Reps of relative mapping class groups via conformal nets.

*André Henriques*, University of Oxford.

## Resumo

Given a surface with boundary $\Sigma$, its relative mapping class group is the quotient of $\operatorname{Diff}(\Sigma)$ by the subgroup of maps which are isotopic to the identity via an isotopy that fixes the boundary pointwise. (If $\Sigma$ has no boundary, then that's the usual mapping class group; if $\Sigma$ is a disc, then that's the group $\operatorname{Diff}(S^1)$ of diffeomorphisms of $S^1$.)

Conformal nets are one of the existing axiomatizations of chiral conformal field theory (vertex operator algebras being another one). We will show that, given an arbitrary conformal net and a surface with boundary $\Sigma$, we get a continuous projective unitary representation of the relative mapping class group (orientation reversing elements act by anti-unitaries). When the conformal net is rational and $\Sigma$ is a closed surface (i.e. $\partial \Sigma = \emptyset$), then these representations are finite dimensional and well known. When the conformal net is not rational, then we must require $\partial \Sigma \neq \emptyset$ for these representations to be defined. We will try to explain what goes wrong when $\Sigma$ is a closed surface and the conformal net is not rational.

The material presented in this talk is partially based on my paper arXiv:1409.8672 with Arthur Bartels and Chris Douglas.

Deformed Airy kernel determinants: from KPZ tails to initial data for KdV.

*Tom Claeys*, Université Catholique de Louvain.

## Resumo

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. I will explain how logarithmic derivatives of the Fredholm determinants can be expressed in terms of a $2\times 2$ Riemann-Hilbert problem.

This Riemann-Hilbert representation can be used to derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal, and it reveals a remarkable connection with a family of unbounded solutions to the Korteweg-de Vries (KdV) equation and with an integro-differential version of the Painlevé II equation.

$G_2$-monopoles (a summary).

*Gonçalo Oliveira*, Universidade Federal Fluminense, Brasil.

## Resumo

This talk is aimed at reviewing what is known about $G_2$-monopoles and motivate their study. After this, I will mention some recent results obtained in collaboration with Ákos Nagy and Daniel Fadel which investigate the asymptotic behavior of $G_2$-monopoles. Time permitting, I will mention a few possible future directions regarding the use of monopoles in $G_2$-geometry.

Topological Data Analysis and Deep Learning.

*Gunnar Carlsson*, Stanford University.

## Resumo

Deep Learning is a powerful collection of techniques for statistical learning, which has shown dramatic applications in many different directions, including including the study of data sets of images, text, and time series. It uses neural networks, specifically convolutional neural networks (CNN's), to produce these results. What we have observed recently is that methods of topology can contribute to this effort, in diagnosing behavior within the CNN's, in the design of neural networks with excellent computational properties, and in improving generalization, i.e. the transfer of results of one neural network from one data set to another of similar type. We'll discuss topological methods in data science, as well as there application to this interesting set of techniques.

On highly anisotropic big bang singularities.

*Hans Ringstrom*, KTH.

## Resumo

In cosmology, the universe is typically modelled by spatially homogeneous and isotropic solutions to Einstein’s equations. However, for large classes of matter models, such solutions are unstable in the direction of the singularity. For this reason, it is of interest to study the anisotropic setting.

The purpose of the talk is to describe a framework for studying highly anisotropic singularities. In particular, for analysing the asymptotics of solutions to linear systems of wave equations on the corresponding backgrounds and deducing information concerning the geometry.

The talk will begin with an overview of existing results. This will serve as a background and motivation for the problem considered, but also as a justification for the assumptions defining the framework we develop.

Following this overview, the talk will conclude with a rough description of the results.

Counting Monster Potentials.

*Davide Masoero*, Group of Mathematical Physics, University of Lisbon.

## Resumo

The monster potentials were introduced by Bazhanov-Lukyanov-Zamolodchikov in the framework of the ODE/IM correspondence. They should in fact be in 1:1 correspondence with excited states of the Quantum KdV model (an Integrable Conformal Field Theory) since they are the most general potentials whose spectral determinant solves the Bethe Ansatz equations of such a theory. By studying the large momentum limit of the monster potentials, I retrieve that

- The poles of the monster potentials asymptotically condensate about the complex equilibria of the ground state potential.
- The leading correction to such asymptotics is described by the roots of Wronskians of Hermite polynomials.

This allows me to associate to each partition of N a unique monster potential with N roots, of which I compute the spectrum. As a consequence, I prove up to a few mathematical technicalities that, fixed an integer N, the number of monster potentials with N roots coincide with the number of integer partitions of N, which is the dimension of the level N subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

This is joint work with Riccardo Conti (Group of Mathematical Physics of Lisbon University).

Localizing the Donaldson-Futaki invariant.

*Éveline Legendre*, Université Paul Sabatier.

## Resumo

We will see how to represent the Donaldson-Futaki invariant as an intersection of equivariant closed forms. We will use it to express this invariant as the intersection on some specific subvarieties of the central fibre of the test configuration. As an application we provide a proof that for Kähler orbifolds the Donaldson-Futaki invariant is the Futaki invariant of the central fiber.

Machine Learning and Scientific Computing.

*Weinan E*, Princeton University.

## Resumo

Neural network-based deep learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the same time, deep learning has also acquired the reputation of being a set of “black box” type of tricks, without fundamental principles. This has been a real obstacle for making further progress in machine learning.

In this talk, I will try to address the following two questions:

- How machine learning will impact computational mathematics and computational science?
- How computational mathematics, particularly numerical analysis, can impact machine learning? We describe some of the most important progresses that have been made on these issues so far. Our hope is to put things into a perspective that will help to integrate machine learning with computational science.

Cluster realization of quantum groups and higher Teichmüller theory.

*Alexander Shapiro*, University of California, Berkeley.

Eigenstate Thermalization, random matrices and (non)local operators in many-body systems.

*Masud Haque*, Maynooth University.

## Resumo

The eigenstate thermalization hypothesis (ETH) is a cornerstone in our understanding of quantum statistical mechanics. The extent to which ETH holds for nonlocal operators (observables) is an open question. I will address this question using an analogy with random matrix theory. The starting point will be the construction of extremely non-local operators, which we call Behemoth operators. The Behemoths turn out to be building blocks for all physical operators. This construction allow us to derive scalings for both local operators and different kinds of nonlocal operators.

On the space of Kähler metrics.

*Xiuxiong Chen*, Stony Brook University.

## Resumo

Inspired by the celebrated $C^0, C^2$ and $C^3$ a priori estimate of Calabi, Yau and others on Kähler Einstein metrics, we will present an expository report of a priori estimates on the constant scalar curvature Kähler metrics. With this estimate, we prove the Donaldson conjecture on geodesic stability and the properness conjecture on Mabuchi energy functional.

This is a joint work with Cheng JingRui.

A anunciar.

*Lindsey Gray*, Fermi National Accelerator Laboratory.

A anunciar.

*Boris Beranger*, School of Mathematics and Statistics, University New South Wales, Sydney.

A anunciar.

*Boris Beranger*, School of Mathematics and Statistics, University New South Wales, Sydney.

Berry's Phase, $\operatorname{TKN}^2$ Integers and All That: My work on Topology in Condensed Matter Physics 1983-1993.

*Barry Simon*, Caltech.

## Resumo

I will give an overview of my work on topological methods in condensed matter physics almost 40 years ago. Include will be Homotopy and $\operatorname{TKN}^2$ integers, holonomy and Berry's phase and quarternions and Berry's phase for Fermions. If time allows, I'll discuss supersymmetry and pairs of projections.

A anunciar.

*Tristan C. Collins*, MIT.

From high dimensional space to a random low dimensional space.

*Conceição Amado*, Instituto Superior Técnico and CEMAT.

Hyperbolic band theory.

*Joseph Maciejko*, University of Alberta.

## Resumo

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.

A anunciar.

*Florent Krzakala*, EPFL.

Superuniversality of superdiffusion.

*Enej Ilievski*, University of Ljubljana.

## Resumo

Emergence of anomalous transport laws in deterministic interacting many-body systems has become a subject of intense study in the past few years. One of the most prominent examples is the unexpected discovery of superdiffusive spin dynamics in the isotropic Heisenberg quantum spin chain with at half filling, which falls into the universality class of the celebrated Kardar-Parisi-Zhang equation. In this talk, we will theoretically justify why the observed superdiffusion of the Noether charges with anomalous dynamical exponent $z=3/2$ is indeed superuniversal, namely it is a feature of all integrable interacting lattice models or quantum field theories which exhibit globally symmetry of simple Lie group $G$, in thermal ensembles that do not break $G$-invariance. The phenomenon can be attributed to thermally dressed giant quasiparticles, whose properties can be traced back to fusion relations amongst characters of quantum groups called Yangians. Giant quasiparticles can be identified with classical solitons, i.e. stable nonlinear solutions to certain integrable PDE representing classical ferromagnet field theories on certain types of coset manifolds. We shall explain why these inherently semi-classical objects are in one-to-one correspondence with the spectrum of Goldstone modes. If time permits, we shall introduce another type of anomalous transport law called undular diffusion that generally occurs amongst the symmetry-broken Noether fields in $G$-invariant dynamical systems at finite charge densities.

A anunciar.

*Andrew Lobb*, Durham University.

Mathematical aspects of neural network learning through measure dynamics.

*Joan Bruna*, Courant Institute and Center for Data Science, NYU.

A anunciar.

*Manuel Scotto*, Instituto Superior Técnico and CEMAT.

A anunciar.

*Benoît Douçot*, LPTHE, Sorbonne Université.

Many more infinite staircases in symplectic embedding functions.

*Ana Rita Pires*, University of Edinburgh.

Learning and Learning to Solve PDEs.

*Bin Dong*, BICMR, Peking University.

Less is more: effective description of topological spin liquids.

*Jiannis K. Pachos*, University of Leeds.

## Resumo

It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents, though we do not know how accurate it is. I will show that the low-energy properties of the Kitaev honeycomb lattice model, such as the shape of Majorana zero modes or the deformations of the correlation length, are faithfully described in terms of Riemann-Cartan geometry. Moreover, I will present how effective axial gauge fields can couple to Majorana fermions, thus giving a unified picture between vortices and lattice dislocations that support Majorana zero modes.

Stability of the symplectomorphism group of rational surfaces.

*Silvia Anjos*, Instituto Superior Técnico and CAMGSD.

Combining knowledge and data driven methods for solving inverse imaging problems - getting the best from both worlds.

*Carola-Bibiane Schönlieb*, DAMTP, University of Cambridge.

## Resumo

Inverse problems in imaging range from tomographic reconstruction (CT, MRI, etc) to image deconvolution, segmentation, and classification, just to name a few. In this talk I will discuss approaches to inverse imaging problems which have both a mathematical modelling (knowledge driven) and a machine learning (data-driven) component. Mathematical modelling is crucial in the presence of ill-posedness, making use of information about the imaging data, for narrowing down the search space. Such an approach results in highly generalizable reconstruction and analysis methods which come with desirable solutions guarantees. Machine learning on the other hand is a powerful tool for customising methods to individual data sets. Highly parametrised models such as deep neural networks in particular, are powerful tools for accurately modelling prior information about solutions. The combination of these two paradigms, getting the best from both of these worlds, is the topic of this talk, furnished with examples for image classification under minimal supervision and for tomographic image reconstruction.

New consequences of convexity beyond dynamical convexity.

*Leonardo Macarini*, Instituto Superior Técnico and CAMGSD.

Dealing with Systematic Uncertainties in HEP Analysis with Machine Learning Methods.

*Tommaso Dorigo*, Italian Institute for Nuclear Physics.

A anunciar.

*Francisco C. Santos*, Departamento do Engenharia Informática, Instituto Superior Técnico.

Universality of dimers via imaginary geometry.

*Gourab Ray*, University of Victoria.

A anunciar.

*Gitta Kutyniok*, Institut für Mathematik - TU Berlin.

A anunciar.

*René Vidal*, Mathematical Institute for Data Science, Johns Hopkins University.

Instituto Superior Técnico
Av. Rovisco Pais,
Lisboa,
PT