Proceedings of the International Congress of Mathematicians (ICM 2018)

The following sections are included:

• Contents
• Preface
• Opening Ceremony — August 1, 2018
• Award Ceremonies for Gauβ Prize and Chern Medal Award — August 4, 2018
• Closing Ceremony — August 9, 2018

We present some highlights of Alessio Figalli mathematical work, which is impressive for its originality, breadth, innovative character and impact to various areas of mathematics. It focus mainly on his contributions to optimal transport theory and its connections to partial differential equations (PDE), the calculus of variations, geometric and functional inequalities and random matrices.

The Leelavati Prize of the International Mathematical Union was awarded during the ICM 2018 in Rio de Janeiro to Ali Nesin for his outstanding contributions towards increasing public awareness of mathematics in Turkey. We review Nesin’s outreach activities, focusing on its founding and development of the “Mathematics Village” as an extraordinary, peaceful place for the exploration of mathematical knowledge dedicated to improving the understanding of mathematics of gifted students at all levels, in the inspiring environment and stimulating atmosphere of a camp.

On Wednesday, August 1st 2018, Cauchcr Birkar was awarded the Fields Medal for his contributions to the minimal model program and his proof of the boundedness of ϵ-log canonical Fano varieties. In this note I will discuss some of Birkar’s main achievements.

He has developed powerful methods in algebraic geometry over p-adic fields, and has proved striking theorems in this area.

Number theory while one of the oldest subjects in mathematics, continues to be one of the most active areas of research. One reason for this is that it borrows from and contributes to many other fields, sometimes quite unexpectedly. Uncovering deeper number theoretic truths often involves advancing techniques across these disciplines resulting in new avenues of research and theories. Venkatesh’s work fits squarely into this mold. His resolution and advancement of a number of long standing problems start with fundamental new insights connecting theories and have led to active research areas. He is both a problem solver and a theory builder of the highest caliber. Due to space limitation and the nature of this report, I have chosen to describe three examples of Venkatesh’s works and end by simply listing a number of his other striking results. Also, in this brief description of some of the results I have omitted some technicalities and the reader should go to the references for precise statements.

Professor Masaki Kashiwara is certainly one of the foremost mathematicians of our time. His influence is spreading over many fields of mathematics and the mathematical community slowly begins to appreciate the importance of the ideas and methods he has introduced.

The following sections are included:

• Introduction
• Past, Present and Future
• A new programme for the future
• New Technology
• Hard Analysis
• References

This is the lecture notes for the author’s Emmy Noether lecture at 2018, ICM, Rio de Janeiro, Brazil. It is a great honor for the author to be invited to give the lecture.

The concept of equilibrium, in its various forms, has played a central role in the development of Game Theory and Economics. The mathematical properties and computational complexity of equilibria are also intimately related to mathematical programming, online learning and fixed point theory. More recently, equilibrium computation has been proposed as a means to learn generative models of high-dimensional distributions.

In this paper, we review fundamental results on minimax equilibrium and its relationship to mathematical programming and online learning. We then turn to Nash equilibrium, reviewing some of our work on its computational intractability. This intractability is of an unusual kind. While computing Nash equilibrium does belong to the well-known complexity class NP, Nash’s theorem—that Nash equilibrium is guaranteed to exist—makes it unlikely that the problem is NP-complete. We show instead that it is as hard computationally as computing Brouwer fixed points, in a precise technical sense, giving rise to the complexity class PPAD, a subclass of total search problems in NP that we will define. The intractability of Nash equilibrium makes it unlikely to always arise in practice, so we study special circumstances where it becomes trouble. We also discuss modern applications of equilibrium computation, presenting recent progress and several open problems in the training of Generative Adversarial Networks. Finally, we take a broader look at the co,mplexity of total search problems in NP, discussing their intimate relationship to fundamental in a range of fields, including Combinatorics, Discrete and Continuous Optimization, Social Choice Theory, Economics, and Cryptography. We overview recent work and present a host of open problems.

Aide Mémoire. We briefly review the contents of the 2018 Gauss Lecture

Esprit de I’Escalier. Some mathematical work that ought to have been mentioned.

Meta Remarks. Comments about the ICM and its awards.

The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems.

After a general overview, we present some recent results on the structure of singular free boundary points. Then, we show some selected applications to the generic smoothness of the free boundary in the stationary obstacle problem (Scha-effer’s conjecture), and to the smoothness of the free boundary in the one-phase Stefan problem for almost every time.

This is a survey of the theory of crystal bases, global bases and cluster algebra structure on the quantum coordinate rings.

I was told I needed to write this about 6 months ago. Since then I’ve been wondering what to write. Along the years I’ve said or written almost everything there is to say about the maths village. Saying the same things twice, three times bores me to death. I can give the same lecture numerous times, even making it better and more fun to give each time. But for ordinary life which isn’t maths? Never mind twice, I don’t even want to tell it once.

The topology of “arithmetic manifolds”, such as the space of lattices in Rⁿ up to rotation, encodes subtle features of the arithmetic of algebraic varieties. In some cases, this can be explained because the arithmetic manifold itself carries the structure of an algebraic variety.

I will talk about some of the phenomena one encounters in the other, “nonalgebraic,” cases.

The theory of curvature-dimension bounds for nonsmooth spaces has several motivations: the study of functional and geometric inequalities in structures which arc very far from being Euclidean, therefore with new non-Riemannian tools, the description of the “closure” of classes of Riemannian manifolds under suitable geometric constraints, the stability of analytic and geometric properties of spaces (e.g. to prove rigidity results). Even though these goals may occasionally be in conflict, in the last few years we have seen spectacular developments in all these directions, and my text is meant both as a survey and as an introduction to this quickly developing research field.

A hundred years ago, Einstein wondered about quantization conditions for classically ergodic systems. Although a mathematical description of the spectrum of Schrödinger operators associated to ergodic classical dynamics is still completely missing, a lot of progress has been made on the delocalization of the associated eigenfunctions.

Machine learning is the subfield of computer science concerned with creating machines that can improve from experience and interaction. It relies upon mathematical optimization, statistics, and algorithm design. Rapid empirical success in this field currently outstrips mathematical understanding. This elementary article sketches the basic framework of machine learning and hints at the open mathematical problems in it.

We describe a recent evolution of Harmonic Analysis to generate analytic tools for the joint organization of the geometry of subsets of ℝⁿ and the analysis of functions and operators on the subsets. In this analysis we establish a duality between the geometry of functions and the geometry of the space. The methods are used to automate various analytic organizations, as well as to enable informative data analysis. These tools extend to higher order tensors, to combine dynamic analysis of changing structures.

This article is a broad-brush survey of two areas in differential geometry. While these two areas are not usually put side-by-side in this way, there are several reasons for discussing them together. First, they both fit into a very general pattern, where one asks about the existence of various differential-geometric structures on a manifold. In one case we consider a complex Kähler manifold and seek a distinguished metric, for example a Kählera–Einsten metric. In the other we seek a metric of exceptional holonomy on a manifold of dimension 7 or 8. Second, as we shall see in more detail below, there are numerous points of contact between these areas at a technical level. Third, there is a pleasant contrast between the state of development in the fields. These questions in Kähler geometry have been studied for more than half a century: there is a huge literature with many deep and far-ranging results. By contrast, the theory of manifolds of exceptional holonomy is a wide-open field: very little is known in the way of general results and the developments so far have focused on examples. In many cases these examples depend on advances in Kähler geometry.

Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical low-rank approximation of time-dependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.

Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathematics has tended to focus on local topics, over a short term-scale, and on the study of ephemeral configurations of mathematicians, theorems or practices. The first part of the paper explains why this change has taken place: a renewed interest in the connections between mathematics and society, an increased attention to the variety of components and aspects of mathematical work, and a critical outlook on historiography itself. The problems of a long-term history are illustrated and tested using a number of episodes in the nineteenth-century history of Hermitian forms, and finally, some open questions aec proposed.

Our topic is the relationship between dynamical systems and optimization. This is a venerable, vast area in mathematics, counting among its many historical threads the study of gradient flow and the variational perspective on mechanics. We aim to build some new connections in this general area, studying aspects of gradient-based optimization from a continuous-time, variational point of view. We go beyond classical gradient flow to focus on second-order dynamics, aiming to show the relevance of such dynamics to optimization algorithms that not only converge, but converge quickly.

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?

Low-dimensional topology is the study of manifolds and cell complexes in dimensions four and below. Input from geometry and analysis has been central to progress in this field over the past four decades, and this article will focus on one aspect of these developments in particular, namely the use of Yang–Mills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4-manifolds, but the past ten years have seen new applications of gauge theory, and new interactions with more recent threads in the subject, particularly in 3-dimensional topology.

This is a field where many mathematical techniques have found applications, and sometimes a theorem has two or more independent proofs, drawing on more than one of these techniques. We will focus primarily on some questions and results where gauge theory plays a special role.

This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject.

There have been incredible progress in the last twenty years in the rigorous analysis of planar statistical mechanics models whose limits are conformally invariant. This paper will not try to survey all the recent advances. Instead, it will discuss some recent results about particular conformally invariant measures on loops and paths.

Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways. In the last decade, a theory of “high dimensional expanders” has begun to emerge. The goal of the current paper is to describe some paths of this new area of study.

We describe in this survey several results relating Fractal Geometry, Dynamical Systems and Diophantine Approximations, including a description of recent results related to geometrical properties of the classical Markov and Lagrange spectra and generalizations in Dynamical Systems and Differential Geometry.

The purpose of this article is to survey some of the context, achievements, challenges and mysteries of the field of metric dimension reduction, including new perspectives on major older results as well as recent advances.

The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts that reflect a certain range of developments, but even in their totality they cannot be taken as a comprehensive survey. In the format of a 30-page contribution aimed at a general mathematical audience, I have decided to illustrate some of the basic ideas in one very interesting example – that of Hilb (ℂ², n), hoping to spark the curiosity of colleagues in those numerous fields of study where one should expect applications.

Cohomological field theories (CohFTs) were defined in the mid 1990s by Kontsevich and Manin to capture the formal properties of the virtual fundamental class in Gromov–Witten theory. A beautiful classification result for semisimple CohFTs (via the action of the Givental group) was proven by Teleman in 2012. The Givental–Teleman classification can be used to explicitly calculate the full CohFT in many interesting cases not approachable by earlier methods.

My goal here is to present an introduction to these ideas together with a survey of the calculations of the CohFTs obtained from

• Witten’s classes on the moduli spaces of r-spin curves,
• Chern characters of the Verlinde bundles on the moduli of curves,
• Gromov–Witten classes of Hilbert schemes of points of ℂ².

The subject is full of basic open questions.

We discuss recent developments in p-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for “compact p-adic manifolds” over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands conjectures, and the construction of “universal” p-adic cohomology theories. We finish with some speculations on how a theory that combines all primes p, including the archimedean prime, might look like.

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and give rise to a variety of questions pertaining to calculus of variations, Partial Differential Equations and probability. We will review these as well as “the mean-field limit” results that allow to derive effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order beyond the mean-field limit, giving information on the system at the microscopic level. In the setting of statistical mechanics, this allows for instance to observe the effect of the temperature and to connect with crystallization questions.

We survey some applications of parity sheaves and Soergel calculus to representation theory.

This is an expanded version of a presentation given at ICM2018. It discusses a number of results taken from a cross-section of the author’s work in Dynamical Systems. The topics include relation between entropy, Lyapunov exponents and fractal dimension, statistical properties of chaotic dynamical systems, physically relevant invariant measures, strange attractors arising from shear-induced chaos, random maps and random attractors. The last section contains two applications of Dynamical Systems ideas to Biology, one to epidemics control and the other to computational neuroscience.

The panel and poster session entitled “Strengthening Mathematics in the Developing World” at the 2018 International Congress of Mathematicians (ICM) in Rio de Janeiro, Brazil was organized by the Commission for Developing Countries (CDC) of the International Mathematical Union (IMU). The objective was to share information about mathematical development activities with mathematicians at the ICM and to serve as a catalyst for interactions between mathematicians, organizations and funding agencies. The panel had representatives from seven organizations including international mathematical unions, commissions supporting the developing world, supporting women, a funding agency and the mathematical society of the host nation to the ICM (Brazil). There were ten other organizations supporting the developing world represented in the poster session, several of which work closely with the organizations in the panel. The panel and poster session had an attendance of approximately 400 and led to a large number of interactions between the representatives of the organizations and the audience.

The panel organised by the Committee for Women in Mathematics (CWM) of the International Mathematical Union (IMU) took place at the International Congress of Mathematicians (ICM) on August 2^nd, 2018. It was attended by about 190 people, with a reasonable gender balance (1/4 men, 3/4 women). The panel was moderated by Caroline Series, President of the London Mathematical Society and Vice-Chair of CWM. Presentations were made by Marie-Françoise Roy, Chair of CWM, June Barrow-Green, Chair of the International Commission on the History of Mathematics, and Silvina Ponce Dawson, Vice-President at Large and Gender Champion of the International Union of Pure and Applied Physics (IUPAP). The presentations were followed by general discussion. Marie-Françoise briefly out-lined the history and activities of CWM and described the ongoing “Gender Gap in Science” project which is being carried out under the leadership of IMU and the International Union of Pure and Applied Chemistry (IUPAC), with the participation of IUPAP and many other scientific unions. June gave some insights into the historical context of the gender gap in mathematics, while Silvina gave an overview of activities undertaken by the IUPAP Working Group on Women in Physics to evaluate and improve the situation of female physicists.

What follows are the authors’ accounts of their presentations together with some notes on the subsequent discussion.

The panel took place on the 7th August 2018. After the moderator had introduced the topic, the panelists presented their experiences and points of view, and then took questions from the floor.

The discussion panel with the topic “How can mathematicians contribute to planetary challenges” met on Tuesday, August 7, 2018.

The following sections are included:

• Introduction
• Overview of Japanese LS
• Introduction into Chinese LS
• Adaptation of LS with pre-service teachers in Malawi
• Adaptation of LS with in-service teachers in Thailand
• Methodological and theoretical issues of studying LS
• Further directions of LS
• References

A Panel at the 2018 International Congress of Mathematicians concerning the efforts to realize the dream of a Global Digital Mathematics Library consisted of Thierry Bouche, Gadadhar Misra, Alf A. Onshuus, Stephen M. Watt and Liu Zheng and was moderated by the writer of the lines below as recorder. This report contains a description setting the stage for the panel, summaries of the panelists’ statements and of some questions and answers at the session as well as at a later open opportunity for further discussion.

The following section is included:

• List of Participants
• Membership by Country
• Author Index

The following section is included:

• Contents

Germs of real-valued functions, surreal numbers, and transseries are three ways to enrich the real continuum by infinitesimal and infinite quantities. Each of these comes with naturally interacting notions of ordering and derivative. The category of H-fields provides a common framework for the relevant algebraic structures. We give an exposition of our results on the model theory of H-fields, and we report on recent progress in unifying germs, surreal numbers, and transseries from the point of view of asymptotic differential algebra.

The subject of descriptive set theory is traditionally concerned with the theory of definable subsets of Polish spaces. By introducing large cardinal/determinacy axioms, a theory of definable subsets of Polish spaces and their associated ordinals has been developed over the last several decades which extends far up in the definability hierarchy. Recently, much interest has been focused on trying to extend the theory of definable objects to more general types of sets, not necessarily subsets of a Polish space or an ordinal. A large class of these objects are represented by equivalence relations on Polish spaces. Even for some of the simpler of these relations, an interesting combinatorial theory is emerging. We consider both problems of extending further the theory of definable subsets of Polish spaces, and that of determining the structure of these new types of definable sets.

This lecture highlights some recent advances on classical decidability issues in local and global fields.

We discuss applications of methods from proof theory, so-called proof interpretations, for the extraction of explicit bounds in convex optimization, fixed point theory, ergodic theory and nonlinear semigroup theory.

The article motivates recent work on saturation of ultrapowers from a general mathematical point of view.

We give an overview of our effort to introduce (dual) semicanonical bases in the setting of symmetrizable Cartan matrices.

This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster) categories of finite dimensional algebras.

We sketch a proof of Weibel’s conjecture on the vanishing of negative algebraic K-groups and we explain an analog of this result for continuous K-theory of non-archimedean algebras.

We report, from an algebraic point of view, on some methods and results on the classification problem of fusion categories over an algebraically closed field of characteristic zero.

Let R be a regular local ring. Let G be a reductive group scheme over R. A well-known conjecture due to Grothendieck and Serre assertes that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of non-abelian cohomology pointed sets

This and other results concerning the conjecture are discussed in the present paper. We illustrate the exposition by many interesting examples. We begin with couple results for complex algebraic varieties and develop the exposition step by step to its full generality.

We discuss some basic problems in representation theory of finite groups, and current approaches and recent progress on some of these problems. We will also outline some applications of these and other results in representation theory of finite groups to various problems in group theory, number theory, and algebraic geometry.

We review the construction of analytic families of Siegel modular cuspforms based on the notion of overconvergent modular forms of p-adic weight. We then present recent developments on the following subjects: the halo conjecture, the construction of p-adic L-functions, and the modularity of irregular motives.

This is a survey of recent advances in commutative algebra, especially in mixed characteristic, obtained by using the theory of perfectoid spaces. An explanation of these techniques and a short account of the author’s proof of the direct summand conjecture are included. One then portrays the progresses made with these (and related) techniques on the so-called homological conjectures.

On présente un résumé de nos travaux sur la courbe que nous avons introduite avec Jean-Marc Fontaine et ses applications en théorie de Hodge p-adique ainsi qu’au programme de Langlands.

Our goal in this note is two-fold. In part I, we motivate and explain the ideas behind a recent theorem of ours.

We discuss recent advances on weak forms of the Prime k-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps between primes.

Estimating the size of automorphic L-functions on the critical line is a central problem in analytic number theory. An easy consequence of the standard analytic properties of the L-function is the convexity bound, whereas the generalised Riemann Hypothesis predicts a much sharper bound. Breaking the convexity barrier is a hard problem. The moment method has been used to surpass convexity in the case of L-functions of degree one and two. In this talk I will discuss a different method, which has been quite successful to settle certain longstanding open problems in the case of degree three.

We describe recent work on the construction of well-behaved arithmetic models for large classes of Shimura varieties and report on progress in the study of these models.

This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over ℚ have rank ≤ 21, which would imply that the rank is uniformly bounded.

Vincent Lafforgue has recently made a spectacular breakthrough in the setting of the global Langlands correspondence for global fields of positive characteristic, by constructing the ‘automorphic–to–Galois’ direction of the correspondence for an arbitrary reductive group G. We discuss a result that starts with Lafforgue’s work and proceeds in the opposite (‘Galois–to–automorphic’) direction.

We survey recent results in functional transcendence theory, and give arithmetic applications to the André-Oort conjecture and other unlikely-intersection problems.

In this talk we will speak about recent progress on the sphere packing problem. The packing problem can be formulated for a wide class of metric spaces equipped with a measure. An interesting feature of this optimization problem is that a slight change of parameters (such as the dimension of the space or radius of the spheres) can dramatically change the properties of optimal configurations. We will focus on those cases when the solution of the packing problem is particularly simple. Namely, we say that a packing problem is sharp if its density attains the so-called linear programming bound. Several such configurations have been known for a long time and we have recently proved that the E₈ lattice sphere packing in ℝ⁸ and the Leech lattice packing in ℝ²⁴ are sharp. Moreover, we will discuss common unusual properties of shared by such configurations and outline possible applications to Fourier analysis.

We provide an informal discussion of the polynomial method. This is a tool of general applicability that can be used to exploit the algebraic structure arising in some problems of arithmetic nature.

This is a report for the author’s talk in ICM-2018. Motivated by the formulas of Gross–Zagier and Waldspurger, we review conjectures and theorems on automorphic period integrals, special cycles on Shimura varieties, and their connection to central values of L-functions and their derivatives. We focus on the global Gan–Gross–Prasad conjectures, their arithmetic versions and some variants in the author’s joint work with Rapoport and Smithling. We discuss the approach of relative trace formulas and the arithmetic fundamental lemma conjecture. In the function field setting, Z. Yun and the author obtain a formula for higher order derivatives of L-functions in terms of special cycles on the moduli space of Drinfeld Shtukas.

We discuss Hironaka’s theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka’s method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jarosław Włodarczyk.

The theory of holomorphic foliations has its origins in the study of differential equations on the complex plane, and has turned into a powerful tool in algebraic geometry. One of the fundamental problems in the theory is to find conditions that guarantee that the leaves of a holomorphic foliation are algebraic. These correspond to algebraic solutions of differential equations. In this paper we discuss algebraic integrability criteria for holomorphic foliations in terms of positivity of its tangent sheaf, and survey the theory of Fano foliations, developed in a series of papers in collaboration with Stéphane Druel. We end by classifying all possible leaves of del Pezzo foliations.

This is a report on some of the main developments in birational geometry in recent years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.

We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature Kähler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the use of pluripotential theory and the interpretation of K-stability in terms of non-Archimedean geometry.

We survey a few results concerning groups of regular or birational transformations of projective varieties, with an emphasis on open questions concerning these groups and their dynamical properties.

In recent years an interesting connection has been established between some moduli spaces of algebro-geometric objects (e.g. algebraic stable curves) and some moduli spaces of polyhedral objects (e.g. tropical curves).

In loose words, this connection expresses the Berkovich skeleton of a given algebro-geometric moduli space as the moduli space of the skeleta of the objects parametrized by the given space; it has been proved to hold in two important cases: the moduli space of stable curves and the moduli space of admissible covers. Partial results are known in other cases.

This connection relies on the study of the boundary of the algebro-geometric moduli spaces and on its recursive, combinatorial properties, some of which have been long known and are now viewed from a new perspective.

In this survey article, we introduce the development of birational geometry associated to pluricanonical maps. Especially, we explain various aspects of explicit studies of threefolds including the key idea of theory of baskets and other applications.

We explain our proof, joint with Mark Gross and Maxim Kontsevich, of conjectures of Fomin–Zelevinsky and Fock–Goncharov on canonical bases of cluster algebras. We interpret a cluster algebra as the ring of global functions on a noncompact Calabi–Yau variety obtained from a toric variety by a blow up construction. We describe a canonical basis of a cluster algebra determined by tropical counts of holomorphic discs on the mirror variety, using the algebraic approach to the Strominger–Yau–Zaslow conjecture due to Gross and Siebert.

These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called ℚ-homology projective planes. Fake projective planes and the complex projective plane are smooth examples. We describe recent progress in the study of such surfaces, singular ones and fake projective planes. We also discuss open questions.

Let X be an algebraic set in ℝⁿ. Real-valued functions, defined on subsets of X, that are continuous and admit a rational representation have some remarkable properties and applications. We discuss recently obtained results on such functions, against the backdrop of previously developed theories of arc-symmetric sets, arc-analytic functions, approximation by regular maps, and algebraic vector bundles.

We discuss several invariants of complex normal surface singularities with a special emphasis on the comparison of analytic–topological pairs of invariants. Additionally we also list several open problems related with them.

It is well known that numerical quantities arising from the theory of 𝒟-modules are related to invariants of singularities in birational geometry. This paper surveys a deeper relationship between the two areas, where the numerical connections are enhanced to sheaf theoretic constructions facilitated by the theory of mixed Hodge modules. The emphasis is placed on the recent theory of Hodge ideals.

We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and the recent progress on the ‘local stability theory’ of an arbitrary Kawamata log terminal singularity.

We discuss several results pertaining to the Hodge and cycle theories of locally symmetric spaces. The unity behind these results is motivated by a vague but fruitful analogy between locally symmetric spaces and projective varieties.

This is a survey of results on positivity of vector bundles, inspired by the Brunn-Minkowski and Prékopa theorems. Applications to complex analysis, Kähler geometry and algebraic geometry are also discussed.

We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch connections to hyperbolic subgroups of hyperbolic groups and end with some open questions.

The goal of this article is to survey some recent developments in the study of groups acting on hyperbolic spaces. We focus on the class of acylindrically hyperbolic groups; it is broad enough to include many examples of interest, yet a significant part of the theory of hyperbolic and relatively hyperbolic groups can be generalized in this context. In particular, we discuss group theoretic Dehn filling and small cancellation theory in acylindrically hyperbolic groups. Many results discussed here rely on the new generalization of relative hyperbolicity based on the notion of a hyperbolically embedded subgroup.

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.

We discuss potential (largely speculative) applications of Bridgeland’s theory of stability conditions to symplectic mapping class groups.

We report some recent progress on studying degenerations and moduli spaces of canonical metrics in Kähler geometry, and the connection with algebraic geometry, with a particular emphasis on the case of Kähler–Einstein metrics.

Riemann surfaces are of fundamental importance in many areas of mathematics and theoretical physics. The study of the moduli space of Riemann surfaces of a fixed topological type is intimately related to the study of the Teichmüller space of that surface, together with the action of the mapping class group. Classical Teichmüller theory has many facets and involves the interplay of various methods from geometry, analysis, dynamics and algebraic geometry. In recent years, higher Te-ichmüller theory emerged as a new field in mathematics. It builds as well on a combination of methods from different areas of mathematics. The goal of my talk is to invite the reader to get to know and to get involved into higher Teichmüller theory by describing some of its many facets.

This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GL_n(ℤ), relative hyperbolic groups and mapping class groups.

Knot contact homology studies symplectic and contact geometric properties of conormals of knots in 3-manifolds using holomorphic curve techniques. It has connections to both mathematical and physical theories. On the mathematical side, we review the theory, show that it gives a complete knot invariant, and discuss its connections to Fukaya categories, string topology, and micro-local sheaves. On the physical side, we describe the connection between the augmentation variety of knot contact homology and Gromov–Witten disk potentials, and discuss the corresponding higher genus relation that quantizes the augmentation variety.

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary hyperbolic groups, CAT(0) groups with rank 1 elements, mapping class groups and the outer automorphism groups of free groups. As an application, we show that mapping class groups act on finite products of Gromov-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. It implies that mapping class groups have finite asymptotic dimension.

We describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics “at infinity” for representations of discrete groups into Lie groups.

We review the construction and context of a stable homotopy refinement of Khovanov homology.

The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin(2)-equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.

We survey recent work on profinite rigidity of residually finite groups.

The evenness conjecture for the equivariant unitary bordism groups states that these bordism groups are free modules over the unitary bordism ring in even dimensional generators. In this paper we review the cases on which the conjecture is known to hold and we highlight the properties that permit to prove the conjecture in these cases.

The little disks operads are classical objects in algebraic topology which have seen a wide range of applications in the past. For example they appear prominently in the Goodwillie-Weiss embedding calculus, which is a program to understand embedding spaces through algebraic properties of the little disks operads, and their action on the spaces of configurations of points (or disks) on manifolds. In this talk we review the recent understanding of the rational homotopy theory of the little disks operads, and how the resulting knowledge can be used to fulfil the promise of the Goodwillie-Weiss calculus, at least in the “simple” setting of long knot spaces and over the rationals. The derivations prominently use and are connected to graph complexes, introduced by Kontsevich and other authors.

We survey a number of results regarding the representation theory of W-algebras and their connection with the resent development of the four dimensional N = 2 superconformal field theories.

We propose a conjectural construction of various slices for double affine Grassmannians as Coulomb branches of 3-dimensional n = 4 supersymmetric affine quiver gauge theories. It generalizes the known construction for the usual affine Grassmannians, and makes sense for arbitrary symmetric Kac-Mody algebras.

Galois algebras allow an effective study of their representation theory based on the invariant skew group structure. In particular, this leads to many remarkable results on Gelfand-Tsetlin representations of the general linear Lie algebra gI_n, quantum gI_n, Yangians of type A and finite W -algebras of type A.

For more than half a century lattices in Lie groups played an important role in geometry, number theory and group theory. Recently the notion of Invariant Random Subgroups (IRS) emerged as a natural generalization of lattices. It is thus intriguing to extend results from the theory of lattices to the context of IRS, and to study lattices by analyzing the compact space of all IRS of a given group. This article focuses on the interplay between lattices and IRS, mainly in the classical case of semisimple analytic groups over local fields.

The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.

We consider the Ext-analogues of branching laws for representations of a group to its subgroups in the context of p-adic groups.

We provide an explicit formula for the following enumerative problem: how many (absolutely) indecomposable vector bundles of a given rank r and degree d are there on a smooth projective curve X of genus g defined over a finite field 𝔽_q? The answer turns out to only depend on the genus g, the rank r and the Weil numbers of the curve X . We then provide several interpretations of these numbers, either as the Betti numbers or counting polynomial of the moduli space of stable Higgs bundles (of same rank r and degree d) over X , or as the character of some infinite dimensional graded Lie algebra. We also relate this to the (cohomological) Hall algebras of Higgs bundles on curves and to the dimension of the space of absolutely cuspidal functions on X .

We give an overview of the theory of local G-shtukas and their moduli spaces that were introduced in joint work of U. Hartl and the author, and in the past years studied by many people. We also discuss relations to moduli of global G-shtukas, properties of their special fiber through affine Deligne-Lusztig varieties and of their generic fiber, such as the period map.

In the study of automorphic representations over a function field, Hitchin moduli stack and its variants naturally appear and their geometry helps the comparison of trace formulae. We give a survey on applications of this observation to a relative fundamental lemma, the arithmetic fundamental lemma and to the higher Gross-Zagier formula.

The following section is included:

• Index of Authors

The following section is included:

• Contents

We outline a general method of constructing ℒ_∞-spaces, based on the ideas of Bourgain and Delbaen, showing how the solution to the Scalar-plus-Compact Problem, the embedding theorem of Freeman, Odell and Schlumprecht and other recent developments fit into this framework.

This is a brief survey of results related to planar harmonic measure, roughly from Makarov’s results of the 1980’s to recent applications involving 4-manifolds, dessins d’enfants and transcendental dynamics. It is non-chronological and rather selective, but I hope that it still illustrates various areas in analysis, topology and algebra that are influenced by harmonic measure, the computational questions that arise, the many open problems that remain, and how these questions bridge the gaps between pure/applied and discrete/continuous mathematics.

We describe a Fourier analytic tool that has found a large number of applications in Number Theory, Harmonic Analysis and PDEs.

Positive closed currents, the analytic counterpart of effective cycles in algebraic geometry, are central objects in pluripotential theory. They were introduced in complex dynamics in the 1990s and become now a powerful tool in the field. Challenging dynamical problems involve currents of any dimension. We will report recent developments on positive closed currents of arbitrary dimension, including the solutions to the regularization problem, the theory of super-potentials and the theory of densities. Applications to dynamics such as properties of dynamical invariants (e.g. dynamical degrees, entropies, currents, measures), solutions to equidistribution problems, and properties of periodic points will be discussed.

Given a semigroup S with zero, which is left-cancellative in the sense that st = sr ≠ 0 implies that t = r, we construct an inverse semigroup called the inverse hull of S, denoted H(S). When S admits least common multiples, in a precise sense defined below, we study the idempotent semilattice of H(S), with a focus on its spectrum. When S arises as the language semigroup for a subsift X on a finite alphabet, we discuss the relationship between H(S) and several C^*-algebras associated to X appearing in the literature.

The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. It is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature. We describe some properties of the NMC and the quasilinear differential operators that are involved when it acts on graphs. We also survey recent results on surfaces having constant NMC and describe their intimate link with some problems arising in the study of overdetermined boundary value problems.

We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We emphasize results which provide classes of (W*-superrigid) actions that can be completely recovered from their von Neumann algebras and II₁ factors that have a unique Cartan subalgebra. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.

In the last few years numerous 20+ year old problems in the geometry of Banach spaces were solved. Some are described herein.

This note describes the impact of disorder or irregularities in the ambient medium on the behavior of stationary solutions to elliptic partial differential equations and on spatial distribution of eigenfunctions, as well as the profound and somewhat surprising connections between these two topics which have been revealed in the past few years.

The famous Banach–Tarski paradox and Hilbert’s third problem are part of story of paradoxical equidecompositions and invariant finitely additive measures. We review some of the classical results in this area including Laczkovich’s solution to Tarski’s circle-squaring problem: the disc of unit area can be cut into finitely many pieces that can be rearranged by translations to form the unit square.

We also discuss the recent developments that in certain cases the pieces can be chosen to be Lebesgue measurable or Borel: namely, a measurable Banach—Tarski ‘paradox’ and the existence of measurable/Borel circle-squaring.

In this article we highlight the interplay of multi-parameter BMO spaces and boundedness of corresponding commutators. In a variety of settings, we discuss two-sided norm estimates for commutators of classical singular operators with a symbol function. In its classical form, this concerns a theorem by Nehari, factorisation of Hardy space, Hankel and Toeplitz forms. We highlight recent results in which a characterization of L^p boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms is obtained, completing a theory on characterisation of BMO spaces begun by Cotlar, Ferguson and Sadosky. In the light of real analysis, we discuss results in a more intricate situation; commutators of multiplication by a symbol function and Calderón-Zygmund or Journé operators. We show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi-parameter BMO class. In this sense the Hilbert or Riesz transforms or their tensor products are a representative testing class for Calderón-Zygmund or Journé operators.

We survey recent progress in the gap and type problems of Fourier analysis obtained via the use of Toeplitz operators in spaces of holomorphic functions. We discuss applications of such methods to spectral problems for differential operators.

In these notes we will survey recent results on various finitary approximation properties of infinite groups. We will discuss various restrictions on groups that are approximated for example by finite solvable groups or finite-dimensional unitary groups with the Frobenius metric. Towards the end, we also briefly discuss various applications of those approximation properties to the understanding of the equational theory of a group.

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the regularity conjecture and its interplay with internal and external approximation properties.

We discuss optimization of Birkhoff averages of real or vectorial functions and of Lyapunov exponents of linear cocycles, emphasizing whenever possible the similarities between the commutative and non-commutative settings.

Sofic entropy theory is a generalization of the classical Kolmogorov-Sinai entropy theory to actions of a large class of non-amenable groups called sofic groups. This is a short introduction with a guide to the literature.

These notes present recent progress on a conjecture about the dynamics of rational maps on ℙ¹(ℂ), connecting critical orbit relations and the structure of the bifurcation locus to the geometry and arithmetic of postcritically finite maps within the moduli space M_d. The conjecture first appeared in a 2013 publication by Baker and DeMarco. Also presented are some related results and open questions.

We discuss some methods for constructing nonhyperbolic ergodic measures and their applications in the setting of nonhyperbolic skew-products, homoclinic classes, and robustly transitive diffeomorphisms.

We propose in these notes a list of some old and new questions related to quasiperiodic dynamics. A main aspect of quasi-periodic dynamics is the crucial influence of arithmetics on the dynamical features, with a strong duality in general between Diophantine and Liouville behavior. We will discuss rigidity and stability in Diophantine dynamics as well as their absence in Liouville ones. Beyond this classical dichotomy between the Diophantine and the Liouville worlds, we discuss some unified approaches and some phenomena that are valid in both worlds. Our focus is mainly on low dimensional dynamics such as circle diffeomorphisms, disc dynamics, quasi-periodic cocycles, or surface flows, as well as finite dimensional Hamiltonian systems.

In an opposite direction, the study of the dynamical properties of some diagonal and unipotent actions on the space of lattices can be applied to arithmetics, namely to the theory of Diophantine approximations. We will mention in the last section some problems related to that topic.

The field of quasi-periodic dynamics is very extensive and has a wide range of interactions with other mathematical domains. The list of questions we propose is naturally far from exhaustive and our choice was often motivated by our research involvements.

Subadditive cocycles are the random version of subadditive sequences. They play an important role in probability and ergodic theory, notably through Kingman’s theorem ensuring their almost sure convergence. We discuss a variation around Kingman’s theorem, showing that a subadditive cocycle is in fact almost additive at many times. This result is motivated by the study of the iterates of deterministic or random semicontractions on metric spaces, and implies the almost sure existence of a horofunction determining the behavior at infinity of such a sequence. In turn, convergence at infinity follows when the geometry of the space has some features of nonpositive curvature.

We report on recent results about the dimension and smoothness properties of self-similar sets and measures. Closely related to these are results on the linear projections of such sets, and dually, their intersections with affine subspaces. We also discuss recent progress on the the Bernoulli convolutions problem.

The ideas of renormalization was introduced into dynamics around 40 years ago. By now renormalization is one of the most powerful tools in the asymptotic analysis of dynamical systems. In this article we discuss the main conceptual features of the renormalization approach, and present a selection of recent results. We also discuss open problems and formulate related conjectures.

We discuss some aspects of the topological dynamics of surface homeomorphisms. In particular, we survey recent results about the dynamics on the boundary of invariant domains, its relationship with the induced dynamics in the prime ends compactification, and its applications in the area-preserving setting following our recent works with P. Le Calvez.

The study of polygonal billiard tables with simple dynamics led to a remarkable class of special subvarieties in the moduli of space of curves called Teichmüller curves, since they are totally geodesic submanifolds for the Teichmüller metric.

We survey the known methods to construct of Teichmüller curves and exhibit structure theorems that might eventually lead towards the complete classification of Teichmüller curves.

Over the last four decades, group actions on manifolds have deserved much attention by people coming from different fields, as for instance group theory, low-dimensional topology, foliation theory, functional analysis, and dynamical systems. This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear panorama on the subject arises from the lecture.

This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with topology. We also present recent examples which introduce a challenge in the classification program and we propose some steps to continue this classification. Finally, we give some suggestions on what to do after classification is achieved.

We survey some results on non-uniform hyperbolicity, geometric pressure and equilibrium states in one-dimensional real and complex dynamics. We present some relations with Hausdorff dimension and measures with refined gauge functions of limit sets for geometric coding trees for rational functions on the Riemann sphere. We discuss fluctuations of iterated sums of the potential −t log |f′| and of radial growth of derivative of univalent functions on the unit disc and the bound-aries of range domains preserved by a holomorphic map f repelling towards the domains.

We survey the recent advances of almost reducibility and its applications in the spectral theory of one dimensional quasi-periodic Schrödinger operators.

We review our works on the nonlinear asymptotic stability and instability of the Couette flow for the 2D incompressible Euler dynamic. In the fits part of the work we prove that perturbations to the Couette flow which are small in Gevrey spaces G^s of class 1/s with s > 1/2 converge strongly in L² to a shear flow which is close to the Couette flow. Moreover in a well chosen coordinate system, the solution converges in the same Gevrey space to some limit profile. In a later work, we proved the existence of small perturbations in G^s with s < 1/2 such that the solution becomes large in Sobolev regularity and hence yields instability. In this note we discuss the most important physical and mathematical aspects of these two results and the key ideas of the proofs.

In these proceedings we are interested in quantitative estimates for advective equations with an anelastic constraint in presence of vacuum. More precisely, we derive a quantitative stability estimate and obtain the existence of renormalized solutions. Our main objective is to show the flexibility of the method introduced recently by the authors for the compressible Navier-Stokes’ system. This method seems to be well adapted in general to provide regularity estimates on the density of compressible transport equations with possible vacuum state and low regularity of the transport velocity field; the advective equation with degenerate anelastic constraint considered here is another good example of that. As a final application we obtain the existence of global renormalized solution to the so-called lake equation with possibly vanishing topography.

In this survey I report on recent progress in the study of the dynamics of the interface in between two incompressible fluids with different characteristics. In particular I focus on the formation of Splash and Splat singularities in two different settings: Euler equations and Darcy’s law.

The aim of this note is to present some recent results on the structure of the singular part of measures satisfying a PDE constraint and to describe some applications.

We present in this talk various results, obtained during the last years by several authors, about the problem of long time existence of solutions of water waves and related equations, with initial data that are small, smooth, and decaying at infinity. After recalling some facts about local existence theory, we shall focus mainly on global existence theorems for gravity waves equations proved by Ionescu–Pusateri, Alazard–Delort and Ifrim–Tataru. We shall describe some of the ideas of the proofs of these theorems, and mention as well related results.

The issue of symmetry and symmetry breaking is fundamental in all areas of science. Symmetry is often assimilated to order and beauty while symmetry breaking is the source of many interesting phenomena such as phase transitions, instabilities, segregation, self-organization, etc. In this contribution we review a series of sharp results of symmetry of nonnegative solutions of nonlinear elliptic differential equation associated with minimization problems on Euclidean spaces or manifolds. Nonnegative solutions of those equations are unique, a property that can also be interpreted as a rigidity result. The method relies on linear and nonlinear flows which reveal deep and robust properties of a large class of variational problems. Local results on linear instability leading to symmetry breaking and the bifurcation of non-symmetric branches of solutions are reinterpreted in a larger, global, variational picture in which our flows characterize directions of descent.

This is essentially a survey paper on a large time behavior of solutions of some simple birth and spread models to describe growth of crystal surfaces. The models discussed here include level-set flow equations of eikonal or eikonal-curvature flow equations with source terms. Large time asymptotic speed called growth rate is studied. As an application, a simple proof is given for asymptotic profile of crystal grown by anisotropic eikonal-curvature flow.

I will review the setting and some of the recent results in the field of singular stochastic partial differential equations (SSPDEs). Since Hairer’s invention of regularity structures this field has experienced a rapid development. SSPDEs are non-linear equations with random and irregular source terms which make them ill-posed in classical sense. Their study involves a tight interplay between stochastic analysis, analysis of PDEs (including paradifferential calculus) and algebra.

In this survey, we review recent results in hyperbolic dynamical systems and in geometric inverse problems using analytic tools, based on spectral theory and microlocal methods.

We review recent advances in understanding singularity and small scales formation in solutions of fluid dynamics equations. The focus is on the Euler and surface quasi-geostrophic (SQG) equations and associated models.

Let u be a solution to an elliptic equation div(A∇u) = 0 with Lipschitz coefficients in ℝⁿ. Assume |u| is bounded by 1 in the ball B = {|x| ≤ 1}. We show that if |u| < ε on a set $E\subset \frac{1}{2}B$ with positive n-dimensional Hausdorf measure, then

For the gradients of the solutions we show that a similar propagation of smallness holds for sets of Hausdorff dimension bigger than n – 1 – c, where c > 0 is a small numerical constant depending on the dimension only.

In this paper we consider the problem: ∂_tu − Δu = f (u), u(0) = u₀ ∈ exp L^p (ℝ^N), where p > 1 and f : ℝ → ℝ having an exponential growth at infinity with f (0) = 0. We prove local well-posedness in $\exp L_0^p\left({ℝ}^N\right)$ for $f\left(u\right)\sim e^{{\left|u\right|}^q},0<q\le p,\left|u\right|\to \infty$. However, if for some λ > 0, $\underset{s\to \infty }{\lim }\inf \left(f\left(s\right)e^{-\lambda s^p}\right)>0$ then non-existence occurs in exp L^p (ℝ^N ). Under smallness condition on the initial data and for exponential nonlinearity f such that |f(u)| ∼ |u|^m as u → 0, $\frac{N\left(m-1\right)}{2}\ge p$, we show that the solution is global. In particular, p – 1 > 0 sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.

We review recent results concerning the interactions of solitary waves for several universal nonlinear dispersive or wave equations. Though using quite different techniques, these results are partly inspired by classical papers based on the inverse scattering theory for integrable models.

We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi–Nash–Moser theorem provides Hölder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. The theory of hypoellipticity of Hörmander provides general “bracket” conditions for regularity of solutions to partial differential equations combining first and second order derivative operators when ellipticity fails in some directions. We discuss recent extensions of the De Giorgi–Nash–Moser theory to hypoelliptic equations of Kolmogorov (kinetic) type with rough coefficients. These equations combine a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives in only part of the variables, and with rough coefficients. We then discuss applications to the Boltzmann and Landau equations in kinetic theory and present a program of research with some open questions.

The study of wave propagation outside bounded obstacles uncovers the existence of resonances for the Laplace operator, which are complex-valued generalized eigenvalues, relevant to estimate the long time asymptotics of the wave. In order to understand distribution of these resonances at high frequency, we employ semiclassical tools, which leads to considering the classical scattering problem, and in particular the set of trapped trajectories. We focus on “chaotic” situations, where this set is a hyperbolic repeller, generally with a fractal geometry. In this context, we derive fractal Weyl upper bounds for the resonance counting; we also obtain dynamical criteria ensuring the presence of a resonance gap. We also address situations where the trapped set is a normally hyperbolic submanifold, a case which can help analyzing the long time properties of (classical) Anosov contact flows through semiclassical methods.

The vanishing viscosity problem consists of understanding the limit, or limits, of solutions of the Navier-Stokes equations, with viscosity v, as v tends to zero. The Navier-Stokes equations are a model for real-world fluids and the parameter v represents the ratio of friction, or resistance to shear, and inertia. Ultimately, the relevant question is whether a real-world fluid with very small viscosity can be approximated by an ideal fluid, which has no viscosity. In this talk we will be primarily concerned with the classical open problem of the vanishing viscosity limit of fluid flows in domains with boundary. We will explore the difficulty of this problem and present some known results. We conclude with a discussion of criteria for the vanishing viscosity limit to be a solution of the ideal fluid equations.

We review our construction of the Teichmüller TQFT. We recall our volume conjecture for this TQFT and the examples for which this conjecture has been established. We end the paper with a brief review of our new formulation of the Teichmüller TQFT together with some anticipated future developments.

The moduli spaces of Calabi–Yau (CY) manifolds are the special Kähler manifolds. The special Kähler geometry determines the low-energy effective theory which arises in Superstring theory after the compactification on a CY manifold. For the cases, where the CY manifold is given as a hypersurface in the weighted projective space, a new procedure for computing the Kähler potential of the moduli space has been proposed by Konstantin Aleshkin and myself. The method is based on the fact that the moduli space of CY manifolds is a marginal subspace of the Frobenius manifold which arises on the deformation space of the corresponding Landau–Ginzburg superpotential. I review this approach and demonstrate its efficiency by computing the Special geometry of the 101-dimensional moduli space of the quintic threefold around the orbifold point.

We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems exactly solvable. We illustrate this with: random surfaces, lattice models, and structure constants in representation theory.

We present recent progress in theory of local conformal nets which is an operator algebraic approach to study chiral conformal field theory. We emphasize representation theoretic aspects and relations to theory of vertex operator algebras which gives a different and algebraic formulation of chiral conformal field theory.

We review recent progress in potential theory of second-order elliptic operators and on the metastable behavior of Markov processes.

I will discuss, from a dynamical systems point of view, some recent attempts to rigorously derive the macroscopic laws of transport (e.g. the heat equation) from deterministic microscopic dynamics.

We consider interacting Bose gases trapped in a box Λ = [0; 1] ³ in the Gross– Pitaevskii limit. Assuming the potential to be weak enough, we establish the validity of Bogoliubov’s prediction for the ground state energy and the low-energy excitation spectrum. These notes are based on a joint work with C. Boccato, C. Brennecke and S. Cenatiempo.

We discuss an application of the transfer operator approach to the analysis of the different spectral characteristics of 1d random band matrices (correlation functions of characteristic polynomials, density of states, spectral correlation functions). We show that when the bandwidth W crosses the threshold W = N^1/2, the model has a kind of phase transition (crossover), whose nature can be explained by the spectral properties of the transfer operator.

We give a rough description of the ‘categories’ formed by quantum field theories. A few recent mathematical conjectures derived from quantum field theories, some of which are now proven theorems, will be presented in this language.

Stochastic interface dynamics serve as mathematical models for diverse time-dependent physical phenomena: the evolution of boundaries between thermodynamic phases, crystal growth, random deposition… Interesting limits arise at large space-time scales: after suitable rescaling, the randomly evolving interface converges to the solution of a deterministic PDE (hydrodynamic limit) and the fluctuation process to a (in general non-Gaussian) limit process. In contrast with the case of (1 + 1)-dimensional models, there are very few mathematical results in dimension (d + 1); d ≥ 2. As far as growth models are concerned, the (2 + 1)-dimensional case is particularly interesting: Dietrich Wolf in 1991 conjectured the existence of two different universality classes (called KPZ and Anisotropic KPZ), with different scaling exponents. Here, we review recent mathematical results on (both reversible and irreversible) dynamics of some (2 + 1)-dimensional discrete interfaces, mostly defined through a mapping to two-dimensional dimer models. In particular, in the irreversible case, we discuss mathematical support and remaining open problems concerning Wolf’s conjecture on the relation between the Hessian of the growth velocity on one side, and the universality class of the model on the other.

The following section is included:

• Index of Authors

The following section is included:

• Contents

We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erdős–Schlein–Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random band matrices and the problem of their Anderson transition. We finally expose a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.

Statistical inference from large-scale data can benefit from sources of heterogeneity. We discuss recent progress of the mathematical formalization and theory for exploiting heterogeneity towards predictive stability and causal inference in high-dimensional models. The topic is directly motivated by a broad range of applications and we will show an illustration from molecular biology with gene knock out experiments.

In this essay, we briefly discuss recent developments, started a decade ago in the seminal work of Smirnov and continued by a number of authors, centered around the conformal invariance of the critical planar Ising model on ℤ² and, more generally, of the critical Z-invariant Ising model on isoradial graphs (rhombic lattices). We also introduce a new class of embeddings of general weighted planar graphs (s-embeddings), which might, in particular, pave the way to true universality results for the planar Ising model.

Percolation models describe the inside of a porous material. The theory emerged timidly in the middle of the twentieth century before becoming one of the major objects of interest in probability and mathematical physics. The golden age of percolation is probably the eighties, during which most of the major results were obtained for the most classical of these models, named Bernoulli percolation, but it is really the two following decades which put percolation theory at the crossroad of several domains of mathematics. In this broad review, we propose to describe briefly some recent progress as well as some famous challenges remaining in the field. This review is not intended to probabilists (and a fortiori not to specialists in percolation theory): the target audience is mathematicians of all kinds.

The last twenty-or-so years have seen spectacular progress in our understanding of the fine spectral properties of large-dimensional random matrices. These results have also shown light on the behavior of various statistical estimators used in multivariate statistics. In this short note, we will describe new strands of results, which show that intuition and techniques built on the theory of random matrices and concentration of measure ideas shed new light and bring to the fore new ideas about an arguably even more important set of statistical tools, namely M-estimators and certain bootstrap methods. All the results are obtained in the large n, large p setting, where both the number of observations and the number of predictors go to infinity.

Wave propagation in random media can be studied by multiscale and stochastic analysis. We review some recent advances and their applications. In particular, in a physically relevant regime of separation of scales, wave propagation is governed by a Schrödinger-type equation driven by a Brownian field. We study the associated moment equations and describe the propagation of coherent and incoherent waves. We quantify the scintillation of the wave and the fluctuations of the Wigner distribution. These results make it possible to introduce and characterize correlation-based imaging methods.

We discuss recent results on asymptotically efficient estimation of smooth functionals of covariance operator Σ of a mean zero Gaussian random vector X in a separable Hilbert space based on n i.i.d. observations of this vector. We are interested in functionals that are of importance in high-dimensional statistics such as linear forms of eigenvectors of Σ (principal components) as well as in more general functionals of the form ⟨f (Σ), B⟩, where f : ℝ ↦ ℝ is a sufficiently smooth function and B is an operator with nuclear norm bounded by a constant. In the case when X takes values in a finite-dimensional space of dimension d ≤ n^α for some α ∈ (0, 1) and f belongs to Besov space $B_{\infty ,1}^s$ (ℝ) for $s>\frac{1}{1-\alpha }$, we develop asymptotically normal estimators of ⟨f (Σ), B⟩ with $\sqrt{n}$ convergence rate and prove asymptotic minimax lower bounds showing their asymptotic efficiency.

Random matrix theory has played an important role in recent work on statistical network analysis. In this paper, we review recent results on regimes of concentration of random graphs around their expectation, showing that dense graphs concentrate and sparse graphs concentrate after regularization. We also review relevant network models that may be of interest to probabilists considering directions for new random matrix theory developments, and random matrix theory tools that may be of interest to statisticians looking to prove properties of network algorithms. Applications of concentration results to the problem of community detection in networks are discussed in detail.

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results. This article surveys a series of works with Sheffield in which it is shown that Liouville quantum gravity (LQG) with parameter $\gamma =\sqrt{8/3}$ is equivalent to the Brownian map. We also briefly describe a series of works with Gwynne which use the $\sqrt{8/3}$-LQG metric to prove the convergence of self-avoiding walks and percolation on random planar maps towards SLE_8/3 and SLE₆, respectively, on a Brownian surface.

Modern data analysis challenges require building complex statistical models with massive numbers of parameters. It is nowadays commonplace to learn models with millions of parameters by using iterative optimization algorithms. What are typical properties of the estimated models? In some cases, the high-dimensional limit of a statistical estimator is analogous to the thermodynamic limit of a certain (disordered) statistical mechanics system. Building on mathematical ideas from the mean-field theory of disordered systems, exact asymptotics can be computed for high-dimensional statistical learning problems.

This theory suggests new practical algorithms and new procedures for statistical inference. Also, it leads to intriguing conjectures about the fundamental computational limits for statistical estimation.

In this article we discuss statistical methods of estimating structured nonparametric regression models. Our discussion is mainly on the additive models where the regression function (map) is expressed as a sum of unknown univariate functions (maps), but it also covers some other non- and semi-parametric models. We present the state of the art in the subject area with the prospect of an extension to non-Euclidean data objects.

It is not difficult to find stories of a crisis in modern science, either in the popular press or in the scientific literature. There are likely multiple sources for this crisis. It is also well documented that one source of this crisis is the misuse of statistical methods in science, with the P-value receiving its fair share of criticism. It could be argued that this misuse of statistical methods is caused by a shift in how data is used in 21st century science compared to its use in the mid-20th century which presumed scientists had formal statistical hypotheses before collecting data. With the advent of sophisticated statistical software available to anybody this paradigm has been shifted to one in which scientists collect data first and ask questions later.

We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ℝ^d or ℤ^d. The first class consists of random walks on ℤ^d in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-root-of-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2017, with some hints to the main ideas of the proofs. No technical details are presented here.

In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of ‘containers’ for the independent sets, each of which contains few edges. We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry, and avoiding technical details as much as possible.

Why do natural and interesting sequences often turn out to be log-concave? We give one of many possible explanations, from the viewpoint of “standard conjectures”. We illustrate with several examples from combinatorics.

We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.

A “limit shape” is a form of the law of large numbers, and happens when a large random system, typically consisting of many interacting particles, can be described, after an appropriate normalization, by a certain nonrandom object. Limit shapes occur in, for example, random integer partitions or in random interface models such as the dimer model. Typically limit shapes can be described by some variational formula based on a large deviations estimate. We discuss limit shapes for certain 2-dimensional interface models, and explain how their underlying analytic structure is related to a (conjectural in some cases) conformal invariance property for the models

We give a broad survey of recent results in enumerative combinatorics and their complexity aspects.

The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial structures appeared in many other areas of mathematics and physics, e.g., in the study of cluster algebras, scattering amplitudes, and solitons. We discuss new ways to think about these structures. In particular, we identify plabic graphs and more general Grassmannian graphs with polyhedral subdivisions induced by 2-dimensional projections of hypersimplices. This implies a close relationship between the positive Grassmannian and the theory of fiber polytopes and the generalized Baues problem. This suggests natural extensions of objects related to the positive Grassmannian.

The so-called graph limit theory is an emerging diverse subject at the meeting point of many different areas of mathematics. It enables us to view finite graphs as approximations of often more perfect infinite objects. In this survey paper we tell the story of some of the fundamental ideas in structural limit theories and how these ideas led to a general algebraic approach (the nilspace approach) to higher order Fourier analysis.

We call simple graphs with a linear order on the vertices ordered graphs. Turán-type extremal graph theory naturally extends to ordered graphs. This is a survey on the ongoing research in the extremal theory of ordered graphs with an emphasis on open problems.

We survey results on counting graphs with given degree sequence, focusing on asymptotic results, and mentioning some of the applications of these results. The main recent development is the proof of a conjecture that facilitates access to the degree sequence of a random graph via a model incorporating independent binomial random variables. The basic method used in the proof was to examine the changes in the counting function when the degrees are perturbed. We compare with several previous uses of this type of method.

Query complexity is a model of computation in which we have to compute a function f(x₁, …, x_N) of variables x_i which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover’s quantum search and a key subroutine of Shor’s factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several long-standing problems in the query complexity. In this talk, we survey these results and related work, including:

• the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring $\text{Ω}\left(\sqrt{N}\right)$ queries classically);
• the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with M queries quantiumly but requiring $\widetilde{\text{Ω}}\left(M^{2.5}\right)$ queries probabilistically);
• the biggest probabilistic-vs-detenninistic gap for total functions (a problem solvable with M queries probabilistically but requiring $\widetilde{\text{Ω}}\left(M^2\right)$ queries deterministically);
• the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions);
• the connections between query algorithms and approximations by low-degree polynomials.

The nearest neighbor problem is defined as follows: Given a set P of n points in some metric space (X, D), build a data structure that, given any point q, returns a point in P that is closest to q (its “nearest neighbor” in P). The data structure stores additional information about the set P, which is then used to find the nearest neighbor without computing all distances between q and P. The problem has a wide range of applications in machine learning, computer vision, databases and other fields.

To reduce the time needed to find nearest neighbors and the amount of memory used by the data structure, one can formulate the approximate nearest neighbor problem, where the the goal is to return any point p′ ∊ P such that the distance from q to p′ is at most c · min_p∊P D(q, p), for some c ≥ 1. Over the last two decades many efficient solutions to this problem were developed. In this article we survey these developments, as well as their connections to questions in geometric functional analysis and combinatorial geometry.

Graph Isomorphism (GI) is one of a small number of natural algorithmic problems with unsettled complexity status in the P / NP theory: not expected to be NP-complete, yet not known to be solvable in polynomial time.

Arguably, the GI problem boils down to filling the gap between symmetry and regularity, the former being defined in terms of automorphisms, the latter in terms of equations satisfied by numerical parameters.

Recent progress on the complexity of GI relies on a combination of the asymptotic theory of permutation groups and asymptotic properties of highly regular combinatorial structures called coherent configurations. Group theory provides the tools to infer either global symmetry or global irregularity from local information, eliminating the symmetry/regularity gap in the relevant scenario; the resulting global structure is the subject of combinatorial analysis. These structural studies are melded in a divide-and-conquer algorithmic framework pioneered in the GI context by Eugene M. Luks (1980).

Efficient verification of computation, also known as delegation of computation, is one of the most fundamental notions in computer science, and in particular it lies at the heart of the P vs. NP question.

This article contains a high level overview of the evolution of proofs in computer science, and shows how this evolution is instrumental to solving the problem of delegating computation. We highlight a curious connection between the problem of delegating computation and the notion of no-signaling strategies from quantum physics.

We use the lens of the maximum flow problem, one of the most fundamental problems in algorithmic graph theory, to describe a new framework for design of graph algorithms. At a high level, this framework casts the graph problem at hand as a convex optimization task and then applies to it an appropriate method from the continuous optimization toolkit. We survey how this new approach led to the first in decades progress on the maximum flow problem and then briefly sketch the challenges that still remain.

Estimation is the computational task of recovering a hidden parameter x associated with a distribution D_x, given a measurement y sampled from the distribution. High dimensional estimation problems can be formulated as system of polynomial equalities and inequalities, and thus give rise to natural probability distributions over polynomial systems.

Sum of squares proofs not only provide a powerful framework to reason about polynomial systems, but they are constructive in that there exist efficient algorithms to search for sum-of-squares proofs. The efficiency of these algorithms degrade exponentially in the degree of the sum-of-squares proofs.

Understanding and characterizing the power of sum-of-squares proofs for estimation problems has been a subject of intense study in recent years. On one hand, there is a growing body of work utilizing sum-of-squares proofs for recovering solutions to polynomial systems whenever the system is feasible. On the other hand, a broad technique referred to as pseudocalibration has been developed towards showing lower bounds on degree of sum-of-squares proofs. Finally, the existence of sum-of-squares refutations of a polynomial system has been shown to be intimately connected to the spectrum of associated low-degree matrix valued functions. This article will survey all of these developments in the context of estimation problems.

We consider the problem of determining whether an Erdős–Rényi random graph contains a subgraph isomorphic to a fixed pattern, such as a clique or cycle of constant size. The computational complexity of this problem is tied to fundamental open questions including P vs. NP and NC¹ vs. L. We give an overview of unconditional average-case lower bounds for this problem (and its colored variant) in a few important restricted classes of Boolean circuits.

In recent years, a new “fine-grained” theory of computational hardness has been developed, based on “fine-grained reductions” that focus on exact running times for problems. Mimicking NP-hardness, the approach is to (1) select a key problem X that for some function t, is conjectured to not be solvable by any O(t(n)^1–ε) time algorithm for ε > 0, and (2) reduce X in a fine-grained way to many important problems, thus giving tight conditional time lower bounds for them. This approach has led to the discovery of many meaningful relationships between problems, and to equivalence classes.

The main key problems used to base hardness on have been: the 3-SUM problem, the CNF-SAT problem (based on the Strong Exponential Time Hypothesis (SETH)) and the All Pairs Shortest Paths Problem. Research on SETH-based lower bounds has flourished in particular in recent years showing that the classical algorithms are optimal for problems such as Approximate Diameter, Edit Distance, Frechet Distance and Longest Common Subsequence.

This paper surveys the current progress in this area, and highlights some exciting new developments.

The sedimentation of a suspension is a unit operation widely used in mineral processing, chemical engineering, wastewater treatment, and other industrial applications. Mathematical models that describe these processes and may be employed for simulation, design and control are usually given as nonlinear, time-dependent partial differential equations that in one space dimension include strongly degenerate convection-diffusion-reaction equations with discontinuous coefficients, and in two or more dimensions, coupled flow-transport problems. These models incorporate non-standard properties that have motivated original research in applied mathematics and numerical analysis. This contribution summarizes recent advances, and presents original numerical results, for three different topics of research: a novel method of flux identification for a scalar conservation law from observation of curved shock trajectories that can be observed in sedimentation in a cone; a new description of continuous sedimentation with reactions including transport and reactions of biological components; and the numerical solution of a multidimensional sedimentation-consolidation system by an augmented mixed-primal method, including an a posteriori error estimation.

In this work a general strategy to design high order well-balanced schemes for hyperbolic system with nonconservative products and/or source terms is reviewed. We briefly recall the theory of Dal Maso-LeFloch-Murat to define weak solutions of nonconservative systems and how it has been used to establish the notion of path-conservative schemes. Next, a family of high order finite volume methods combining a reconstruction operator and a first order path-conservative scheme is described. Then, the well-balanced property of the proposed methods is analyzed. Finally, some challenging examples on tsunami modeling are shown.

This lecture serves as an invitation to further studies on nonlocal models, their mathematics, computation, and applications. We sample our recent attempts in the development of a systematic mathematical framework for nonlocal models, including basic elements of nonlocal vector calculus, well-posedness of nonlocal variational problems, coupling to local models, convergence and compatibility of numerical approximations, and applications to nonlocal mechanics and diffusion. We also draw connections with traditional models and other relevant mathematical subjects.

In recent years there has been very substantial growth in stochastic modelling in many application areas, and this has led to much greater use of Monte Carlo methods to estimate expected values of output quantities from stochastic simulation. However, such calculations can be expensive when the cost of individual stochastic simulations is very high. Multilevel Monte Carlo greatly reduces the computational cost by performing most simulations with low accuracy at a correspondingly low cost, with relatively few being performed at high accuracy and a high cost.

This article reviews the key ideas behind the multilevel Monte Carlo method. Some applications are discussed to illustrate the flexibility and generality of the approach, and the challenges in its numerical analysis.

Quasicrystals are one kind of fascinating aperiodic structures, and give a strong impact on material science, solid state chemistry, condensed matter physics and soft matters. The theory of quasicrystals, included in aperiodic order, has grown rapidly in mathematical and physical areas over the past few decades. Many scientific problems have been explored with the efforts of physicists and mathematicians. However, there are still lots of open problems which might to be solved by the close collaboration of physicists, mathematicians and computational mathematicians. In this article, we would like to bridge the physical quasicrystals and mathematical quasicrystals from the perspective of numerical mathematics.

Kinetic modeling and computation face the challenges of multiple scales and uncertainties. Developing efficient multiscale computational methods, and quantifying uncertainties arising in their collision kernels or scattering coefficients, initial or boundary data, forcing terms, geometry, etc. have important engineering and industrial applications. In this article we will report our recent progress in the study of multiscale kinetic equations with uncertainties modelled by random inputs. We first study the mathematical properties of uncertain kinetic equations, including their regularity and long-time behavior in the random space, and sensitivity of their solutions with respect to the input and scaling parameters. Using the hypocoercivity of kinetic operators, we provide a general framework to study these mathematical properties for general class of linear and nonlinear kinetic equations in various asymptotic regimes. We then approximate these equations in random space by the stochastic Galerkin methods, study the numerical accuracy and long-time behavior of the methods, and furthermore, make the methods “stochastically asymptotic preserving”, in order to handle the multiple scales efficiently.

A large variety of efficient numerical methods, of the finite volume, finite difference and DG type, have been developed for approximating hyperbolic systems of conservation laws. However, very few rigorous convergence results for these methods are available. We survey the state of the art on this crucial question of numerical analysis by summarizing classical results of convergence to entropy solutions for scalar conservation laws. Very recent results on convergence of ensemble Monte Carlo methods to the measure-valued and statistical solutions of multi-dimensional systems of conservation laws are also presented.

In this article, we overview recent developments of modern computational methods for the approximate solution of phase-field problems. The main difficulty for developing a numerical method for phase field equations is a severe stability restriction on the time step due to nonlinearity and high order differential terms. It is known that the phase field models satisfy a nonlinear stability relationship called gradient stability, usually expressed as a time-decreasing free-energy functional. This property has been used recently to derive numerical schemes that inherit the gradient stability. The first part of the article will discuss implicit-explicit time discretizations which satisfy the energy stability. The second part is to discuss time-adaptive strategies for solving the phase-field problems, which is motivated by the observation that the energy functionals decay with time smoothly except at a few critical time levels. The classical operator-splitting method is a useful tool in time discrtization. In the final part, we will provide some preliminary results using operator-splitting approach.

In this paper, we review a set of fast and spectrally accurate methods for rapid evaluation of three dimensional electrostatic and Stokes potentials. The algorithms use the so-called Ewald decomposition and are FFT-based, which makes them naturally most efficient for the triply periodic case. Two key ideas have allowed efficient extension of these Spectral Ewald (SE) methods to problems with periodicity in only one or two dimensions: an adaptive 3D FFT that apply different upsampling rates locally combined with a new method for FFT based solutions of free space harmonic and biharmonic problems. The latter approach is also used to extend to the free space case, with no periodicity. For the non-radial kernels of Stokes flow, the structure of their Fourier transform is exploited to extend the applicability from the radial harmonic and biharmonic kernels.

A window function is convolved with the point charges to assign values on the FTT grid. Spectral accuracy is attained with a variable number of points in the support of the window function, tuning a shape parameter according to this choice. A new window function, recently introduced in the context of a non-uniform FFT algorithm, allows for further reduction in the computational time as compared to the truncated Gaussians previously used in the SE method.

We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing our previous results. To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton’s method as well as of their linesearch variants. For each method in this class and arbitrary accuracy threshold ∊ ∈ (0, 1), we exhibit a smooth objective function with bounded range, whose gradient is globally Lipschitz continuous and whose Hessian is α–Hölder continuous (for given α ∈ [0, 1]), for which the method in question takes at least ⌊∊^{−(2+α)/(1+α)}⌋ function evaluations to generate a first iterate whose gradient is smaller than ∊ in norm. Moreover, we also construct another function on which Newton’s takes ⌊∊⁻²⌋ evaluations, but whose Hessian is Lipschitz continuous on the path of iterates. These examples provide lower bounds on the worst-case evaluation complexity of methods in our class when applied to smooth problems satisfying the relevant assumptions. Furthermore, for α = 1, this lower bound is of the same order in ∊ as the upper bound on the worst-case evaluation complexity of the cubic regularization method and other algorithms in a class of methods recently proposed by Curtis, Robinson and Samadi or by Royer and Wright, thus implying that these methods have optimal worst-case evaluation complexity within a wider class of second-order methods, and that Newton’s method is suboptimal.

We consider inverse problems for hyperbolic equations and systems and the solutions of these problems based on the focusing of waves. Several inverse problems for linear equations can be solved using control theory. When the coefficients of the modelling equation are unknown, the construction of the point sources requires solving blind control problems. For non-linear equations we consider a new artificial point source method that applies the non-linear interaction of waves to create microlocal points sources inside the unknown medium. The novel feature of this method is that it utilizes the non-linearity as a tool in imaging, instead of considering it as a difficult perturbation of the system. To demonstrate the method, we consider the non-linear wave equation and the coupled Einstein and scalar field equations.

The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on K. It turns out that this methodology can be also applied to solve problems with positivity constraints “f (x) ≥ 0 for all x ∊ K” and/or linear constraints on Borel measures. Such problems can be viewed as specific instances of the “Generalized Problem of Moments” (GPM) whose list of important applications in various domains is endless. We describe this methodology and outline some of its applications in various domains.

The realization that many nondifferentiable functions exhibit some form of structured nonsmoothness has been atracting the efforts of many researchers in the last decades. Identifying theoretically and computationally certain manifolds where a nonsmooth function behaves smoothly poses challenges for the nonsmooth optimization community. We review a sequence of milestones in the area that led to the development of algorithms of the bundle type that can track the region of smoothnes and mimic a Newton algorithm to converge with superlinear speed. The new generation of bundle methods is sufficiently versatile to deal with structured objective functions, even when the available information is inexact.

Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expressed as the projection of a much simpler set in higher dimensions called a lift of the original set. This is a brief survey of recent developments in the topic of lifts of convex sets. Our focus will be on lifts that arise from affine slices of real positive semidefinite cones known as psd or spectrahedral lifts. The main result is that projection representations of a convex set are controlled by factorizations, through closed convex cones, of an operator that comes from the convex set. This leads to several research directions and results that lie at the intersection of convex geometry, combinatorics, real algebraic geometry, optimization, computer science and more.