# 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

#### , Mathematical Relativity

Melanie Graf, University of Vienna.

The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics.

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

### , Monday

#### , String Theory

Maximilian Schwick, Instituto Superior Técnico.

Introduction to resurgence.

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

### , Friday

#### , Geometria em Lisboa

Pedro Freitas, Instituto Superior Técnico.

The spectral determinant of the quantum harmonic oscillator in arbitrary dimensions.

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

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

### , Tuesday

#### , Geometria em Lisboa

Matias Del Hoyo, Universidade Federal Fluminense.

Discrete dynamics and differentiable stacks.

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

### , Tuesday

#### , String Theory

Panagiotis Betzios, University of Crete.

Matrix Quantum Mechanics and the $S^1/\mathbb{Z}_2$ orbifold.

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

### , Monday

#### , String Theory

Olga Papadoulaki, University of Southampton.

FZZT branes and non-singlets of Matrix Quantum Mechanics.

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

### , Thursday

#### , Algebra

Julien Ducoulombier, ETH Zurich.

Swiss Cheese operad and applications to embedding spaces.

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

### , Wednesday

#### , Mathematical Relativity

Oliver Lindblad Petersen, University of Potsdam.

Wave equations with initial data on compact Cauchy horizons.

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

### , Monday

#### , String Theory

Masazumi Honda, Weizmann Institute of Science.

Resurgent transseries and Lefschetz thimble in 3d $\mathcal{N}=2$ supersymmetric Chern-Simons matter theories.

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

### , Thursday

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

Anastasiia Panchuk, Academia Nacional das Ciências de Kiev.

A piecewise linear map with two discontinuities: bifurcation structures in the chaotic domain.

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

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

### , Monday

#### , String Theory

Roberto Vega, Instituto Superior Técnico.

Introduction to resurgence.

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

### , Tuesday

#### , Geometria em Lisboa

Pedro Boavida, CAMGSD, Instituto Superior Técnico, Universidade de Lisboa.

Spaces of smooth embeddings and the little disks operad.

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

### , Friday

#### , Mathematical Relativity

Katharina Radermacher, KTH Royal Institute of Technology.

On the Cosmic No-Hair Conjecture in $\mathbb{T}^2$-symmetric non-linear scalar field spacetimes.

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

This talk is based on arXiv:1712.01801.

### , Wednesday

#### , Mathematical Relativity

Masashi Kimura, Instituto Superior Técnico.

Mass Ladder Operators.

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

### , Wednesday

#### , Mathematical Relativity

Pedro Oliveira, Instituto Superior Técnico.

Cosmic no-hair in spherically symmetric black hole spacetimes.

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

### , Wednesday

#### , Mathematical Relativity

David Hilditch, Instituto Superior Tecnico.

Free-evolution formulations of GR for numerical relativity.

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

### , Tuesday

#### , Mathematical Relativity

Claes Uggla, Karlstads Universitet.

Aspects of cosmological perturbation theory.

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

### , Tuesday

#### , Geometria em Lisboa

Zuoqin Wang, University of Science and Technology of China Heifei.

Equivariant Eigenvalues on Manifolds with Large Symmetry.

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

### , Wednesday

#### , Mathematical Relativity

Moritz Reintjes, Instituto Superior Técnico.

The Question of Essential Metric Regularity at General Relativistic Shock Waves.

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