# 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

Paul Tod, University of Oxford.

Penrose’s Weyl Curvature Hypothesis and his Conformal Cyclic Cosmology.

Penrose’s Weyl Curvature Hypothesis, which dates from the late 70s, is a hypothesis, motivated by observation, about the nature of the Big Bang as a singularity of the space-time manifold. His Conformal Cyclic Cosmology is a remarkable suggestion, made a few years ago and still being explored, about the nature of the universe, in the light of the current consensus among cosmologists (and the Nobel Committee) that there is a positive cosmological constant. I shall review both sets of ideas within the framework of general relativity, emphasise how the second set solves a problem posed by the first, and say something about predictions of CCC.

### , Tuesday

#### , Geometria em Lisboa

Marcos Mariño, University of Geneva.

Spectral theory and enumerative geometry.

In recent years, a surprising correspondence has been found between the spectral theory of certain trace class operators, and the enumerative geometry of certain Calabi-Yau threefolds. This correspondence leads to a new family of exactly solvable operators in spectral theory, as well as to a new point of view on Gromov-Witten theory. In this overview talk I will introduce the conjecture and some developments inspired by it.

### , Monday

#### , String Theory

Junya Yagi, Perimeter Institute.

String theory and integrable lattice models.

I will discuss a string theoretic approach to integrable lattice models. This approach provides a unified perspective on various important notions in lattice models, and relates these notions to four-dimensional N =1 supersymmetric field theories and their surface operators. I will also explain how my construction connects to Costello's work and the Nekrasov-Shatashvili correspondence.

### , Wednesday

#### , Topological Quantum Field Theory

João Miguel Nogueira, Universidade de Coimbra.

Meridional essential surfaces of unbounded Euler characteristics in knot exteriors.

In this talk we will discuss further the existence of knot exteriors with essential surfaces of unbounded Euler characteristics. More precisely, we show the existence of a knot with an essential tangle decomposition for any number of strings. We also show the existence of knots where each exterior contains meridional essential surfaces of simultaneously unbounded genus and number of boundary components. In particular, we construct examples of knot exteriors each of which having all possible compact surfaces embedded as meridional essential surfaces.

### , Tuesday

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

Leonardo de Carlo, CAMGSD, Instituto Superior Técnico.

Geometric and combinatoric structures in stationary Markov chains.

We study the combinatoric and geometric structure of stationary non-reversible Markov chains defined on graphs, in particular in applications we focus on non-equilibrium states for interacting particle systems from a microscopic viewpoint. We discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field. This decomposition applied to the instantaneous current of any interacting particle system on a finite torus tell us that it can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components can be computed and are associated with functions on the configuration space.

### , Monday

#### , String Theory

Michele Cirafici, Instituto Superior Técnico.

On two applications of persistent homology to string theory vacua.

Persistent homology studies which homological features of a topological space persist over a long range of scales. I will discuss two applications of this formalism to the study of vacua in string theory. In the first application, I will discuss how to adapt such techniques to address the presence/absence of structure in a series of string compactifications (for example flux vacua in type IIB or heterotic vacua). In the second application, I will address the problem of studying vacua of certain two-dimensional Landau-Ginzburg models and see what information we can get about their algebraic structures.

### , Friday

#### , Topological Quantum Field Theory

Carlos Florentino, Universidade de Lisboa.

Geometry, Topology and Arithmetic of character varieties.

Character varieties are spaces of representations of finitely presented groups $F$ into Lie groups $G$. When $F$ is the fundamental group of a surface, these spaces play a key role both in Chern-Simons theory and in 2d conformal field theory. In some cases, they are also interpreted as moduli spaces of $G$-Higgs bundles over Kähler manifolds, and were recently studied in connection with the geometric Langlands program, and with mirror symmetry. When $G$ is a complex algebraic group, character varieties are algebraic and have interesting geometry and topology. We can also consider more refined invariants such as Deligne's mixed Hodge structures, which are typically very difficult to compute, but also provide relevant arithmetic information.

In this seminar, we present some explicit computations of the mixed Hodge-Deligne polynomials, and the so-called E-polynomials, of $G$-character varieties of free, and free abelian groups, when $G$ is a group such as $\operatorname{SL}(n,\mathbb{C})$, $(P)\operatorname{GL}(n,\mathbb{C})$ or $\operatorname{Sp}(n,\mathbb{C})$. We also comment on interesting relations between the free case and some explicit formulas by Reineke-Mozgovoy on counting quiver representations over finite fields.

This is joint work with A. Nozad, J. Silva and A. Zamora.

### , Thursday

#### , String Theory

Suresh Nampuri, Instituto Superior Técnico.

Hot Attractors and Area Laws.

We introduce a notion of attractors and attractive geometry in the context of static four-dimensional non-extremal black holes, and we establish constraints on the flow of scalars in these black hole backgrounds to ultimately derive area laws pertaining to horizons in these backgrounds that were hitherto known empirically.

### , Thursday

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

Alexandre Boritchev, Institut Camille Jordan, Université Lyon 1.

A random particle system and nonentropy solutions of the Burgers equation on the circle.

We consider a particle system which is equivalent to a process valued on the space of nonentropy solutions of the inviscid Burgers equation. Such solutions are conjectured to be relevant for the study of the KPZ fixed point. We prove ergodicity and obtain some properties of the stationary measure.

Joint work with C.-E. Bréhier (Lyon) and M. Mariani (Rome).

### , Thursday

#### , Partial Differential Equations

Martin Taylor, Imperial College London.

Global nonlinear stability of Minkowski space for the massive and massless Einstein-Vlasov systems.

The Einstein-Vlasov system describes an ensemble of collisionless particles interacting via gravity, as modelled by general relativity. Under the assumption that all particles have equal mass there are two qualitatively different cases according to whether this mass is zero or nonzero. I will present two theorems concerning the global dispersive properties of small data solutions in both cases. The massive case is joint with Hans Lindblad.

### , Thursday

#### , String Theory

Gabriel Lopes Cardoso, Instituto Superior Técnico.

Exact results in the STU model.

We consider a specific type II string theory model with $N=2$ local supersymmetry, the so-called STU model. This is a model with exact duality symmetries. Using holomorphy and duality, we obtain exact results for this model that go beyond the perturbative formulation of topological string theory.

### , Tuesday

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

Daniel Rodrigues, University of Groningen.

Percolation on the stationary distribution of the voter model on $\mathbb{Z}^d$.

The voter model on ${\mathbb Z}^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When the model is considered in dimension 3 or higher, its set of (extremal) stationary distributions is equal to a family of measures $\mu_\alpha$, for $\alpha$ between 0 and 1. A configuration sampled from $\mu_\alpha$ is a field of 0's and 1's on ${\mathbb Z}^d$ in which the density of 1's is $\alpha$. We consider such a configuration from the point of view of site percolation on ${\mathbb Z}^d$. We prove that in dimensions 5 and higher, the probability of existence of an infinite percolation cluster exhibits a phase transition in $\alpha$. If the voter model is allowed to have long range, we prove the same result for dimensions 3 and higher. Joint work with Balázs Ráth.

### , Monday

#### , Topological Quantum Field Theory

Alissa Crans, Loyola Marymount University, USA.

Crossed Modules of Racks.

A rack is a set equipped with two binary operations satisfying axioms that capture the essential properties of group conjugation and algebraically encode two of the three Reidemeister moves. We will begin by generalizing Whitehead's notion of a crossed module of groups to that of a crossed module of racks. Motivated by the relationship between crossed modules of groups and strict 2-groups, we then will investigate connections between our rack crossed modules and categorified structures including strict 2-racks and trunk-like objects in the category of racks. We will conclude by considering topological applications, such as fundamental racks. This is joint work with Friedrich Wagemann.

### , Thursday

#### , Algebra

Geoffroy Horel, Université Paris XIII.

Mixed Hodge structure, Galois actions and formality.

Given a dg-algebra or any algebraic structure in chain complexes, one may ask if it is quasi-isomorphic to its homology equipped with the zero differential. This property is called formality and has important consequences in algebraic topology. For example it forces the collapse of certain spectral sequences. In this talk I will explain how mixed Hodge structures can be used to prove formality when working with rational coefficients. I will also explain work in progress using Galois actions as a replacement for mixed Hodge structures in the case of torsion coefficients. This is joint work with Joana Cirici.

### , Wednesday

#### , Partial Differential Equations

Blake Temple, University of California, Davis.

An Instability of the Standard Model of Cosmology Creates the Anomalous Acceleration Without Dark Energy.

We identify the condition for smoothness at the center of spherically symmetric solutions of Einstein’s original equations (without the cosmological constant), and use this to derive a universal phase portrait which describes general, smooth, spherically symmetric solutions near the center of symmetry when the pressure $p=0$. In this phase portrait, the critical $k=0$, $p=0$ Friedmann spacetime appears as an unstable saddle rest point. This phase portrait tells us that the Friedmann spacetime is unstable to spherical perturbations when the pressure drops to zero, no matter what point is taken to be the center, and in this sense it is unobservable by redshift vs luminosity measurements looking outward from any point. Moreover, the unstable manifold of the saddle rest point corresponding to Friedmann describes the evolution of smooth perturbations, and we show that this evolution creates local uniformly expanding spacetimes which, in the under-dense case, create precisely the same range of quadratic corrections to redshift vs luminosity as does Dark Energy in the form of a nonzero cosmological constant. Thus the anomalous accelerations inferred from the supernova data are not only consistent with Einstein’s original theory of GR, but are a prediction of it.

This is joint work with Joel Smoller and Zeke Vogler.

### , Tuesday

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

David Krejcirik, Czech Technical University in Prague.

Absence of eigenvalues of Schrödinger operators with complex potentials.

We prove that the spectrum of Schrödinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of multipliers, we also establish the absence of point spectrum for electromagnetic Schrödinger operators in all dimensions under various alternative hypotheses, still allowing complex-valued potentials with critical singularities. This is joint work with Luca Fanelli and Luis Vega.

### , Thursday

#### , Partial Differential Equations

Levon Nurbekyan, Kaust.

First-order, stationary mean-field games with congestion.

Mean-field games (MFGs) are models for large populations of competing, rational agents that seek to optimize a suitable functional. In the case of congestion, this functional takes into account the difficulty of moving in high-density areas.

In this talk, I will present a recent contribution on MFGs with congestion with power-like Hamiltonians. First, using explicit examples, I will illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing density. Next, I will present our main contribution - a new variational formulation for MFGs with congestion. This formulation was not previously known, and, thanks to it, we prove the existence and uniqueness of solutions. Consequently, we show that the MFG congestion model under consideration is well-posed for a far larger set of parameters than it was previously known.

Furthermore, I will discuss various new transformations for MFGs with congestion that, in some cases, significantly simplify the problem. Finally, I will present some numerical applications.

### , Tuesday

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

Adriana Neumann, Universidade Federal do Rio Grande do Sul.

Asymptotic behavior of the exclusion process with slow boundary.

The system of interacting particles that will be presented (the exclusion process with slow boundary) arouses considerable interest in its applicability, for modeling mass transfer between reservoirs with different densities. But it also arouses interest in its theoretical part because of its non-triviality, for example: the invariant measure is given through matrices of Ansatz, see Derrida. Another interesting theoretical aspect, which will be the main focus of this talk, is the behavior of particles density (hydrodynamic limit) is given by the heat equation with boundary conditions. These boundary conditions have phase transition, which depends on how slow the behavior at the border is. More specifically, if the boundary has transfer rate of the order of $N^{- a}$, where $N$ is the scale parameter and $a$ is a fixed non-negative real number, then we get for $a$ in $[0,1)$, Dirichlet boundary conditions, for $a\gt 1$ Neumann boundary conditions and the critical case is when $a=1$, which has Robin boundary conditions. In this talk, in addition to the hydrodynamic limit, other results for the scale limits of this model will be presented, such as fluctuations and large deviations.

### , Wednesday

#### , Topological Quantum Field Theory

Luis Miguel Pereira, IPFN, Instituto Superior Tecnico.

Techniques for the summation of hypergeometric series and the quantum pendulum.

In this informal seminar we will give a presentation based on practical examples of some of the several methods that can be used to sum hypergeometric series. These series include several known special functions and almost all combinatorial sums. Questions from the public will be welcomed. The goal of the seminar will ultimately be to set the stage for the preparation of a strategy to attack the problem of finding closed form solutions for the problem of the quantum pendulum (that is, to find closed form solutions for the Fourier coefficients of Mathieu functions), from stationary solutions to this problem in the Wigner formalism that were obtained by the speaker.

### , Friday

#### , Partial Differential Equations

Heinrich Freistühler, University of Konstanz.

A Causal Five-Field Theory of Dissipative Relativistic Fluid Dynamics.

In the absence of dissipation, relativistic fluid dynamics is governed by the Euler equations which are a five-field theory: Its constituents, the energy-momentum tensor and the particle number density current are given in terms of the fluid’s velocity $U$, energy density $\rho$, pressure $p$, and particle number density $n$. The state description $(U, ρ, p, n)$ has five degrees of freedom, and the Euler equations are five partial differential equations which determine the spatiotemporal evolution of these five fields from general initial data.

Regarding the modeling of dissipation, i.e., viscosity and heat conduction, various theories have been suggested over the last almost eight decades. This talk deals with the question of whether dissipative relativistic fluid dynamics can be properly modeled by a causal five-field theory with dissipation tensors $\Delta T$, $\Delta N$ that are linear in the gradients of the five fields (“relativistic Navier-Stokes”). The question is answered in the affirmative. The proposed formulation is intimately related to the theory of second-order symmetric hyperbolic systems developed by Hughes, Kato, and Marsden on the one hand and to the classical (non-causal) descriptions given by Eckart and Landau on the other.

Joint work with B. Temple, UC Davis

### , Monday

#### , Geometria em Lisboa

Ailsa Keating, Cambridge University.

Examples of monotone Lagrangians.

Joint work with Mohammed Abouzaid. We present some methods for constructing examples of compact monotone Lagrangians in families of $(n-1)$-dimensional affine hypersurfaces; they can be upgraded to Lagrangians in $C^n$. We will then explain different strategies for telling them apart, using Floer homology, and, time allowing, some counts of holomorphic annuli. The talk will only assume minimal knowledge of symplectic geometry.

### , Thursday

#### , Topological Quantum Field Theory

Luis Miguel Pereira, IPFN, Instituto Superior Tecnico.

The quantum pendulum in the Wigner formalism and Mathieu functions.

The time-independent Schrödinger equation with the pendulum's potential is the Mathieu equation from 19th century mathematical physics. Though there are many ways to approximate its solutions there are no known closed formulas for these solutions. In this talk we will show that with João Pedro Bizarro's modification of the Wigner-Berry transform it is possible to obtain closed formulas for several families of transforms of stationary observables.

### , Wednesday

#### , Topological Quantum Field Theory

Beatriz Elizaga de Navascués, Instituto de Estructura de la Materia, Madrid.

Unitary dynamics as a uniqueness criterion for the quantization of Dirac fields.

It is well known that linear canonical transformations are not generally implemented as unitary operators in QFT. Such transformations include the dynamics that arises from linear field equations on the background spacetime. This evolution is specially relevant in nonstationary backgrounds, where there is no time-translational symmetry that can be exploited to select a quantum theory. We investigate whether it is possible to find a Fock representation for the canonical anticommutation relations of a Dirac field, propagating on homogeneous and isotropic cosmological backgrounds, on the one hand, and on tridimensional conformally ultrastatic spacetimes, on the other hand, such that the field evolution is unitarily implementable. First, we restrict our attention to Fock representations that are invariant under the group of symmetries of the system. Then, we prove that there indeed exist Fock representations such that the dynamics is implementable as a unitary operator. Finally, once a convention for the notion of particles and antiparticles is set, we show that these representations are all unitarily equivalent.

### , Wednesday

#### , Geometria em Lisboa

Miguel Abreu, Centro de Análise Matemática Geometria e Sistemas Dinâmicos, Instituto Superior Técnico.

Contact topology of Gorenstein toric isolated singularities.

Links of Gorenstein toric isolated singularities are good toric contact manifolds with zero first Chern class. In this talk I will present some results relating contact and singularity invariants in this particular toric context. Namely,

- I will explain why the contact mean Euler characteristic is equal to the Euler characteristic of any crepant toric smooth resolution of the singularity (joint work with Leonardo Macarini).
- I will discuss applications of contact invariants of Lens spaces that arise as links of Gorenstein cyclic quotient singularities (joint work with Leonardo Macarini and Miguel Moreira).

### , Wednesday

#### , Geometria em Lisboa

Marta Batoréo, Universidade Federal do Espírito Santo.

On periodic points of symplectomorphisms on closed manifolds.

In this talk, we will discuss symplectomorphisms on closed manifolds with periodic orbits. We will present some results on the existence of (infinitely many) periodic orbits of certain symplectomorphisms on closed manifolds. Moreover, we will give a construction of a symplectic flow on a closed surface of genus $g$ greater than $1$ with exactly $2g-2$ fixed points and no other periodic orbits.

### , Friday

#### , Geometria em Lisboa

Ana Rita Pires, Fordham University.

Symplectic embedding problems and infinite staircases, with some proofs.

### , Friday

#### , Geometria em Lisboa

Nick Sheridan, Princeton University.

Cubic fourfolds, K3 surfaces, and mirror symmetry.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Tuesday

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

Rachid El Harti, Univ. Hassan I.

C*-algebra valued numerical range for adjointable operators and some applications.

Let $A$ be C*-algebra and $E$ a Hilbert C*-module over $A$. For an adjointable operator $T$ on $E$, we define an $A$-valued numerical range $W(T)$ of $T$. We derive properties of $W(T)$ which are the analogs of the classical numerical range for operators acting on Hilbert space, including the Toeplitz-Hausdorff theorem and the equality between the numerical radius and the operator norm for normal adjointable operators.

### , Monday

#### , String Theory

João Rodrigues, University of the Witwatersrand.

Constructing $AdS_4$ from $3$ dimensional vector valued fields.

The singlet sectors of $O(N)$ vector field theories in three dimensions have been conjectured to be dual to theories of higher spins in $AdS_4$, in their large $N$ limit. We use the bilocal description of this invariant sector to obtain a constructive description of this correspondence.

### , Wednesday

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

Francesco Russo, University of Catania.

Some loci of rational cubic fourfolds.

We shall report on joint work with Michele Bolognesi and Giovanni Staglianò on the irreducible divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$. A general point of $\mathcal C_{14}$ is, by definition, a smooth cubic fourfold containing a smooth quartic rational normal scroll (or, equivalently, a smooth quintic del Pezzo surfaces) so that it is rational. We shall prove that **every** cubic fourfold contained in $\mathcal C_{14}$ is rational.

In passing we shall review and put in modern terms some ideas of Fano, yielding a geometric insight to some known results on cubic fourfolds, e.g. the Beauville-Donagi isomorphism, and discuss also the connections of our results with the recent examples about the bad behavior of rationality in smooth families of fourfolds.

### , Monday

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

Francesco Russo, University of Catania.

On (special versions of) the Hartshorne Conjecture on Complete Intersections.

We shall present some general techniques for studying projective embedded manifolds uniruled by lines, based on the Hilbert scheme of lines passing through a general point of the manifold and contained in it. The main applications will be the proofs of Hartshorne Conjecture for quadratic manifolds, of the classification of quadratic Hartshorne varieties, of the classification of Severi varieties. Our approach will show many connections between these problems, which were overlooked before, and also a uniform way of solving them. If time allows, we shall also discuss some open problems including the Barth-Ionescu Conjecture.

### , Wednesday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Tuesday

#### , Geometria em Lisboa

Gonçalo Oliveira, Duke University.

$G_2$-instantons on noncompact $G_2$-manifolds.

I will report on joint work with Jason Lotay on some existence and nonexistence results for $G_2$-instantons. I shall compare the behavior of $G_2$-instantons for two distinct $G_2$-holonomy metrics on $\mathbb{R}^4\times S^3$.

### , Tuesday

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

Louis H. Kauffman, University of Illinois at Chicago.

Knotoids and Virtual Knot Theory.

Knotoids are open-ended knot diagrams whose endpoints can be in different regions of the diagram. Two knotoids are said to be isotopic if there is a sequence of Reidemeister moves that connects one diagram to the other without moving arcs across endpoints. The definition is due to Turaev. We will discuss three dimensional interpretations of knotoids in terms of projections of open-ended embeddings of intervals into three dimensional space, and we shall discuss a number of invariants of knotoids based on concepts from virtual knot theory. Knotoids are a new branch of classical knot theory and they promise to provide a way to measure the “knottiness” of open interval embeddings in three space. This talk is joint work with Neslihan Gugumcu.

### , Wednesday

#### , Topological Quantum Field Theory

Louis H. Kauffman, University of Illinois, Chicago, USA.

Majorana Fermions, Braiding and Quantum Computing.

We will discuss the mathematics of Majorana Fermions and the structure of representations of the Artin Braid group that are associated with them. We will discuss how one class of representations is related to the Temperley Lieb algebra, and another class of representations is related to a Hamiltonian constructed from the Bell-Basis solution of the Yang-Baxter equation, and the relationship of this Hamiltonian with the Kitaev spin chain. We will discuss how these braiding representations are related to topological quantum computing.

### , Tuesday

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

Louis H. Kauffman, University of Illinois at Chicago.

Reconnection, Vortex Knots and the Fourth Dimension.

Vortex knots tend to unravel into collections of unlinked circles by writhe preserving reconnections. We can model this unravelling by examining the world line of the knot, viewing each reconnection as a saddle point transition. The world line is then seen as an oriented cobordism of the knot to a disjoint collection of circles. Cap each circle with a disk (for the mathematics) and the world line becomes an oriented surface in four-space whose genus can be no more than one-half the number of recombinations. Now turn to knot theory between dimensions three and four and find that this genus can be no less than one-half the Murasugi signature of the knot. Thus the number of recombinations needed to unravel a vortex knot $K$ is greater than or equal to the signature of the knot $K$. This talk will review the backgrounds that make the above description intelligible and we will illustrate with video of vortex knots and discuss other bounds related to the Rasmussen invariant. This talk is joint work with William Irvine.

### , Friday

#### , Partial Differential Equations

Jorge Drumond Silva, Instituto Superior Técnico.

Minicourse on Microlocal Analysis.

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

### , Wednesday

#### , Topological Quantum Field Theory

Emmanuel Wagner, Université de Bourgogne, France.

Trivalent TQFT and applications.

MOY calculus has been introduced in the 90s to compute combinatorially the quantum link invariant associated with the Hopf algebra $U_q(\mathfrak{sl}(N))$. It associates to any decorated graph a Laurent polynomial in $q$. I will describe a TQFT-like functor which categorifies the MOY calculus and provides a new description of the $\mathfrak{sl}(N)$-homology.

(joint work with L.-H. Robert)

### , 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.