Showing papers in "Oberwolfach Reports in 2015"

Subfactors and Conformal Field Theory

Abstract: We discuss relations between the combinatorial structure of subfactors, solvable lattice models, (rational) conformal field theory, and topological quantum field theory.

History of Mathematics: Models and Visualization in the Mathematical and Physical Sciences

Abstract: This workshop brings together historians of mathematics and science as well as mathematicians to explore important historical developments connected with models and visual elements in the mathematical and physical sciences. It will address the larger question of what mathematicians mean by a model, a term that has been used in a variety of contexts, both within pure mathematics as well as in applications to other fields. Most of the talks will present case studies from the period 1800 to 1950 that deal with the modelling of analytical, geometrical, mechanical, astronomical, and physical phenomena. Some speakers will also show how computergenerated models and animations can be used to enhance visual understanding. This workshop will also consider the role of visual thinking as a component of mathematical creativity and understanding. For the period in view, we hope to form a provisional picture of how models and visual thinking shaped important historical developments.

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

Abstract: The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms. Mathematics Subject Classification (2010): 34-99, 35F25, 35L65, 41A10, 76-99. Introduction by the Organisers The workshop Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, organised by Rémi Abgrall (Zürich), Willem Hundsdorfer (Amsterdam), Andreas Meister (Kassel) and Thomas Sonar (Braunschweig) was held September 14th–September 19th, 2015. This meeting was well attended with over 50 participants with broad geographic representation from all continents. Since modern numerical methods like Discontinuous Galerkin or Spectral Element Finite Difference methods are based on orthogonal polynomials on simplices and use modal filters as well as methods for edge detection and many more mathematical devices from different areas of research we decided to invite renowned 2400 Oberwolfach Report 41/2015 researchers from numerical methods for partial differential equations and approximation theory. Furthermore, to couple mathematical precision with a large range of applicability we also invited scientist from engineering departments working in the field of numerical schemes. The talks ranged from new Runge-Kutta solvers, new filters and edge detection algorithms, Discontinuous Galerkin methods, Spectral Difference methods, Finite Difference operators, implicit solvers, and finite volume methods to the modeling of shallow water flow, viscous as well as inviscid fluid flow and solid mechanics. Discussions were lively and many different research areas met for the first time resulting in interesting talks and contacts. The workshop was a tremendous success and we are looking forward to repeat this kind of conference in Oberwolfach again in a few years. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Numerics of Nonlinear Hyperbolic Conservation Laws 2401 Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

Geometric and Algebraic Combinatorics

Abstract: The 2015 Oberwolfach meeting "Geometric and Algebraic Com- binatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) counterexamples to the topological Tverberg con- jecture, and (2) the latest results around the Heron-Rota-Welsh conjecture.

Mathematical Methods in Quantum Molecular Dynamics

Abstract: The workshop brought together chemists developing mathematical and computational tools for studying the motion of atoms in polyatomic molecules and mathematicians interested in numerical methods for highdimensional problems and semiclassical mechanics. Quantum and semiclassical methods applicable to diatomic molecules are well known and widely used, but the outstanding problem in this field is devising new mathematical and computation tools for studying larger molecules. This is difficult due to the dimensionality of the problem. In principle, molecular dynamics can be understood by solving the time-dependent Schrodinger equation. However, because 3N coordinates are required to specify the configuration of the nuclei in a molecular or reacting system with N atoms, quantum molecular dynamics calculations must deal with very high dimension. This is typically referred to as “the curse of dimensionality.” Effective computational approaches exist for solving differential equations in up to three dimensions, but for a molecule with 6 atoms one must deal with 18 dimensions! Three of the 18 coordinates can be chosen to specify the position of the centre of mass of the system, and are therefore easy to separate. It is common to select coordinates so that three others describe the rotational orientation of the system, and if this is done there are 3N − 6 coordinates describing the shape of the molecule or reacting system. Rotation, however, does not separate because of Coriolis and centrifugal coupling. Although one can easily write down molecular Schrodinger equations, one cannot solve them. So, one resorts to various approximations, primarily to deal with the very high dimensionality of the problem. At the workshop, mathematicians learned what theoretical chemists are doing and what difficulties they must overcome. Chemists learned about rigorous mathematical results obtained recently by mathematicians. This primarily involved theoretical work, but also included ways to deal with high dimensionality when using computers for approximations. In prior conferences and workshops in this subject, there have been significant difficulties getting chemists and mathematicians to talk to one another in a meaningful way. There are differences in nomenclature, and people in the two disciplines often have different aims and priorities. A main goal of this workshop was to facilitate as much interaction between the two groups of individuals as possible, and in this regard, the workshop was very successful.

Mini-Workshop: Modern Applications of $s$-numbers and Operator Ideals

Abstract: The main aim of this mini-workshop was to present and discuss some modern applications of the functional-analytic concepts of s-numbers and operator ideals in areas like Numerical Analysis, Theory of Function Spaces, Signal Processing, Approximation Theory, Probability on Banach Spaces, and Statistical Learning Theory. Mathematics Subject Classification (2010): Primary: 47B06, 47L20; Secondary: 41A65, 41A63, 60G15, 68T05, 94A20. Introduction by the Organisers The mini-workshop was devoted to modern applications of s-numbers and operator ideals in a various more applied areas. It was attended by 16 mathematicians from Germany (10), Spain (2), Austria (1), Canada (1), Finland (1) and France (1), the participants were a mixture of experienced senior scientists and younger researchers, with different mathematical backgrounds. The theories of s-numbers and operator ideals, which are both closely related to geometry and local theory of Banach spaces, and also to probability on Banach spaces, were already developed in the 1970s and 1980s, with main contributions due to Albrecht Pietsch. During the last 15 years these by now almost classical abstract functional-analytic concepts appeared quite naturally in several, more applied branches of mathematics. In particluar, they have found important applications in areas such as 370 Oberwolfach Report 6/2015 • Compressed Sensing and Image Processing (Gelfand numbers, JohnsonLindenstrauss lemma) • Numerical Analysis and Information-based Complexity (approximation and entropy numbers, 2-summing operators, Banach spaces of type) • Function Spaces (various s-numbers, operator ideal techniques) • Approximation Theory (abstract approximation spaces) • Small Deviations of Gaussian Processes (entropy numbers) • Statistical Learning Theory (covering numbers) The main aims of the Mini-Workshop were to • bring together experienced functional analysts and younger researchers from applied areas, • present some modern applications of s-numbers and operator ideals in the above-mentioned areas, • discuss open problems and identify directions for future research, • initiate exchange and co-operation between different communities. In order to achieve these goals the mini-workshop was organized as follows. Each participant gave a 50-minutes talk on her/his field of research, pointing out in particular the use of s-number and operator ideal techniques, open questions and relations to other fields. In this way the participants from different communities could learn from each other, and the ground was laid for further discussions. The first talk was given by Albrecht Pietsch, who presented an overview over important problems that have been left open in the area of s-numbers and operator ideals itself. Let us mention just one, which is related to the famous counterexample by Enflo, who showed that there are Banach spaces without the approximation property. In the language of operators, the problem is to quantify the gap between compact and approximable operators, that means to determine the smallest entropy ideal that contains non-approximable operators. The four organizers gave survey talks on the role of s-numbers and operator ideals in the theory of function spaces (Dorothee Haroske), approximation theory (Fernando Cobos), signal processing and numerical analysis (Tino Ullrich) and Gaussian processes (Thomas Kuhn). In the talks of the remaining participants several other interesting topics were presented, e.g. Khintchine-type inequalities (Hermann Konig, Gilles Pisier), entropy inequalities (Nicole Tomczak-Jaegermann), tractability problems in information-based complexity (Stefan Heinrich), polynomials on Banach spaces (Andreas Defant), entropy numbers in statistical learning theory (Ingo Steinwart), singular traces (Albrecht Pietsch). Moreover, apart from these talks which were already scheduled in advance, we spontaneously organized an informal session on Thursday afternoon, in which Hermann Konig lectured on new developments concerning the famous Grothendieck constant. Since the works of Krivine in 1975 in the real case and Haagerup in 1986 in the complex case, there has been no progress for many years. Only very recently Assaf Naor introduced new averaging processes which led to an improvement. But still the problem of the exact value of the Grothendieck constant is wide open. Mini-Workshop: Modern Applications of s-numbers and Operator Ideals 371 This talk was followed by a problem session, where several really important and quite challenging problems were presented and discussed. These problems covered a wide range, e.g. Schur multipliers on Schatten p-classes, cotype of projective tensor products, tractability of star-discrepancy, approximation vs. sampling numbers, entropy numbers in learning theory. During the whole week of the mini-workshop there was an intensive scientific interaction between the participants from different communities. There were lot of discussions in smaller groups on specific problems, which led to several new mathematical contacts and to first plans for concrete projects of future co-operation. Throughout the mini-workshop the atmosphere was very inspiring. As usual, on Wednesday afternoon we had an excursion, this time consisting of a walk to Oberwolfach and a visit of the MiMa. The guided tour through the museum with so many beautiful minerals and the interactive mathematics part was very interesting and enjoyable, thanks to our expert tour guide Stephan Klaus. Last but not least we would like to thank – on behalf of all participants – the director, administration and staff of the MFO for their excellent professional work and kind support before and during the mini-workshop. This made it possible to create the fruitful scientific atmosphere which is so typical for Oberwolfach meetings, and to make our mini-workshop a full success, an impression shared by all participants. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Mini-Workshop: Modern Applications of s-numbers and Operator Ideals 373 Mini-Workshop: Modern Applications of s-numbers and Operator Ideals

Mixed-integer Nonlinear Optimization: A Hatchery for Modern Mathematics

Abstract: The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special “open problems” session. Mathematics Subject Classification (2010): 9006, 90C11, 90C22, 90C26, 90C30. Introduction by the Organisers This report refers to the second workshop on MINLP organized at Oberwolfach. We refer to the report following the first workshop [4] for a somewhat longer definition of MINLP. In summary, MINLP is one of the most general classes of MP, which is itself a formal language used to describe optimization problems in terms of parameters (the input of the problem), decision variables (which will contain the output after the solution procedure), an objective function to be optimized, and some constraints to be satisfied. The workshop was organized in 5 tutorial talks (one per day, one hour long, including 15 minutes for questions), 21 “normal” talks (45 minutes long, including 15 minutes for questions), 11 short research announcements (SRA — 15 minutes 2 Oberwolfach Report 26/2019 long, including 5 minutes for questions), and one open problems session proposing 9 new open problems in the field of MINLP, and attended by everyone at the workshop. The discussion after practically all talks was lively and filled with questions from many attendees. As Oberwolfach tradition warrants, we spent Wednesday afternoon hiking towards a scrumptious Schwarzwälderkirschtorte in St. Roman, a little more than 8km away from the Institute.

Geometric Partial Differential Equations: Surface and Bulk Processes

Abstract: The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems.

Mini-Workshop: Discrete p-Laplacians: Spectral Theory and Variational Methods in Mathematics and Computer Science

Abstract: The p-Laplacian operators have a rich analytical theory and in the last few years they have also offered efficient tools to tackle several tasks in machine learning. During the workshop mathematicians and theoretical computer scientists working on models based on p-Laplacians on graphs and manifolds have presented the latest theoretical developments and have shared their knowledge.

Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems

Abstract: It is a fundamental challenge for many problems of significant current interest in algebraic geometry and commutative algebra to understand symbolic powers I(m) of homogeneous ideals I in polynomial rings, particularly ideals of linear varieties. Such problems include computing Waring ranks of polynomials, determining the occurrence of equality I(m) = I (or, more generally, of containments I(m) ⊆ I), computing Waldschmidt constants (i.e., determining the limit of the ratios of the least degree of an element in I(m) to the least degree of an element of I), and studying major conjectures such as Nagata’s Conjecture and the uniform SHGH Conjecture (which respectively specify the Waldschmidt constant of ideals of generic points in the plane and the Hilbert functions of their symbolic powers). Mathematics Subject Classification (2010): Primary: 14N20, 11P05; Secondary: 13F20, 13P10. Introduction by the Organisers The mini-workshop, Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems, involved 11 men and 7 women (one of whom, due to medical issues, did not attend in person but presented her talk by skype). The participants represented 6 different countries, and were drawn from all career ranks (2 postdocs, 3 early career researchers, 6 midcareer researchers and 7 senior researchers), covering a range of specialties and expertise. This variety of expertise not only generated stimulating discussions during the workshop, but the working group sessions have led to at least three on-going research collaborations which are expected to be the basis for a number of research articles in the near future. 490 Oberwolfach Report 9/2015 The theme of the workshop Ideals of linear subspaces, and points in particular, have long held a prominent position in algebraic geometry. They have, in particular, played a prime role in recent progress on the Waring Problem for forms, which deals with power sum representations of forms, i.e., expressions of the type F = L1 + . . . + L d r , where F is a form of degree d and the Li are forms of degree 1. A crucial quantity for this problem is the Waring rank rk(F ) of F , defined as the least r for which F can be written as such a sum of powers. In the 90s, results of Alexander and Hirschowitz [1] for ideals of points in projective space gave the dimension of all secant varieties of Veronese varieties, which in turn determined rk(F ) for generic forms F of any degree in any number of variables. But the Waring rank for a specific form can be larger than this generic value; obtaining bounds for rk(F ) is an active area of research for which ideals of points have played a crucial role. For example, Carlini, Catalisano and Geramita [9] use the geometry of reduced points to compute the Waring rank of monomials and of the sum of coprime monomials. Before this result, the Waring rank was explicitly known only for quadratic forms, binary forms and cubic ternary forms. Ideals of linear subspaces have also been a focus of attention in recent research on the question of which symbolic powers of an ideal are equal to or at least contained in specific ordinary powers of the ideal. Interest in which powers are symbolic goes back at least to work of Hochster [28] and more recently has gotten attention in the work of Morey [32] and Li and Swanson [31]. In a talk in the late 00s, Huneke asked whether I = I for all m ≥ 1 if I = I, given any homogeneous ideal I of big height c. Recent work of Guardo, Harbourne and Van Tuyl [21, 23] has exploited the fact that ideals of arrangements of points in P1×P1 can be regarded as defining arrangements of lines in P to give a negative answer to Huneke’s question. The question of containment of symbolic powers of an ideal in specific ordinary powers of the ideal has seen even more explosive growth, starting with a paper of Swanson [34] which prompted the seminal papers of Ein-Lazarsfeld-Smith and Hochster-Huneke [16, 29] showing (as one minor consequence) that the symbolic fourth power I of any radical ideal of points in the projective plane is contained in I. (In the case of the radical ideal I of points p1, . . . , ps ∈ P, we note that I = ∩1≤i≤s(I(pi)) where I(pi) is the ideal generated by all forms that vanish at pi.) Further stimulation came from the following question of Huneke. Question: If I is the radical ideal of points in the projective plane, must it be true that I ⊆ I? While a number of important basic results are now available (see [6, 7, 8, 11, 12, 13, 21]), it was only two years ago that a special configuration of points was discovered giving a negative answer to Huneke’s question [14]. Since then additional mainly sporadic examples have been found [27, 2, 33] giving counterexamples to the containment in Huneke’s question (and in the case of [27], also to certain Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems 491 related containment conjectures posed in [3, 26]), but many questions and conjectures remain. In addition, new avenues of research have opened up (see, e.g., [26]) and old problems that had become quiescent have been given new life. One such problem that has been resurrected is that of computing Waldschmidt constants. In the late 70s, in work related to transcendence questions in number theory, Waldschmidt introduced an asymptotic quantity α̂(I) [36] for homogeneous ideals I, now known as a Waldschmidt constant [15]. It is defined as limm→∞ α(I) m , where for any homogeneous ideal J 6= (0), α(J) is the degree of a generator of J of least degree. Efforts to compute or estimate Waldschmidt constants started soon after its introduction [10, 17]. The fact that it, and variants of it, are closely related to the problem of which symbolic powers of an ideal I are contained in given ordinary powers of I [7, 8, 22] has caused a resurgence of interest in computing Waldschmidt constants; see for example [3, 15, 19, 18]. Additional related work [22, 23] which became a focus of discussion at the workshop used the connection between points in multi-projective spaces and higher dimensional linear varieties in single projective spaces to study Waldschmidt constants. The foundation for these papers was understanding points in P × P; previous work on points in P × P, such as, for example, [20, 24, 25, 35] provided important tools relied on in [21]. Moreover, the recent attention given to Waldschmidt constants has led to additional new questions, starting with the paper [5] of Bocci and Chiantini, related to fattenings of linear subvarieties in projective space (see [4, 30]). The structure of the workshop The design of the workshop was successful in prompting a lot of research interaction. Short talks (35 minutes each) by participants were scheduled for the mornings, with afternoons and evenings reserved for working on specific problems raised by the participants. Potential problems for workshopping were solicited from the participants in advance of the meeting. On the first day of the workshop, the participants by acclimation settled on three main problems to focus on during the workshop. Participants were free to move from one discussion group to another and to change the focus of the discussions, as warranted by individual interest and by the potential for progress. The topics selected for focused discussions in the discussion groups were as follows: • H-constants and ideal containments; • Waring rank problems; and • computing Waldschmidt constants and stability questions (how many powers of an ideal must be symbolic for all of them to be symbolic). 492 Oberwolfach Report 9/2015

Mini-Workshop: Singular Curves on $K3$ Surfaces and Hyperkähler Manifolds

Abstract: The workshop focused on Severi varieties on K3 surfaces, hyperkähler manifolds and their automorphisms. The main aim was to bring researchers in deformation theory of curves and singularities together with researchers studying hyperkähler manifolds for mutual learning and interaction, and to discuss recent developments and open problems. Mathematics Subject Classification (2010): Primary: 14H10, 14H20, 14H51, 14C20, 14J28, 14J50; Secondary: 14B07, 14E30. Introduction by the Organisers The workshop was attended by 15 participants with broad geographic and thematic representation. Its main aim was to bring together researchers in deformation theory of curves and singularities, especially working on Severi varieties of singular curves on K3 surfaces, together with researchers studying hyperkähler manifolds and their automorphisms. Severi varieties take their name from the mathematician who introduced them at the beginning of last century. Let S be a smooth complex projective surface and |D| a linear system on S containing smooth irreducible curves. The Severi variety of δ-nodal curves V S |D|,δ ⊆ |D| is defined as the locally closed subset of |D| parametrizing irreducible curves with only δ nodes as singularities. Curves on smooth surfaces, their moduli and their enumerative geometry have been fundamental topics of algebraic geometry from the beginning of the previous century until today, thanks to the contribution of Severi, Segre, Zeuthen, Albanese, Enriques, Castelnuovo, Zariski, Arbarello, Cornalba, Harris, Shustin, Greuel and 2940 Oberwolfach Report 51/2015 many others. An important breakthrough was made by Harris [18], who proved that Severi varieties of nodal plane curves are irreducible, as stated by Severi. Some years later, Kontsevich and Manin [23], by using Gromow-Witten theory, computed the degree of the Severi variety of rational plane curves. Their formulas were generalized by Caporaso and Harris [10], who found a recursive formula for the degree of Severi varieties of nodal plane curves of any genus, using only classical techniques. Later on, great progress was made in the study of the enumerative geometry of V S |D|,δ, by among others Pandharipande, Vakil, Ran, Göttsche, Yau, Zaslow, Vainsencher, Tzeng and Thomas. Although a lot of work has been made on Severi varieties, many interesting problems remain open, especially in the case of K3 surfaces, as explained in the abstracts of Ciliberto–Flamini and Dedieu. At the same time, the Brill-Noether theory of smooth curves on K3 surfaces has received a lot of attention in the last couple of decades, from the seminal papers of Lazarsfeld and Green [24, 17] to the more recent works on the Green conjecture and divisors on the moduli space of curves of Voisin, Farkas, Popa and Aprodu [26, 25, 14, 1]. Very recently, two conjectures about syzygies of curves, the Green-Lazarsfeld secant conjecture and the Prym-Green conjecture were (essentially) solved by Farkas and Kemeny in [12, 13] using curves on K3 surfaces, and an account of this is given in Kemeny’s abstract. Similarly, two outstanding conjectures by Wahl were established in [2], where it is proved that a Brill-NoetherPetri curve of genus ≥ 12 lies on a polarised K3 surface or on a limit of such if and only if the Wahl map for C is not surjective. An account of related open problems is made in Sernesi’s abstract. The recent paper [11] starts the study of Brill-Noether theory of singular curves on a K3 surface S. Besides its intrinsic interest, the study is related to Mori theory of hyperkähler manifolds: indeed, curves on S with normalizations carrying pencils of degree k define rational curves on the Hilbert scheme S of k points on the surface, one of the few examples known (together with its deformations) of hyperkähler manifolds. The other known examples are Albanese fibers of Hilbert schemes of points on abelian surfaces, called generalized Kummer varieties, (and their deformations), as well as two examples of O’Grady in dimensions 6 and 10. We recall that a (compact) hyperkähler manifold (or irreducible holomorphic symplectic manifold) is a simply-connected compact complex Kähler manifold X such that H(X,ΩX) is spanned by a nowhere degenerate two-form. The interest in hyperkähler manifolds stems from Bogomolov’s decomposition theorem for compact, complex Kähler manifolds with trivial canonical bundle in the 70s: up to finite étale cover they all decompose into products of Calabi-Yau, hyperkähler manifolds and tori. The birational geometry of hyperkähler manifolds is determined by their rational curves; in particular, rational curves determine their nef and ample cones, just like for K3s. Many years of research on this topic, passing in particular through several works and conjectures of Hassett and Tschinkel, culminated recently in the work of Bayer and Macr̀ı [5] using Bridgeland stability, which determines (up to numerical computations) the extremal rays of the Mori cone of the Hilbert schemes of points on a K3 surface. Singular Curves on K3 Surfaces and Hyperkähler Manifolds 2941 Despite recent advances by different methods, the study of curves on K3 or abelian surfaces with normalizations carrying special pencils still seems to be the most efficient way of concretely producing rational curves on hyperkähler manifolds. The results in [11] were recently extended to abelian surfaces in [21]. Some consequences of the results in [11, 21] on the birational geometry of the associated hyperkähler manifolds are obtained in [22] and the results and some open problems are given in Knutsen’s abstract. Many of the recent results on singular curves on K3 (and abelian) surfaces have been proved by degenerating the surfaces. It is therefore natural to ask whether one can find similar degenerations of hyperkähler manifolds, as is done in Galati’s abstract, which also gives a brief account on the K3 case. Another way of producing rational curves on S is through automorphisms, as in e.g. [15]: the idea is to start with a special K3 surface such that S contains a family of rational curves not present on the general projective deformation of it, use an automorphism of S to produce another family of rational curves, and prove that the latter can be preserved under deformation. This is an interesting point of view, but one needs automorphisms of S not coming from automorphisms of S, i.e. non-natural, and at the moment only one such example is known: the involution of Beauville on S when S is a quartic. Thus one is in need of new such constructions. But the construction of new non-natural automorphisms on S and more generally on other hyperkähler manifolds is an interesting and very active research topic on its own. The interest in automorphisms of hyperkähler manifolds has grown tremendously the last years. The foundational work on K3 surfaces by Nikulin, Mukai and Morrison was followed by classification results of Sarti with coauthors [3, 4, 16] and the recent work of Huybrechts [20]. Finally, the study of non-symplectic automorphisms on K3 surfaces has found a recent application in the study of Chow groups of K3 surfaces in particular in relation to the study of rational curves and the Bloch-Beilinson conjecture [19, 20]. Very little is known in higher dimensions, again there are results of Sarti, Boissière and coauthors [6, 7, 8, 9]. The abstract of Boissière gives an overview of results on automorphisms of special hyperkähler manifolds; more precise results and some open problems are formulated in the abstracts of Camere and Cattaneo, concerning existence of automorphisms and moduli spaces. The abstracts of Lehn, Saccà and Markushevich explain other fundamental topics related to hyperkähler manifolds such as the construction of new manifolds, computation of Hodge numbers and Lagrangian fibrations. Finally, the abstract of Ohashi explains results on the automorphism group of Enriques surfaces and curve configurations. The study of the automorphism group of Enriques surfaces is very natural when studying automorphisms of K3 surfaces. To promote interaction, the participants were asked to focus their talks on background results and open problems. Most talks were given in the first two days of the workshop to have time to discuss the proposed problems. We present the abstracts in chronological order and end with a few lines about the discussed open questions. 2942 Oberwolfach Report 51/2015

Scaling Limits in Models of Statistical Mechanics

Abstract: This conference (part of a long running series) aims to cover the interplay between probability and mathematical statistical mechanics. Specific topics addressed during the 22 talks include: Universality and critical phenomena, disordered models, Gaussian free field (GFF), random planar graphs and unimodular planar maps, reinforced random walks and non-linear σ-models, non-equilibrium dynamics. Less stress is given to topics which have running series of Oberwolfach conferences devoted to them specifically, such as random matrices or integrable models and KPZ universality class. There were 50 participants, including 9 postdocs and graduate students, working in diverse intertwining areas of probability, statistical mechanics and mathematical physics. Subject classification: MSC: 60,82; IMU: 10,13. Introduction by the Organisers This workshop was a sequel to a MFO conference, by the same organizers, which took place in 2015. More broadly, it is a sequel to MFO conferences in 2006, 2009 and 2012, organised by Ken Alexander, Marek Biskup, Remco van der Hofstad and Vladas Sidoravicius. The main focus of the conference remained on probabilistic and analytic methods of non-integrable statistical mechanics. With respect to the previous editions, greater emphasis was put on statistical mechanics models on groups and general graphs, as a lot has happened in this arena recently. The list of 50 participants reflects our attempts to maintain an optimal balance between diverse fields, leading experts and promising young researchers. Nine participants were on postdoctoral and graduate level. 2536 Oberwolfach Report 41/2018 In our choice of 22 talks we tried to illuminate major recent advances in the field and to expose and address at least some aspects of the works for each and every one of the participants. A more detailed account of the presentations is given below. Due to an intended intertwining of topics and themes it is hard to give an unambiguous classification. Statistical mechanical models on groups and general graphs. Tom Hutchcroft described an improved proof of the Aizenman-Kesten-Newman arm exponent estimate, and applications to percolation on hyperbolic groups. Aran Raoufi described joint work with Duminil-Copin, Goswami, Severo and Yadin in which they showed that percolation on every graph with isoperimetric dimension at least 4 has a non-trivial phase transition. Christoph Garban described work on the inverted orbit of random walk on interval exchange transformation groups, related to the Thompson group. Two dimensional models. The understanding of two dimensional models proceeds rapidly and we heard 5 talks on the topic. Vincent Tassion discussed a renormalisation scheme which allows to show a quadrachotomy for the two dimensional Potts model. Béatrice de Tilière talked about massive Laplacians on isoradial graphs. Giambattista Giacomin revisited a classic paper of McCoy and Wu about analyticity of the pressure of the Ising model with columnar quenched disorder. Two more talks were on Gaussian multiplicative chaos, a model representing the scaling limit of planar random graphs: Hubert Lacoin talked about its fluctuations and Jason Miller investigated random walks and Brownian motion on it. Statistical mechanics models with quenched disorder. We heard 3 talks on Processes in random environment: Marek Biskup analyzed degenerate dynamical random environment motivated by the Helffer-Sjöstrand representation. Bálint Töth talked about the Lorentz gas with random obstacles, on time-scales beyond the Boltzmann-Grad limit. Quentin Berger studied directed polymers in heavy-tailed environment. Two talks were devoted to spin glasses: David Belius discussed the TAP-Plefka equations for the spherical Sherrington-Kirkpatrick model, and Aukosh Jagannath discussed the Langevin dynamics and long equilibration time for mean-field generalised spin glasses. Other models inspired by statistical physics. Finally, a few topics were covered by a single talk: • Silke Rolles strengthened the connection between the vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. • Roland Bauerschmidt talked about renormalisation for hierarchical spin models and implications on the dynamical spectral gap. • Sébastien Ott applied Ornstein-Zernike theory to study the Potts model with a defect line. Scaling Limits in Models of Statistical Mechanics 2537 • Nicholas Crawford discussed the eigenvectors of a non-Hermitian random matrix model. • Wendelin Werner talked about loop-soups and critical percolation in dimensions 7 and above. • Lisa Hartung talked about the extremal set of branching Brownian motion. • Perla Sousi talked about the capacity of the range of simple random walk in different space dimensions. • Wioletta Ruszel investigated the scaling limit of the odometer function in sandpile models. • Frank den Hollander talked about a population model with seed-bank and spatial structure. Summary. The workshop was an obvious success. In particular, it helped to update the participants on the state of the art and on the important pending open problems in the fields related to their domain of research, facilitated exchange of ideas between researchers in technically disconnected areas, and it gave rise to many interesting and informative discussions. In particular, we had a lively evening session focused mostly on open problems. Some new collaborations arose, notably Hutchcroft and Pete solved a long-standing problem on the cost of Kazhdan groups (this was announced a few weeks after the conference ended). We would like to thank the MFO personnel for the help and for the invaluable logistic support, as well as for creating a friendly and stimulating environment throughout the entire meeting. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1641185, “US Junior Oberwolfach Fellows”. Moreover, the MFO and the workshop organizers would like to thank the Simons Foundation for supporting Hubert Lacoin in the “Simons Visiting Professors” program at the MFO. Scaling Limits in Models of Statistical Mechanics 2539 Workshop: Scaling Limits in Models of Statistical Mechanics

Non-Archimedean Geometry and Applications

Abstract: The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields, in particular to number theory and algebraic geometry. These applications included Mirror Symmetry, the Langlands program, p-adic Hodge theory, tropical geometry, resolution of singularities and the geometry of moduli spaces. Much emphasis was put on making the list of talks to reflect this diversity, thereby fostering the mutual inspiration which comes from such interactions. Mathematics Subject Classification (2010): 03 C 98, 11 G 20, 11 G 25, 14 F 20, 14 G 20, 14 G

Mini-Workshop: Deformation quantization: between formal to strict

Abstract: The philosophy of deformation was proposed by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer in the seventies and since then, many developments occurred. Deformation quantization is based on such a philosophy in order to provide a mathematical procedure to pass from classical mechanics to quantum mechanics. Basically, it consists in deforming the pointwise product of functions to get a non-commutative one, which encodes the quantum mechanics behaviour. In formal deformation quantization, the non-commutative product (also said, star product) is given by a formal deformation of the pointwise product, i.e. by a formal power series in the deformation parameter which physically play the role of Planck’s constant ̵ h. From a physical point of view this is clearly not sufficient to provide a reasonable quantum mechanical description and hence one needs to overcome the formal power series aspects in some way. One option is strict deformation quantization, which produces quantum algebras not just in the space of formal power series but in terms of C∗-algebras, as suggested by Rieffel, with e.g. a continuous dependence on ̵ h. There are several other options in between continuous and formal dependence on ̵ h like analytic or smooth deformations. The Oberwolfach workshop Deformation quantization: between formal to strict consolidated, continued, and extended these research activities with a focus on the study of the connection between formal and strict deformation quantization in their various flavours and their applications in particular those in quantum physics and non-commutative geometry. It brought together specialists in differential geometry, operator algebras, non-commutative geometry, and quantum field theory with research interests in the mentioned quantization procedures. The aim of the workshop was to develop a coherent viewpoint of the many recent diverse developments in the field and to initiate new lines of research. 572 Oberwolfach Report 11/2015 Mathematics Subject Classification (2010): 53D55, 14D15. Introduction by the Organisers Formal deformation quantization as introduced by Bayen et al. has reached by now a very satisfying state: with the highly non-trivial formality theorem of Kontsevich the questions on existence as well as on classification of formal star products on general Poisson manifolds have been settled and answered in the positive in 1997. Alternative approaches to the globalization of Kontsevich’s result were also obtained by Cattaneo, Felder, and Tomassini in 2002 as well as by Dolgushev 2005. Before, the symplectic case was investigated by various groups. Here the existence of star products was shown by Lecomte and DeWilde already 1983, later independently by Fedosov in 1986 and by Omori, Maeda, and Yoshioka in 1991. The classification of star products in the symplectic case was obtained by Nest and Tsygan in 1995 and independently by Deligne in 1995 and Bertelson, Cahen, and Gutt in 1997. The representation theory of the deformed algebras, which is crucial for a physical application, has been investigated in detail by many people: among other things, the full classification of the star product algebras up to Morita equivalence was obtained by Waldmann, Bursztyn and Dolgushev in 2012. For a physical interpretation of the star product algebras as observable algebras of a quantized physical system, the formal parameter has to be identified with Planck’s constant h̵. Hence a convergence of the formal series in h̵ is crucial. In the early era of deformation quantization the formal star products have been constructed by means of asymptotic expansions of other quantizations like Berezin-Toeplitz quantizations on quantizable Kahler manifolds or symbol calculus quantizations on cotangent bundles. Beside producing rather explicit examples like the constructions of Cahen, Gutt and Rawnsley case as well as Karabegov in the Kahler case or Bordemann, Neumaier, Pflaum and Waldmann in the cotangent bundle case, the good understanding of the formal star products also led to interesting results on the convergent origins: here the computations of characteristic classes by Karabegov and Schlichenmaier or the index theorems of Fedosov as well as Nest and Tsygan should be mentioned. For the whole world beyond smooth Poisson manifolds the works of Pflaum, Posthuma, and Tang show first deep results on deformation quantization also in this case. On a more analytic oriented approach based on a C∗-algebraic formulation using continuous fields of C∗-algebras, Rieffel showed how an action of R on a C∗-algebra can be used to deform this C∗-algebra in a continuous way. Applied to the bounded continuous functions on a manifold, this ultimately leads again to a formal star product by an asymptotic expansion of the continuous deformation for h̵ Ð→ 0, at least on sufficiently smooth vectors of the action. Ever since, Rieffel’s paradigma of deformation by group actions was studied in many contexts and substantially extended recently to other (non-abelian) Lie groups than R by Bieliavsky and Gayral and coworkers. On a more abstract level, Natsume, Nest, and Peter considered symplectic manifolds with a topological condition (trivial Mini-Workshop: Deformation quantization: between formal to strict 573 second fundamental group) and showed that a strict quantization always exists, based on the usage of Darboux charts and a Čech cohomological argument. The relation between formal and strict deformation quantization has been subject of several studies, but there still remain deep open questions. Since the approach of formal deformation quantization is universal, as proved by Kontsevich, it is natural to try to find the way back: from the easy formal situation to the more complicated convergent one. Since the above mentioned quantization schemes all use particular geometric features, one can hope to recover not only a convergent quantization as required by physics, but also interesting information about the underlying geometry. There are only few examples where this way backwards was investigated: in the flat case, Beiser, Rmer and Waldmann considered the convergence of the Wick star product on C and recovered the full symmetry, coherent states, and the Bargmann-Fock representation from the convergence conditions. While this example is still geometrically rather trivial, it already shows a rich structure beyond the locally multiplicatively convex theory. It can be extended to infinite dimensions in a rather conceptual way as recently shown by Waldmann. The relations to the approches of Dito’s star products on Hilbert spaces still remain to be investigated. Later, Beiser and Waldmann considered a Wick-type star product on the Poincaré disk. Here the underlying geometry is topologically still trivial but enjoys a curved Kahler structure. Again, in this example the full symmetry of the problem is recovered and the foundations of a representation theory to establish the relations with the Berezin-Toeplitz quantizations are formed. Bieliavsky, Detournay, and Spindel gave a deformation of the Poincaré disk in a C∗-algebraic approach thus complementing the picture from the other side. However, the precise relations between the different versions of convergence remain unclear. Even though these examples seem to be isolated at the moment,they can be seen as a proof of concept that investigating the convergence of formal star products gives both physically relevant and manageable observable algebras and interesting information about the underlying geometry. Understanding the analytic aspects of deformation quantization has led to many non-trivial and surprising applications beyond the field of deformation quantization itself . Here we only want to mention a few: the works of Anderson and coworkers on the mapping class group where the results of Bordemann, Meinrenken, and Schlichenmaier on the asymptotic properties of Berezin-Toeplitz quantization enter in a crucial way. The works of Lechner show how one can use Rieffel’s deformations to construction new examples of quantum field theories as deformations of free theories. In some sense they can be seen as quantum field theories on a non-commutative Minkowski spacetime. Quantum deformations of classical geometries lead to interesting spaces in non-commutative geometry, here the quantum spheres of Connes and Landi provide a non-trivial and rich class where concepts of non-commutative geometry can be tested explicitly. Still many questions remain open: first, the above mentioned examples have to be investigated further to understand their relations and connections. Moreover, the quest for convergence of star products in order to produce (ultimately) a 574 Oberwolfach Report 11/2015 continuous field of C∗-algebras has to be extended beyond the above examples. Here one can think of other types of algebras between the formal power series on the one hand and the C∗-algebras on the other hand: in particular locally convex algebras and also bornological algebras may provide a good bridge. Here the techniques developed by Meyer on bornological algebras will play a crucial role. The overall goal of the workshop was to develop a coherent viewpoint of the many recent developments on the analytic aspects of deformation quantization as described above with particular emphasis on the connection between formal and strict and their potential applications in physics. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Mini-Workshop: Deformation quantization: between formal to strict 575 Mini-Workshop: Deformation quantization: between formal to strict

Factor copulas constructed from stochastic processes

Abstract: Randomizing the parameter(s) of a given parametric family of univariate random variables is a popular technique to enrich the distribution in concern with additional stochastic properties and to create new probability laws. On a multivariate level, another motivation is to introduce dependence to originally independent objects by means of a joint mixture variable affecting multiple random variables in a similar way. A well-known example are extendible Archimedean copulas that can be interpreted as the survival copulas arising from a two step experiment: Firstly, a positive random variable M is simulated. Secondly, a sequence of exponential random variables with rate parameter M is drawn independently. Other examples are credit-risk models where a joint (random) default probability p ∈ (0, 1) is used as mixture variable in a sequence of Bernoulli(p) experiments, or loss models for insurance claims based on Poisson-distributed count variables with joint (random) intensities. Factor models created in this way are popular due to, among others, the following facts: They enjoy a great level of interpretability, they are straightforward to simulate in large dimensions, the dimension of the considered problem is flexible, convenient limit results for large-dimensional random vectors (X1, . . . , Xd) for (d↗∞) are often computable, parametric families of mixture variables imply parametric families of copulas, and extensions beyond conditional iid (i.e. homogeneous one-factor models) are typically easy to find by, e.g., using multiple factors or inhomogeneous marginal laws. Moreover, hierarchical constructions are immediate in many cases, see [10]. Most factor models currently considered – in theory as well as in practice – are based on the aforementioned idea of using random parameters. This ansatz, however, can be extended to more involved random objects, e.g. stochastic processes. Providing more mathematical structure, a famous result by Bruno de Finetti (see [2]) shows that an infinite sequence of random variables {Xk}k∈N on (Ω,F ,P) is exchangeable if and only if it is conditionally iid, i.e. there exists a sub-σ-algebra G ⊂ F s.t. for all d ≥ 2:

Open problem : Strict Dissipativity and the Turnpike Property

Abstract: s.t. x(k + 1) = f(x(k), u(k)), x(k) 2 X, u(k) 2 U. Here f : X ⇥ U ! Rn is the dynamics, ` : X ⇥ U ! R is the stage cost and X ⇢ Rn and U ⇢ Rm are the state and control constraint set, respectively, which for siplicity of exposition we assume to be compact. Optimal trajectories (which we neither assume to exist nor to be unique) will be denoted by x⇤(·) The turnpike property now demands that there exists a point xe 2 X such that any optimal trajectory, regardless of its initial value, stays in a neighborhood of this point xe 2 X for a time which is independent of N . Formally this can be expressed as follows.

Mini-Workshop: Recent Developments on Approximation Methods for Controlled Evolution Equations

Abstract: This mini-workshop brought together mathematicians engaged in partial differential equations, functional analysis, numerical analysis and systems theory in order to address a number of current problems in the approximation of controlled evolution equations. Mathematics Subject Classification (2010): 93C20, 93C25, 65Mxx. Introduction by the Organisers The mini-workshop Recent Developments on Approximation Methods for Controlled Evolution Equations, organised by Birgit Jacob (Wuppertal), Enrique Zuazua (Bilbao) and Hans Zwart (Twente) was held November 1st – 7th, 2015. This meeting was well attended with 16 participants with broad geographic representation. Systems modelled by linear ordinary differential equations have long been studied and there exists a wide body of theory and design algorithms dealing with their control. The state describing such a system lies in a finite-dimensional vector space. This setting has its limitations, as many systems of interest, from the point of view of applications to industry and other disciplines, do not fall into this class. A more interesting generalisation is that to systems with an infinitedimensional state space. This class includes delay systems, and systems modelled by functional differential equations and partial differential equations (PDEs), generally called evolution equations. This field finds applications in such diverse areas as aeronautics, mechanical and electrical engineering. Since they appear frequently 2912 Oberwolfach Report 50/2015 as models in these fields of applications, evolution equations with boundary control and boundary observation are of particular interest. One of the key issues when addressing real applications is the effective control of those systems, which requires of significant effort from the point of view of mathematical analysis. The talks where grouped into three main themes: • Modeling and control of real-live problems • Numerical analysis of PDE control • Theoretical aspects of controller design and approximations for systems described by PDEs In the first theme the following participants gave talks: Rob Fey, Aitziber Ibañez, Jarmo Malinen, George Weiss. Furthermore, Athanasios Antoulas, András Bátkai, Umberto Biccari, Nicolae Cı̂ndae, Weiwei Hu, Orest Iftime, Kirsten Morris, Timo Reis and Hans Zwart were the speakers of the second theme. The last theme was covered by Björn Augner, Birgit Jacob, Felix Schwenninger and Hans Zwart. Although we have grouped them according to our themes, there was significant overlap between the approaches which stimulated many productive discussions. The organizers and participants thank the Mathematisches Forschungsinstitut Oberwolfach for providing an inspiring setting for this mini-workshop, which allowed us to concentrate on the mathematics. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Approximation Methods for Controlled Evolution Equations 2913 Mini-Workshop: Recent Developments on Approximation Methods for Controlled Evolution Equations

Mathematical Milky Way models from Kelvin and Kapteyn to Poincaré, Jeans and Einstein

Abstract: Contribution to a workshop that brought together historians of mathematics and science as well as mathematicians to explore important historical developments connected with models and visual elements in the mathematical and physical sciences. It addressed the larger question of what has been meant by a model, a notion that has seldom been subjected to careful historical study. Most of the talks dealt with case studies from the period 1800 to 1950 that covered a number of analytical, geometrical, mechanical, astronomical, and physical phenomena. The workshop also considered the role of visual thinking as a component of mathematical creativity and understanding. Mathematics Subject Classification (2010): 01A55, 01A60.

Mirror Symmetry, Hodge Theory and Differential Equations

Abstract: The following is the report on the Oberwolfach workshop Mirror Symmetry, Hodge Theory and Differential Equations (April 2015), which brought together researchers from various areas such as quantum cohomology, complex algebraic geometry, Hodge theory representation theory etc. Mathematics Subject Classification (2010): 14J33, 32S40, 14D07, 34Mxx, 53D45. Introduction by the Organisers The workshop Mirror Symmetry, Hodge Theory and Differential Equations, which took place from April 20 to 24, 2015 aimed at reporting on recent developments on research topics related to various areas of pure mathematics such as Hodge theory, linear differential equations, quantum cohomology, category theory and representation theory, to name only a few. The workshop had 25 participants, with a wide geographical horizon. The list of participants included several young postdocs, the workshop was an excellent opportunity for them to present their results to a larger audience. The meeting has highlighted the intense activity in these mathematical domains, as well as the strong interaction with other mathematical topics, such as Langlands correspondence, representation theory and irregular differential equations. The workshop consisted of 19 talks and many informal discussions, supplemented with some evening discussion sessions. Some talks were meant to give an overview on a particular field relevant for the main subject of the workshop, others reported on precise new results and and a few ones mainly contained new ideas or work in progress. For example, Duco van Straten talked on joint work with 1202 Oberwolfach Report 22/2015 A. Mellit and V. Golyshev on geometric Langlands correspondence and congruence differential equations, motivated by the classification of Fano manifolds by using quantum cohomology and tools from Langlands correspondence. Two talks, by Clélia Pech and Konstanze Rietsch (partly on joint work with R. Marsh and L. Williams) were on mirror symmetry statements for non-toric varieties such as homogeneous spaces and more specifically Grassmannians (including equivariant aspects). A whole day was concerned with talks related to various kinds of hypergeometric equations. Uli Walther (based on joint work with L. Matusevich and E. Miller as well as M. Schulze) introduced us to the techniques of Euler-Koszul homology, Thomas Reichelt used these to describe how the formalism of mixed Hodge modules can be applied to the study of Gelfand-Kapranov-Zelevinsky differential systems. Takuro Mochizuki explained his recent work on twistor modules and GKZ-systems, and Alberto Castaño Domı́nguez explained results on hypergeometric description of the cohomology of the Dwork family. Hiroshi Iritani reported on his recent work on mirror statements involving the big quantum cohomology rings, whereas Alessandro Chiodo overviewed his results (with Y. Ruan and H. Iritani) on the Landau-Ginzburg/Calabi-Yau correspondence. The talk of Lev Borisov showed a surprising application of mirror symmetry: From some specific examples of so-called double mirror Calabi-Yau families (i.e., families having the same mirror) one can derive that the class of the affine line is a zero divisor in the Grothendieck ring. Categorical aspects of quantum cohomology and mirror symmetry have been discussed in several talks: Dmytro Shklyarov explained how to (re)construct GaußManin cohomology from categories (like matrix factorizations), and Etienne Mann talked about his joint work with M. Robalo on categorification of Gromov-Witten invariants. Todor Milanov reported on a recent construction which builds on the oscillating integrals as well as the period integrals in singularity theory and Landau-Ginzburg models: It establishes a vertex operator algebra structure on a certain Fock space associated to these data. Alexey Basalaev explained how to endow cohomological field theories with an action of SL(2,C) and how to show modularity properties of certain potentials. In a somewhat different spirit, Emmanuel Scheidegger talked on joint work with M. Alim, H. Movasati and S.T. Yau on the construction of a certain Lie algebra of vector fields on the moduli space of Calabi-Yau threefolds and its relation to the holomorphic anomaly equations. Ana Ros Camacho introduced a new equivalence relation for homogeneous polynomials, called orbifold equivalence, by imposing the existence of a matrix factorization with nonzero quantum dimension for the difference of the two polynomials, and considers the case of ADE singularities. Helge Ruddat reported on joint work with B. Siebert on the construction of canonical coordinates for a degenerating family of Calabi-Yau varieties. Mirror Symmetry, Hodge Theory and Differential Equations 1203 In an overview talk, Philip Boalch described examples of wild character varieties as finite dimensional multiplicative symplectic quotients. He related these examples to a 1764 paper of Euler, and showed that Euler’s continuant polynomials are group valued moment maps. The last talk of the conference by Marco Hien was concerned with a topological approach to the recent work of d’Agnolo-Kashiwara on Riemann-Hilbert correspondence for arbitrary differential systems. Summarizing, we feel that we had an extremely interesting meeting with many beautiful talks covering a large variety of subjects. The discussions that took place between the talks (as well as during the traditional hike to St. Roman on Wednesday afternoon) were quite stimulating. The enthusiasm of the participants as well as the great atmosphere at MFO largely contributed to the success of the workshop. The meeting showed that the subject of mirror symmetry, in all its ramifications, is as vibrant as ever and many open questions are still ahead of us. Acknowledgements: The MFO and the workshop organizers would like to thank the Simons Foundation for supporting Takuro Mochizuki in the “Simons Visiting Professors” program at the MFO. The organizers also acknowledge the financial support of the ANR-DFG program SISYPH (ANR-13-IS01-0001-01/02, DFG No HE 2287/4-1 & SE 1114/5-1). Mirror Symmetry, Hodge Theory and Differential Equations 1205 Workshop: Mirror Symmetry, Hodge Theory and Differential Equations