## Search

## Mathematical Relativity

Decay of solutions to the Klein-Gordon equation on some expanding cosmological spacetimes.

*Amol Sasane*, London School of Economics.

## Abstract

The decay of solutions to the Klein-Gordon equation is studied in two expanding cosmological spacetimes, namely the de Sitter universe in flat Friedmann-Lemaître-Robertson-Walker (FLRW) form, and the cosmological region of the Reissner-Nordström-de Sitter (RNdS) model. Using energy methods, for initial data with finite higher order energies, decay rates for the solution are obtained. Also, a previously established decay rate of the time derivative of the solution to the wave equation, in an expanding de Sitter universe in flat FLRW form, is improved, proving Rendall's conjecture. A similar improvement is also given for the wave equation in the cosmological region of the RNdS spacetime.

## Analysis, Geometry, and Dynamical Systems

A Mini-course in large deviations (I).

*Tertuliano Franco*, Universidade Federal da Bahia.

## Abstract

Large deviations has importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations corresponds to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lowerbound large deviations.

## String Theory

Supersymmetric line operators and their spectral problem.

*Michele Cirafici*, University of Trieste.

## Abstract

I will discuss BPS invariants associated with quantum line operators in certain supersymmetric quantum field theories. Such operators can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR they are described by representation theory data. I will discuss the associated BPS spectral problem and the relevant indices.

Note: unusual date

## Topological Quantum Field Theory

Topological Field Theory in dimension 3.

*Manuel Araújo*, Instituto Superior Técnico.

## Abstract

I will give an overview of some aspects of 3d TFT, from the Turaev-Viro and Reshetikin-Turaev invariants of oriented 3-manifolds, to the more recent classifications of fully extended theories in terms of fusion categories and once extended theories in terms of Modular Tensor Categories.

## Algebra

Derived Geometry and its applications.

*Jorge António*, Université Paul Sabatier (Toulouse III).

## Abstract

In this talk, we are going to present a user-friendly approach to derived geometry. One of our goals is to convince the audience that the notions of derived manifold/scheme/space/stack are just as natural as their classical counterparts. After having introduced the basic techniques, we will apply them to study certain moduli spaces of geometrical origin. Derived geometry also has recently found applications in arithmetics, which we will try to explain in the last part of the talk.

## Analysis, Geometry, and Dynamical Systems

A Mini-course in large deviations (II).

*Tertuliano Franco*, Universidade Federal da Bahia.

## Abstract

Large deviations has importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations corresponds to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lower bound large deviations.

## Algebra

Dirac's theorem for random regular graphs.

*António Girão*, University of Birmingham.

## Abstract

In 1952, Dirac proved that any graph on $n$ vertices with minimum degree $n/2$ contains a Hamiltonian cycle, i.e. a cycle which passes through every vertex of the graph exactly once. We prove a resilience version of Dirac’s Theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\epsilon \gt 0$, a.a.s. the following holds: let $G_0$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2 + \epsilon)d$. Then, $G_0$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved. This is joint work with Padraig Condon, Alberto Espuny Díaz, Daniela Kuhn and Deryk Osthus.

## Analysis, Geometry, and Dynamical Systems

A Mini-course in large deviations (III).

*Tertuliano Franco*, Universidade Federal da Bahia.

## Abstract

Large deviations has importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations corresponds to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lower bound large deviations.

## Probability and Statistics

Monitoring Image Processes.

*Wolfgang Schmid*, European University Viadrina, Department of Statistics, Frankfurt, Germany.

## Abstract

In recent years we observe dramatic changes in the way in which quality features of manufactured products are designed and inspected. The modeling and monitoring problems obtained by new inspection methods and fast multi-stream high-speed sensors are quite complex. These measurement tools are used in emerging technologies like, e.g., additive manufacturing. It has been shown that in these fields other types of quality characteristics have to be monitored. It is mainly not the mean, the variance, the covariance matrix or a simple profile which reflects the behavior of the quality characteristics but the shape, surfaces and images, etc. This is a new area for SPC. Note that more complicated characteristics arise in other fields of applications as well like, e.g., the monitoring of optimal portfolio weights in finance. Since in the last years many new approaches have been developed in the fields of image analysis, spatial statistics and for spatio-temporal modeling a huge amount of tools are available to model the underlying processes. Thus the main problem lies on the development of monitoring schemes for such structures.

In this talk new procedures for monitoring image processes are introduced. They are based on multivariate exponential smoothing and cumulative sums taking into account the local correlation structure. A comparison is given with existing methods. Within an extensive simulation study the performance of the analyzed methods is discussed.

The presented results are based on a joint work with Yarema Okhrin and Ivan Semeniuk.

## Analysis, Geometry, and Dynamical Systems

A short KPZ story.

*Alessandra Occelli*, Instituto Superior Técnico.

## Abstract

The aim of this talk is to present a few models in the Kardar–Parisi–Zhang (KPZ) universality class, a class of stochastic growth models that have been widely studied in the last 30 years. We will focus in particular on last passage percolation (LPP) models. They provide a *physical* description of combinatorial problems, such as Ulam's problem, in terms of zero temperature directed polymers; but also a geometric interpretation of an interacting particle system, the totally asymmetric simple exclusion process (TASEP); and of a system of queues and servers. Moreover, in the large time limit, they share statistical features with certain ensembles of random matrices.

## $QM^3$ Quantum Matter meets Maths

The geometry and topology of free fermions.

*Bruno Mera*, Security and Quantum Information Group of Instituto de Telecomunicações.

## Abstract

In this talk, rather than presenting new results, I will discuss relatively recent results in condensed matter which bring geometry and topology to the realm of quantum matter. In particular, I will focus on systems of free fermions. The ground state of a gapped translation invariant charge conserving free fermion Hamiltonian on a $d-$dimensional lattice can be described by a smooth map from a $d-$dimensional torus of quasi-momenta to a Grassmannian manifold. This map gives rise to a vector bundle over the torus whose isomorphism class determines the topological phase of the system. In particular, in $d = 2$, the Chern class can be naturally associated with the transverse Hall conductivity of the system. By considering families of systems of free fermions one can see phase transitions associated to the gap closing points in momentum space. The change in topology can be understood in terms of a transversal crossing of the image of the Hamiltonian in the space of Hermitian matrices with the subvariety formed by those matrices having multiple eigenvalues. Further geometric aspects will be discussed if time permits to do so.

## Analysis, Geometry, and Dynamical Systems

KPZ universality for last passage percolation models.

*Alessandra Occelli*, Instituto Superior Técnico.

## Abstract

In this seminar we consider last passage percolation on $\mathbb{Z}^2$, a model in the Kardar–Parisi–Zhang (KPZ) universality class. We will investigate the universality of the limit distributions of the last passage time for different settings. In the first part we analyze the correlations of two last passage times for different ending points in a neighbourhood of the characteristic. For a general class of random initial conditions, we prove the universality of the first order correction when the two observation times are close. In the second part we consider a model of last passage percolation in half-space and we obtain the distribution of the last passage time for the stationary initial condition.

## Geometria em Lisboa

Introduction to Mirror Symmetry on the Hitchin System.

*Emilio Franco*, CAMGSD, Instituto Superior Técnico.

## Abstract

This will be an introductory talk for the Working Seminar on Mirror Symmetry on the Hitchin System. During this minicourse, organized by T. Sutherland and myself, we aim to understand Mirror Symmetry on Higgs moduli spaces as a classical limit of the Geometric Langlands program. In this talk I will briefly describe the geometrical objects involved in this program and provide a motivation for it coming from mathematical physics. The structure of the working seminar will also be discussed.

## Geometria em Lisboa

To be announced.

*Michela Zedda*, Università di Parma.

## $QM^3$ Quantum Matter meets Maths

Continuous-Time Crystals in Open Quantum Systems.

*Angelo Carollo*, University of Palermo.

## Abstract

I will briefly review the idea, proposed by F.Wilczek, of "time crystals" as

phases that spontaneously break the continuous time translation into a discrete subgroup. Recently, it has been demonstrated that the possibility of time crystals at equilibrium, i.e. in the canonical ensemble of "not-too-long-range" Hamiltonians, is ruled out. The search for time crystals in non-equilibrium systems remains still open. I will show that in the general context of a stable state of a "not-too-long-range" Markovian open dynamics, spontaneous break of continuous time symmetry is also forbidden.

## String Theory

$3d$ Modularity.

*Francesca Ferrari*, SISSA Trieste.

## Abstract

We find and propose an explanation for a large variety of modularity-related symmetries in problems of $3$-manifold topology and physics of $3d$ $N=2$ theories, where such structures a priori are not manifest.

## Geometria em Lisboa

To be announced.

*Bruno Colbois*, Université de Neuchâtel.

## $QM^3$ Quantum Matter meets Maths

To be announced.

*Alex Bullivant*, University of Leeds.

## String Theory

An SLE approach to four dimensional black hole microstate entropy.

*Paolo Benincasa*, Niels Bohr Institute.

## Abstract

We model the Bekenstein-Hawking entropy of a four dimensional extremal black hole in terms of classifying particles moving in its near horizon $AdS_2$ geometry. We use the framework of SLE curves in $AdS_2$ to classify these particle trajectories in terms of their boundary conditions.

## Mathematical Relativity

Mode stability for the Teukolsky equation on extremal Kerr black hole spacetimes.

*Rita Teixeira da Costa*, University of Cambridge.

## Abstract

We prove that there are no exponentially growing modes nor modes on the real axis for the Teukolsky equation on extremal Kerr black hole spacetimes. While the result was previously known for subextremal spacetimes, we show that the proof for the latter cannot be extended to the extremal case as the nature of the event horizon changes radically in the extremal limit.

Finally, we explain how mode stability could serve as a preliminary step towards understanding boundedness, scattering and decay properties of general solutions to the Teukolsky equation on extremal Kerr black holes.