# 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

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

Brian Hall, University of Notre Dame.

Eigenvalues of random matrices in the general linear group.

I will begin by discussing the two standard prototype random matrix models, one for Hermitian matrices and one for general matrices. For large matrices, the eigenvalues follow the "semicircular law" in the first case and the "circular law" in the second case. Furthermore, there is a simple relationship between these two laws.

I will then discuss two "multiplicative" analogs of these models, in which the random matrices are chosen from the unitary group and the general linear group, respectively. In the unitary case, the limiting eigenvalue distribution was computed by Biane and exhibits an interesting phase transition when a certain scaling parameter equals 4. I will then describe recent results of mine with Driver and Kemp on the general linear case. The limiting distribution again undergoes a phase transition and turns out to have a remarkably simple structure. The talk will be self-contained with lots of pictures and possibly even a few jokes.

### , Tuesday

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

Brian Hall, University of Notre Dame.

Large-$N$ Segal-Bargmann transform with application to random matrices.

I will describe the Segal-Bargmann transform for compact Liegroups, with emphasis on the case of the unitary group $U(N)$. In this case, the transform is a unitary map from the space of $L^2$ functions on $U(N)$ to the space of $L^2$ holomorphic functions on the "complexified" group $\operatorname{GL}(N;\mathbb{C})$. I will then discuss what happens in the limit as $N$ tends to infinity. Finally, I will describe an application to the eigenvalues of random matrices in $\operatorname{GL}(N;\mathbb{C})$. The talk will be self-contained and have lots of pictures.

### , Monday

#### , Room P3.10, Mathematics Building, Geometria em Lisboa

Richard Laugesen, University of Illinois at Urbana-Champaign.

Spectrum of the Robin Laplacian: recent results, and open problems.

I will discuss two recent theorems and one recent conjecture about maximizing

or minimizing the first three eigenvalues of the Robin Laplacian of a simply

connected planar domain. Conformal mappings and winding numbers play a key

role in the geometric constructions. In physical terms, these eigenvalues

represent decay rates for heat flow assuming a “partially insulating” boundary.

### , Thursday

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

Simão Correia, Faculdade de Ciências, Universidade de Lisboa.

Critical well-posedness for the modified Korteweg-de Vries equation and self-similar dynamics.

We consider the modified Korteweg-de Vries equation over $\mathbb{R}$ $$ u_t + u_{xxx}=\pm(u^3)_x. $$ This equation arises, for example, in the theory of water waves and vortex filaments in fluid dynamics. A particular class of solutions to (mKdV) are those which do not change under scaling transformations, the so-called *self-similar* solutions. Self-similar solutions blow-up when $t\to 0$ and determine the asymptotic behaviour of the evolution problem at $t=+\infty$. The known local well-posedness results for the (mKdV) fail when one considers critical spaces, where the norm is scaling-invariant. This means that self-similar solutions lie outside of the scope of these results. Consequently, the dynamics of (mKdV) around self-similar solutions are currently unknown. In this talk, we will show existence and uniqueness of solutions to the (mKdV) lying on a critical space which includes both regular and self-similar solutions. Afterwards, we present several results regarding global existence, asymptotic behaviour at $t=+\infty$ and blow-up phenomena at $t=0$. This is joint work with Raphaël Côte and Luis Vega.

### , Tuesday

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

Diogo Arsénio, New York University Abu Dhabi.

Recent progress on the mathematical theory of plasmas.

The incompressible Navier–Stokes–Maxwell system is a classical model describing the evolution of a plasma (i.e. an electrically conducting fluid). Although small smooth solutions to this system (in the spirit of Fujita–Kato) are known to exist, the existence of large weak solutions (in the spirit of Leray) in the energy space remains unknown. This defect can be attributed to the difficulty of coupling the Navier–Stokes equations with a hyperbolic system. In this talk, we will describe recent results aiming at building solutions to Navier–Stokes–Maxwell systems in large functional spaces. In particular, we will show, for any initial data with finite energy, how a smallness condition on the electromagnetic field alone is sufficient to grant the existence of global solutions.

### , Wednesday

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

Phillipo Lappicy, Universidade de São Paulo.

Space of initial data for self-similar Schwarzschild solutions.

The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic equation on the sphere with various singularities and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar Schwarzschild solutions. Those describe, for example, the initial data for the interior of black holes. We construct the space of initial data for such solutions and show that the metric at the event horizon is constrained to the global attractors of such parabolic equations. Lastly, some properties of those attractors and its solutions are explored.

### , Tuesday

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

Cédric Bernardin, University of Nice Sophia-Antipolis.

Microscopic models for multicomponents SPDE’s with a KPZ flavor.

The usual KPZ equation is the scaling limit of weakly asymmetric microscopic models with one conserved quantity. In this talk I will present some weakly asymmetric microscopic models with several conserved quantities for which it is possible to derive macroscopic SPDEs with a KPZ flavor.

Joint work with R. Ahmed, T. Funaki, P. Gonçalves, S. Sethuraman and M. Simon.

### , Monday

#### , Room P3.10, Mathematics Building, String Theory

Vishnu Jejjala, University of the Witwatersrand.

Experiments with Machine Learning in Geometry & Physics.

Identifying patterns in data enables us to formulate questions that can lead to exact results. Since many of the patterns are subtle, machine learning has emerged as a useful tool in discovering these relationships. We show that topological features of Calabi–Yau geometries are machine learnable. We indicate the broad applicability of our methods to existing large data sets by finding relations between knot invariants, in particular, the hyperbolic volume of the knot complement and the Jones polynomial.

### , Wednesday

#### , Room P4.35, Mathematics Building, Mathematical Relativity

Carlos Herdeiro, Instituto Superior Técnico.

Light ring stability in ultra-compact objects.

We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities. Our result implies that smooth, physically and dynamically reasonable ultracompact objects are not viable as observational alternatives to black holes whenever these instabilities occur on astrophysically short time scales. The proof of the theorem has two parts: (i) We show that light rings always come in pairs, one being a saddle point and the other a local extremum of an effective potential. This result follows from a topological argument based on the Brouwer degree of a continuous map, with no assumptions on the spacetime dynamics, and hence it is applicable to any metric gravity theory where photons follow null geodesics. (ii) Assuming Einstein's equations, we show that the extremum is a local minimum of the potential (i.e., a stable light ring) if the energy-momentum tensor satisfies the null energy condition.

### , Tuesday

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

Conrado Costa, Leiden University.

Random walks in cooling random environments: stable and unstable behaviors under regular diverging cooling maps.

Random Walks in Cooling Random Environments (RWCRE), a model introduced by L. Avena, F. den Hollander, is a dynamic version of Random Walk in Random Environment (RWRE) in which the environment is fully resampled along a sequence of deterministic times, called refreshing times. In this talk I will consider effects of the ressampling map on the fluctuations associated with the annealed law and the Large Deviation principle under the quenched measure. I conclude clarifying the paradox of different fluctuations and identical LDP for RWCRE and RWRE. This is a joint work with L. Avena, Y. Chino, and F. den Hollander.

### , Thursday

#### , Room P4.35, Mathematics Building, Geometria em Lisboa

Isabelle Charton, University of Cologne.

Hamiltonian $S^1$-spaces with large equivariant pseudo-index.

Let \((M,\omega)\) be a compact symplectic manifold of dimension \(2n\) endowed with a Hamiltonian circle action with only isolated fixed points. Whenever \(M\) admits a toric \(1\)-skeleton \(\mathcal{S}\), which is a special collection of embedded \(2\)-spheres in \(M\), we define the notion of equivariant pseudo-index of \(\mathcal{S}\): this is the minimum of the evaluation of the first Chern class \(c_1\) on the spheres of \(\mathcal{S}\).

In this talk we will discuss upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of \(M\) are unimodal, we prove that it is at most \(n+1\) . Moreover, when it is exactly \(n+1\), \(M\) must be homotopically equivalent to \(\mathbb{C}P^n\).

### , Thursday

#### , Room P3.10, Mathematics Building, Algebra

Mark Lawson, Heriot-Watt University.

Non-commutative Boolean algebras.

In this talk, I shall explain how the classical theory of Stone duality may be generalized to a non-commutative setting. This theory has connections with étale groupoids, quantales, groups and inverse semigroups.

Some of the work was joint with Alina Vdovina. I shall assume no prior exposure to this theory.

### , Tuesday

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

Federico Cantero, University of Barcelona, Spain.

Higher Steenrod squares for Khovanov homology.

We describe stable cup-$i$ products on the cochain complex with $\mathbb{F}_2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of any link.

### , Tuesday

#### , Room P5.18, Mathematics Building, String Theory

Nils Carqueville, University of Vienna.

TQFTS, Orbifolds and Topological Quantum Computation.

I will review basic notions and results in topological quantum field theory and discuss its orbifolds, with the aim to apply them in the context of topological quantum computation.

### , Monday

#### , Room P3.10, Mathematics Building, String Theory

Ceyda Simsek, University of Groningen.

Spacetime geometry of non-relativistic string theory.

Non-relativistic string theory is described by a sigma model that maps a two dimensional string worldsheet to a non-relativistic spacetime geometry. We discuss recent developments in understanding the spacetime geometry of non-relativistic string theory trying to provide several new insights. We show that the non-relativistic string action admits a surprisingly large number of symmetries. We introduce a non-relativistic limit to obtain the non-relativistic string action which also provides us the non-relativistic T-duality transformation rules and spacetime equations of motion.

### , Wednesday

#### , Room P4.35, Mathematics Building, Algebra

João Fontinha, ETH Zurich.

A primer on the Section Conjecture — a bridge between arithmetic and homotopy.

In 1983, Grothendieck wrote a letter to Faltings in which he formulated a conjecture for hyperbolic curves over fields which are finitely generated over the rationals. Remaining open to date, it carries the study of rational points on an algebraic variety to the realm of profinite groups. Assuming only a working knowledge of basic Algebraic Geometry, we formulate and motivate the Section Conjecture and outline some modern attempts to tackle it.

### , Tuesday

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

Phillipo Lappicy, ICMC, Universidade de São Paulo e CAMGSD-IST, Universidade de Lisboa.

A nonautonomous Chafee-Infante attractor: a connection matrix approach.

The goal of this talk is to present the construction of the global attractor for a genuine nonautonomous variant of the Chafee-Infante parabolic equation in one spatial dimension. In particular, the attractor consists of asymptotic profiles (which correspond to the equilibria in the autonomous counterpart) and heteroclinic solutions between those profiles. We prove the existence of heteroclinic connections between periodic and almost periodic asymptotic profiles, yielding the same connection structure as the well-known Chafee-Infante attractor. This work is still an ongoing project with Alexandre N. Carvalho (ICMC - Universidade de São Paulo).

### , Thursday

#### , Room P3.10, Mathematics Building, Geometria em Lisboa

Gleb Smirnov, ETH Zurich.

Symplectic triangle inequality.

This talk will be concerned with handling problems about embedding Lagrangians in symplectic four-manifolds where the target manifold is rational. In particular, we will determine those three-fold blow-ups of the symplectic ball which admit an embedded Lagrangian projective plane.

### , Tuesday

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

Clement Erignoux, Università Roma Tre.

Hydrodynamics for a non-ergodic facilitated exclusion process.

The Entropy Method introduced by Guo, Papanicolaou and Varadhan (1988) has been used with great sucess to derive the scaling hydrodynamic behavior of wide ranges of conserved lattice gases (CLG). It requires to estimate the entropy of the measure of the studied process w.r.t. some good, usually product measure. In this talk, I will present an exclusion model inspired by a model introduced by Gonçalves, Landim, Toninelli (2008), with a dynamical constraint, where a particle at site $x$ can only jump to $x+\delta$ iff site $x-\delta$ is occupied as well. I will give some insight on the different microscopic and macroscopic situations that can occur for this model, and briefly describe the steps to derive the hydrodynamic limit for this model by adapting the Entropy Method to non-product reference measures. I will also expand on the challenges and question raised by this model and on some of its nice mapping features. Joint work with O. Blondel, M. Sasada, and M. Simon.

### , Monday

#### , Room P3.10, Mathematics Building, String Theory

Davide Masoero, Faculdade de Ciências, Universidade de Lisboa.

Meromorphic opers and the Bethe Ansatz.

The Bethe Ansatz equations were initially conceived as a method to solve some particular Quantum Integrable Models (IM), but are nowadays a central tool of investigation in a variety of physical and mathematical theories such as string theory, supersymmetric gauge theories, and Donaldson-Thomas invariants. Surprisingly, it has been observed, in several examples, that the solutions of the same Bethe Ansatz equations are provided by the monodromy data of some ordinary differential operators with an irregular singularity (ODE/IM correspondence).

In this talk I will present the results of my investigation on the ODE/IM correspondence in quantum $g$-KdV models, where $g$ is an untwisted affine Kac-Moody algebra. I will construct solutions of the corresponding Bethe Ansatz equations, as the (irregular) monodromy data of a meromorphic $L(g)$-oper, where $L(g)$ denotes the Langlands dual algebra of $g$.

The talk is based on:

- D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case. Comm. Math. Phys. (2016)
- D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections II: The nonsimply-laced case. Comm. Math. Phys. (2017)
- D Masoero, A Raimondo, Opers corresponding to Higher States of the $g$-Quantum KdV model. arXiv 2018.

### , Tuesday

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

Ofer Busani, University of Bristol.

Transversal fluctuations in last passage percolation.

In Last Passage Percolation(LPP) we assign i.i.d Exponential weights on the lattice points of the first quadrant of $\mathbb{Z}^2$. We then look for the up-right path going from $(0,0)$ to $(n,n)$ that collects the most weights along the way. One is then often interested in questions regarding (1) the total weight collected along the maximal path, and (2) the behavior of the maximal path. It is known that this path's fluctuations around the diagonal is of order $n^{2/3}$. The proof, however, is only given in the context of integrable probability theory where one relies on some algebraic properties satisfied by the Exponential Distribution. We give a probabilistic proof for this phenomenon where the main novelty is the probabilistic proof for the lower bound. Joint work with Marton Balazs and Timo Seppalainen

### , Thursday

#### , Room P3.10, Mathematics Building, Algebra

Rachid El Harti, Univ. Hassan I, Morocco.

Amenable algebras: algebraic and analytical perspectives.

In this talk, we investigate the amenability of algebras from algebraic and analytical viewpoints.

We also consider its relationship with the

- semi-simplicity of operator algebras and
- crossed product Banach algebras associated with a class of $C^\ast$-dynamical systems.

### , Wednesday

#### , Room P4.35, Mathematics Building, Algebra

Christopher Deninger, University of Muenster.

Dynamical systems for arithmetic schemes - the higher dimensional case.

Extending the colloquium lecture, which essentially deals with $\operatorname{spec} \mathbb{Z}$ we discuss the general case of our construction of dynamical systems for arithmetic schemes. Functoriality and the relation to rational Witt vectors and Fontaine's $p$-adic period ring $A_\inf$ will also be explained if time permits.

### , Wednesday

#### , Room P5.18, Mathematics Building, Mathematical Relativity

José Natário, Instituto Superior Técnico.

Elastic shocks in relativistic rigid rods and balls.

We study the free boundary problem for the "hard phase" material introduced by Christodoulou, both for rods in $(1+1)$-dimensional Minkowski spacetime and for spherically symmetric balls in $(3+1)$-dimensional Minkowski spacetime. Unlike Christodoulou, we do not consider a "soft phase", and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks, defined as hypersurfaces where the material's density, pressure and velocity are discontinuous, must be null hypersurfaces. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

### , Tuesday

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

Nicola Vassena, Free University Berlin.

Introduction to sensitivity of chemical reaction networks.

This talk is an introductory overview of my research topic: Sensitivity of Networks.

We address the following questions: How does a dynamical network respond to perturbations of equilibrium - qualitatively? How does a perturbation of a targeted component spread in the network? What is the sign of the response?

In more detail, we consider general systems of differential equations inspired from chemical reaction networks: $dx/dt = S r(x)$. Here, $x$ might be interpreted as the vector of the concentrations of chemicals, $S$ is the stoichiometric matrix and $r(x)$ is the vector of reaction functions, which we consider as positive given parameters. Abstractly - for a given directed network: the vector $x$ represents the vertices, the matrix $S$ is the incidence matrix and the vector $r(x)$ refers to the directed arrows.

Sensitivity studies the response of equilibrium solutions to perturbations of reaction rate functions, using the network structure as ONLY data. We give here an introduction of the results and techniques developed through this structural approach.

### , Monday

#### , Room P3.10, Mathematics Building, Algebra

Paulo Lima-Filho, Texas A&M University.

Equidimensional algebraic cycles and current transforms.

In this talk we show how equidimensional algebraic correspondences between complex algebraic varieties can be used to construct pull-backs and transforms of a class of currents representable by integration. As a main application we exhibit explicit formulas at the level of complexes for a regulator map from the Higher Chow groups of smooth quasi-projective complex algebraic varieties to Deligne-Beilinson with integral coefficients.

We exhibit a few examples and indicate how this can be applied to Voevodsky’s motivic complexes. This is joint work with Pedro dos Santos and Robert Hardt.

### , Thursday

#### , Room P3.10, Mathematics Building, Geometria em Lisboa

Adela Mihai, Technical University of Civil Engineering Bucharest.

On Einstein spaces.

A Riemannian manifold $(M,g)$ of dimension $n \ge 3$ is called an Einstein space if $\operatorname{Ric} = \lambda \cdot \operatorname{id}$, where trivially $\lambda = \kappa$, with $\kappa$ the (normalized) scalar curvature; in this case one easily proves that $\lambda = \kappa = \operatorname{constant}$.

We recall the fact that any $2$-dimensional Riemannian manifold satisfies the relation $\operatorname{Ric} = \lambda \cdot \operatorname{id}$, but the function $\lambda= \kappa $ is not necessarily a constant. It is well known that any $3$-dimensional Einstein space has constant sectional curvature. Thus the interest in Einstein spaces starts with dimension $n=4$.

Singer and Thorpe discovered a symmetry of sectional curvatures which characterizes $4$-dimensional Einstein spaces. Later, this result was generalized to Einstein spaces of even dimensions $n = 2k \ge 4.$ We established curvature symmetries for Einstein spaces of arbitrary dimension $n \ge 4$.

### , Thursday

#### , Seminar room (2.8.3), Physics Building, Mathematical Relativity

Jarrod Williams, Queen Mary, University of London.

The Friedrich-Butscher method for the construction of initial data in General Relativity.

The construction of initial data for the Cauchy problem in General Relativity is an interesting problem from both the mathematical and physical points of view. As such, there have been numerous methods studied in the literature the "Conformal Method" of Lichnerowicz-Choquet-Bruhat-York and the "gluing" method of Corvino-Schoen being perhaps the best-explored. In this talk I will describe an alternative, perturbative, approach proposed by A. Butscher and H. Friedrich, and show how it can be used to construct non-linear perturbations of initial data for spatially-closed analogues of the $k = -1$ FLRW spacetime. Time permitting, I will discuss possible renements/extensions of the method, along with its generalisation to the full Conformal Constraint Equations of H. Friedrich.

### , Tuesday

#### , Room P3.10, Mathematics Building, Geometria em Lisboa

Claude LeBrun, Stonybrook.

Einstein Metrics, Harmonic Forms, and Conformally Kaehler Geometry.

There are certain compact 4-manifolds, such as real and complex hyperbolic 4-manifolds, 4-tori, and K3, where we completely understand the moduli space of Einstein metrics. But there are vast numbers of other 4-manifolds where we know that Einstein metrics exist, but cannot currently determine whether or not there are other Einstein metrics on them that are quite different from the currently-known ones. In this lecture, I will first present a characterization of the known Einstein metrics on Del Pezzo surfaces which I proved several years ago, and then describe an improved version which I obtained only quite recently.

### , Friday

#### , 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 PDEs; 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.

### , Thursday

#### , Room P3.10, Mathematics Building, Geometria em Lisboa

Lars Setktnan, UQUAM Montréal.

Blowing up extremal Poincaré type manifolds.

Metrics of Poincaré type are Kähler metrics defined on the complement $X\setminus D$ of a smooth divisor $D$ in a compact Kähler manifold $X$ which near $D$ are modeled on the product of a smooth metric on $D$ with the standard cusp metric on a punctured disk in $\mathbb{C}$. In this talk I will discuss an Arezzo-Pacard type theorem for such metrics. A key feature is an obstruction which has no analogue in the compact case, coming from additional cokernel elements for the linearisation of the scalar curvature operator. I will discuss that even in the toric case, this gives an obstruction to blowing up fixed points (which is unobstructed in the compact case). This additional condition is conjecturally related to ensuring the metrics remain of Poincaré type.

### , Wednesday

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

Artur Alho, Instituto Superior Técnico.

Multi-body spherically symmetric steady states of Newtonian self-gravitating elastic matter.

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

### , Wednesday

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

Gonçalo Quinta & Rui André, Physics of Information and Quantum Technologies Group - IST (GQ); Center for Astrophysics and Gravitation - IST (RA).

Topological Links and Quantum Entanglement.

We present a classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for any given number of rings. We demonstrate the use of this new classification scheme for three and four qubits and its potential in the context of qubit networks.

### , Tuesday

#### , Room P3.10, Mathematics Building, Geometria em Lisboa

Bruno Oliveira, University of Miami.

Big jet-bundles on resolution of orbifold surfaces of general type.

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

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

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

### , Friday

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

Pedro Girão, Instituto Superior Técnico.

Solutions of the wave equation bounded at the Big Bang.

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

### , Wednesday

#### , Seminar room (2.8.3), Physics Building, Mathematical Relativity

Juan Antonio Valiente Kroon, Queen Mary, University of London.

Construction of anti de Sitter-like spacetimes using the metric conformal field equations.

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

### , Tuesday

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

Federico Sau, Delft University.

Self-duality for conservative interacting particle systems.

### , Thursday

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

Noa Zilberman, Technion.

Quantum effects near the inner horizon of a black hole.

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

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

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

### , Wednesday

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

Anne Franzen, Instituto Superior Técnico.

Flat FLRW and Kasner Big Bang singularities analyzed on the level of scalar waves.

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

### , Wednesday

#### , Room P3.10, Mathematics Building, Algebra

Ismar Volic, Wellesley College.

Cohomology of braids, graph complexes, and configuration space integrals.

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

### , Thursday

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

Marco Mackaay, Universidade do Algarve.

The 2-representation theory of Soergel bimodules of finite Coxeter type: a road map to the complete classification of all simple transitive 2-representations.

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

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

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

### , Monday

#### , Room P3.10, Mathematics Building, Geometria em Lisboa

Hugues Auvray, Université Paris-Sud.

Complete extremal metrics and stability of pairs on Hirzebruch surfaces.

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

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

This is joint work with Vestislav Apostolov and Lars Sektnan.