Floquet higher-order topological insulators: principles and path towards realizations.

*Gil Refael*, Institute for Quantum Information and Matter.

## Abstract

The co-existence of spatial and non-spatial symmetries together with appropriate commutation/anticommutation relations between them can give rise to static higher-order topological phases, which host gapless boundary modes of co-dimension higher than one. Alternatively, space-time symmetries in a Floquet system can also lead to anomalous Floquet boundary modes of higher co-dimensions, with different commutation/anticommutation relations with respect to non-spatial symmetries. In my talk I will review how these dynamical analogs of the static HOTI's emerge, and also show how a coherently excited phonon mode can be used to support non-trivial Floquet higher-order topological phases. If time allows, I will also review recent work on Floquet engineering and band flattening of twisted-bilayer graphene.

Weak SYZ conjecture for hypersurfaces in the Fermat family.

*Yang Li*, Institute for Advanced Study.

## Abstract

The SYZ conjecture predicts that for polarised Calabi-Yau manifolds undergoing the large complex structure limit, there should be a special Lagrangian torus fibration. A weak version asks if this fibration can be found in the generic region. I will discuss my recent work proving this weak SYZ conjecture for the degenerating hypersurfaces in the Fermat family. Although these examples are quite special, this is the first construction of generic SYZ fibrations that works uniformly in all complex dimensions.

Reps of relative mapping class groups via conformal nets.

*André Henriques*, University of Oxford.

## Abstract

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.

## Abstract

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.

Topology and the Yang Mills functional.

*Gonçalo Oliveira*, Universidade Federal Fluminense, Brasil.

## Abstract

The Yang-Mills functional is a physics-inspired functional for connections on vector/principal bundles. It is now almost 40 years since the, then groundbreaking, work of Atiyah and Bott extensively studying it on vector bundles over Riemann surfaces. The major outcome of this study was the relationship of its critical levels with the moduli spaces of holomorphic bundles, which allowed for results to flow in both directions of the relationship. Despite its success in that 2 dimensional setting and the 40 years that have since passed, few attempts at exploring the functional, and its critical points, in 3 dimensions were made. I will report on ongoing work with Alex Waldron and Thomas Walpuski towards a Morse theoretic approach for the Yang-Mills functional in 3 dimensional oriented Riemannian manifolds.

(joint work with Alex Waldron and Thomas Walpuski)

To be announced.

*Gunnar Carlsson*, Stanford University.

From high dimensional space to a random low dimensional space.

*Conceição Amado*, Instituto Superior Técnico and CEMAT.

On highly anisotropic big bang singularities.

*Hans Ringstrom*, KTH.

## Abstract

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.

A solution of the Riemann-Hilbert problem on the $A_2$ quiver.

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

Localizing the Donaldson-Futaki invariant.

*Éveline Legendre*, Université Paul Sabatier.

## Abstract

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.

To be announced.

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

Machine Learning and Scientific Computing.

*Weinan E*, Princeton University.

## Abstract

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.

To be announced.

*Manuel Scotto*, Instituto Superior Técnico and CEMAT.

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.

## Abstract

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.

## Abstract

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.

To be announced.

*Lindsey Gray*, Fermi National Accelerator Laboratory.

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

*Barry Simon*, Caltech.

## Abstract

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.

To be announced.

*Tristan C. Collins*, MIT.

Hyperbolic band theory.

*Joseph Maciejko*, University of Alberta.

## Abstract

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.

To be announced.

*Florent Krzakala*, EPFL.

To be announced.

*Andrew Lobb*, Durham University.

Mathematical aspects of neural network learning through measure dynamics.

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

To be announced.

*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.

To be announced.

*Jiannis K. Pachos*, University of Leeds.

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.

## Abstract

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.

Universality of dimers via imaginary geometry.

*Gourab Ray*, University of Victoria.

To be announced.

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

Instituto Superior Técnico
Av. Rovisco Pais,
Lisboa,
PT