difficult. A. Dranishnikov has recently established that the topological complexity of the non-orientable surfaces of genus at least 4 is maximal. In this talk, we will determine the topological complexity of the Klein bottle and extend Dranishnikov's result to all the non-orientable surfaces of genus at least 2. This is a work in collaboration with Daniel C. Cohen.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Monday

#### , String Theory

Vishnu Jejjala, University of the Witwatersrand.

On the Shape of Things: From holography to elastica.

We explore the question of which shape a manifold is compelled to take when immersed in another one, provided it must be the extremum of some functional. We consider a family of functionals which depend quadratically on the extrinsic curvatures and on projections of the ambient curvatures. These functionals capture a number of physical setups ranging from holography to the study of membranes and elastica.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Wednesday

#### , Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.

M-theory from the superpoint revisited.

The last talk we gave on this topic (in the meeting Iberian Strings 2017) was largely about the physics; here we focus on the mathematics. No prior knowledge will be assumed.

We define the process of invariant central extension: taking central extensions by cocycles invariant under a given subgroup of automorphisms of a Lie superalgebra. We give conditions that allow us to carve out the Lorentz group inside the automorphisms of Minkowski superspacetime, and prove that by successive invariant central extensions of the superpoint, we construct all superspacetimes up to dimension 11.

### , Tuesday

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

Juliana Pimentel, UFABC (Brazil).

Unbounded Attractors Under Perturbations.

We put forward the recently introduced notion of unbounded attractors. These objects will be addressed in the context of a class of 1-D semilinear parabolic equations. The nonlinearities are assumed to be non-dissipative and, in addition, defined in such a way that the equation possesses unbounded solutions as time goes to infinity. Small autonomous and non-autonomous perturbations of these equations will be treated. This is based on joint work with A. Carvalho and S. Bruschi.

### , Monday

#### , Algebra

Ricardo Campos, ETH Zurich.

Configuration spaces of points and their homotopy type.

Given a manifold $M$, one can study the configuration space of $n$ points on the manifold, which is the subspace of $M^n$ in which two points cannot be in the same position. The study of these spaces from a homotopical perspective is of interest in very distinct areas such as algebraic topology or quantum field theory. However, even if we started with a simple manifold $M$, despite the apparent simplicity such configuration spaces are remarkably complicated; even the homology of these spaces is reasonably unknown, let alone their (rational/real) homotopy type.

In this talk, I will give an introduction to the problem of understanding configuration spaces and present a combinatorial/algebraic model of these spaces using graph complexes. I will explain how these models allow us to answer fundamental questions about the dependence on the homotopy type of $M$. I will explain how these models give us new tools to address other problems such as understanding embedding spaces or computing factorization homology.

This is joint work with Thomas Willwacher based on arXiv:1604.02043.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Wednesday

#### , Topological Quantum Field Theory

Pedro Boavida, Dep. Matemática, Instituto Superior Técnico.

Operads of genus zero curves and the Grothendieck-Teichmuller group.

In Esquisse d’un programme, Grothendieck made the fascinating suggestion that the absolute Galois group of the rationals could be understood via its action on certain geometric objects, the (profinite) mapping class groups of surfaces of all genera. The collection of these objects, and the natural relations between them, he called the "Teichmuller tower”.

In this talk, I plan to describe a genus zero analogue of this story from the point of view of operad theory. The result is that the group of automorphisms of the (profinite) genus zero Teichmuller tower agrees with the Grothendieck-Teichmuller group, an object which is closely related to the absolute Galois group of the rationals. This is joint work with Geoffroy Horel and Marcy Robertson.

### , Monday

#### , String Theory

Suresh Nampuri, Instituto Superior Técnico.

A Riemann-Hilbert approach to rotating attractors.

We study rotating attractor solutions from the point of view of a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. We describe an explicit vectorial Riemann-Hilbert factorization method which we use to show that the near-horizon limit of these extremal solutions can be constructed by Riemann-Hilbert factorization of monodromy matrices with poles of second order.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Monday

#### , Geometria em Lisboa

Cristiano Spotti, Centre for Quantum Geometry of Moduli Spaces, Aarhus.

Kähler-Einstein Fano varieties and their moduli spaces.

Possibly singular Fano varieties which admit Kähler-Einstein metrics are of particular interest since, among other things, they form compact separated moduli spaces. In the seminar, I will talk about existence results for these canonical metrics and describe examples of compact moduli spaces of these special varieties, explaining how the existence and moduli problems are intimately related to each other when looking for explicit examples of such Kähler-Einstein Fano varieties.

### , Friday

#### , Partial Differential Equations

Nicola Abatangelo, Université libre de Bruxelles.

Keller-Osserman type solutions for the fractional laplacian.

Large or boundary blow-up solutions — namely, solutions to elliptic Dirichlet problems prescribed to attain the value $+\infty$ at the boundary — are a useful tool in the analysis of nonlinear PDEs, whereas they show a deep connection with geometrical and probabilistic problems.

A systematic study of these solutions originates by the independent works of Keller (1957) and Osserman (1957), but the topic is even more classical and dates back to Bieberbach (1930).

We want to study whether and under what assumptions this boundary explosion can be spotted also in a fractional nonlocal framework, in which the *fractional Laplacian* operator is known to lose smoothness at the boundary. We will also give some characterization of the asymptotic behaviour and we will compare it with the one coming from the classical theory.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Wednesday

#### , Topological Quantum Field Theory

Aleksandar Mikovic, Universidade Lusófona.

Hamiltonian analysis of the BFCG theory for a generic Lie 2-group.

We perform a complete Hamiltonian analysis of the BFCG action for a general Lie 2-group by using the Dirac procedure. We show that the resulting dynamical constraints eliminate all local degrees of freedom which implies that the BFCG theory is a topological field theory.

### , Monday

#### , Partial Differential Equations

Peter Hintz, University of California, Berkeley.

Non-linear stability of Kerr-de Sitter black holes.

I will explain some ideas behind the proof of the stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will explain the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein's equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges; I will also address the issue of finding the mass and the angular momentum of the final black hole. This talk is based on joint work with András Vasy.

### , 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

#### , 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

#### , 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

#### , 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.