Open Problem Session Research Articles

Ruler Wrapping

In 1985 Hopcroft, Joseph and Whitesides showed it is NP-complete to decide whether a carpenter’s ruler with segments of given positive lengths can be folded into an interval of at most a given length, such that the folded hinges alternate between 180 degrees clockwise and 180 degrees counter-clockwise. At the open-problem session of 33rd Canadian Conference on Computational Geometry (CCCG ’21), O’Rourke proposed a natural variation of this problem called ruler wrapping, in which all folded hinges must be folded the same way. In this paper we show O’Rourke’s variation has a linear-time solution.

Densities for sets of natural numbers vanishing on a given family

Abstract upper densities are monotone and subadditive functions from the power set of positive integers into the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density.At the open problem session of the Workshop “Densities and their application”, held at St. Étienne in July 2013, G. Grekos asked a question whether there is a “nice” abstract upper density, whose the family of null sets is precisely a given ideal of subsets of N, where “nice” would mean the properties of the familiar densities consider in number theory.M. Di Nasso, R. Jin (2018) [3] showed that the answer is positive for the summable ideals (for instance, the family of finite sets and the family of sequences whose series of reciprocals converge) when “nice” density means translation invariant and rich density (i.e. density which is onto the unit interval).In this paper we extend their result to all ideals with the Baire property.

Variational Analysis Down Under Open Problem Session

We state the problems discussed in the open problem session at Variational Analysis Down Under conference held in honour of Prof. Asen Dontchev on 19–21 February 2018 at Federation University Australia.

Introduction

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

A Note on Edge Isoperimetric Numbers and Regular Graphs

This note resolves an open problem asked by Bezrukov in the open problem session of IWOCA 2014. It shows an equivalence between regular graphs and graphs for which a sequence of invariants presents some symmetric property. We extend this result to a few other sequences.

Report from the Open Problems Session at OPSFA13.

These are the open problems presented at the 13th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA13), Gaithersburg, Maryland, on June 4, 2015.

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry.

Mini-Workshop: Differentiable Ergodic Theory, Dimension Theory and Stable Foliations

The mini-workshop Differentiable Ergodic Theory, Dimension Theory and Stable Foliations brought together experts in thermodynamical formalism, hyperbolic dynamics and dimension theory from several countries. The geographic representation was broad, from Europe, USA and Japan. All participants gave interesting 1-hour talks, and there was organized also an open problem session, where directions for future work and many open problems were discussed. Among the topics presented/discussed in the workshop, there were ones related to dimension theory and probability measures on fractals, various types of hyperbolicity, systems with overlaps, complex dynamics and iterated function systems.

Introduction

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 14th to 20th April 2013. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

Discrete Differential Geometry

This is the collection of extended abstracts for the 24 lectures and the open problems session at the third Oberwolfach workshop on Discrete Differential Geometry.

Geometric Aspects of Spectral Theory

The workshop “Geometric Aspects of Spectral Theory” brought together leading researchers working in various areas of this vast field of mathematics. The meeting featured presentations on some of the most fascinating recent developments in the subject, including five survey talks given by top experts, as well as reports on the progress made by graduate students and postdocs. A number of new stimulating questions were formulated during the open problem session.

Report on the open problems session

This report summarizes the problems presented during the last session of the “Algebra meets Topology: Advances and Applications” conference in honor of Dikran Dikranjan on his 60th birthday, Barcelona, July 20–23, 2010.

Teichmüller Theory

This is a report on the workshop on Teichmüller theory held in Oberwolfach, from November 28 to December 4, 2010. The workshop brought together people working in various aspects of the field, with a focus on recent developments. The topics discussed included higher Teichmüller theory, moduli spaces of flat connections, cluster algebras, quantization of Teichmüller spaces, the dynamical aspects of the Teichmüller and Weil-Petersson geodesic flows, the metric and the boundary theory of Teichmüller space including the new developments on Thurston’s asymmetric metric, string topology, geometric analysis on moduli spaces, and relations with three-manifold topology and with minimal surface theory were also highlighted. The mapping class group was also discussed in detail, from various points of view, including its actions on simplicial complexes and on infinite-dimensional Teichmüller spaces, its asymptotic dimension, the relation with the arc operad, the generalizations of the Johnson homomorphisms to the monoid of homology cylinders, making contact with knot theory and with the Casson invariant and other 3-manifolds invariants. There was an open problem session, which is also reported on here.

Introduction

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from 26 April to 2 May. This meeting focused on the common themes of Combinatorics, Discrete Probability and Theoretical Computer Science, and the lectures, many of which were given by young participants, stimulated fruitful discussions. The open problems session held during the meeting, and the fact that the participants work in different and related topics, encouraged interesting discussions and collaborations.

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problems session at the second Oberwolfach workshop on Discrete Differential Geometry.

Von Neumann Algebras and Ergodic Theory of Group Actions

The workshop \emph{Von Neumann Algebras and Ergodic Theory of Group Actions} was organized by Dietmar Bisch (Vanderbilt University, Nashville), Damien Gaboriau (ENS Lyon), Vaughan Jones (UC Berkeley) and Sorin Popa (UC Los Angeles). It was held in Oberwolfach from October 26 to November 1, 2008. This workshop was the first Oberwolfach meeting on von Neumann algebras and orbit equivalence ergodic theory. The organizers took special care to invite many young mathematicians and more than half of the 28 talks were given by them. The meeting was very well attended by over 40 participants, leading senior researchers and junior mathematicians in the field alike. Participants came from about a dozen different countries including Belgium, Canada, Denmark, France, Germany, Great Britain, Japan, Poland, Switzerland and the USA. The first day of the workshop featured beautiful introductory talks to orbit equivalence and von Neumann algebras (Gaboriau), Popa's deformation/rigidity techniques and applications to rigidity in II _1 factors (Vaes), subfactors and planar algebras (Bisch), random matrices, free probability and subfactors (Shlyakhtenko), subfactor lattices and conformal field theory (Xu) and an open problem session (Popa). There were many excellent lectures during the subsequent days of the conference and many new results were presented, some for the first time during this meeting. A few of the highlights of the workshop were Vaes' report on a new cocycle superrigidity result for non-singular actions of lattices in SL(n, \mathbb R ) on {\mathbb R}^n and on other homogeneous spaces (joint with Popa), Ioana's result showing that every sub-equivalence relation of the equivalence relation arising from the standard SL(2, \mathbb Z )-action on the 2 -torus {\mathbb T}^2 is either hyperfinite, or has relative property (T), and Epstein's report on her result that every countable, non-amenable group admits continuum many non-orbit equivalent, free, measure preserving, ergodic actions on a standard probability space. Other talks discussed new results on fundamental groups of II _1 factors, L ^2 -rigidity in von Neumann algebras, II _1 factors with at most one Cartan subalgebra, subfactors from Hadamard matrices, a new construction of subfactors from a planar algebra and new results on topological rigidity and the Atiyah conjecture. Many interactions and stimulating discussions took place at this workshop, which is of course exactly what the organizers had intended. The organizers would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing the splendid environnment for holding this conference. Special thanks go to the very helpful and competent staff of the institute.

Discrete Geometry

Discrete Geometry deals with the structure and complexity of discrete geometric objects ranging from finite point sets in the plane to more complex structures like arrangements of n -dimensional convex bodies. Classical problems such as Kepler's conjecture and Hilbert's third problem on decomposing polyhedra, as well as classical works by mathematicians such as Minkowski, Steinitz, Hadwiger and Erdös are part of the heritage of this area. By its nature, this area is interdisciplinary and has relations to many other vital mathematical fields, such as algebraic geometry, topology, combinatorics, computational geometry, convexity, and probability theory. At the same time it is on the cutting edge of applications such as geographic information systems, mathematical programming, coding theory, solid modelling, computational structural biology and crystallography. The workshop was attended by 40 participants. There was a series of 12 survey talks giving an overview of developments in Discrete Geometry and related fields: In addition, there were 18 shorter talks and an open problem session chaired by János Pach on Wednesday evening — a collection of open problems resulting from this session can be found in this report. The program left ample time for research and discussions in the stimulating atmosphere of the Oberwolfach Institute.

Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry

The workshop “Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry”, organized by Gerhard Knieper, Leonid Polterovich and Leonid Potyagailo took place the second time in Oberwolfach. The general idea to organize a joint meeting of specialists belonging to three different but close and interacting fields turned out to be really fruitful. The aim of our workshop is to give a possibility to people working in the above fields to discuss their results and open problems from different points of view. We are sure that such conferences need to exist not only on the level of World or European Congresses but also in much less official and restrictive atmosphere inherent for Oberwolfach conferences. Our last meeting brought together world renown senior experts in our fields as well as talented mathematicians in the beginning in their careers. Let us describe shortly the directions of the given talks. The presented talks were devoted to: Historically, one of the motivating subjects being in the center of our joint interests was the theory of quasi-morphisms and bounded cohomology. In the present time there exist many other subjects which are in our common interests: study of groups of diffeomorphisms using methods of symplectic topology (e.g. the talk of F. Le Roux where he constructed a new candidate for a non-trivial normal subgroup of the group of area-preserving homeomorphisms); dynamical methods describing the geometry of the mapping class group and the group of outer automorphisms of free groups (G. Levitt); probabilistic methods in the geometric group theory (A. Erschler and K. Fujiwara); solving equations in free groups and diophantine geometry (F. Dahmani); strong Tits alternative for linear groups and its consequences: uniform exponential growth and uniform decay of return probability (E. Breuillard); spectral theory for combinatorial Laplacian of Cayley graphs of finite abelian groups (A. Karlsson) etc. Another important issue of our meetings is that they attract many students working in these subjects. Several students (or recent students) gave 30 minutes talks during our informal evening sessions attended by all participants. The students have been actively participating in the discussions during the conference. We continued our tradition by organizing the open problem session on Wednesday night successfully chaired by Anatole Katok. We enclose below the problems formulated during this session and hope that they will stimulate further research in our fields. We strongly believe that the tradition of such joint meetings representing several research directions should continue in Oberwolfach.

Mini-Workshop: Time Series with Sudden Structural Changes

Time series with nonstationarities can be found in a variety of applications from economics to engineering to biological and medical sciences to geosciences. During the past 20 years, much effort in investigating nonstationary time series has been focused on models with slowly changing structure. Such examples include autoregressive processes with slowly changing parameters and, more generally, locally stationary processes with a time-dependent spectrum. Research on these models has led to considerable insight into the structure of stochastic processes as well as parametric and nonparametric statistical methods. In particular, the fact that time series data are no longer assumed to be identically distributed has resulted in some challenging and interesting statistical problems. Typically locally stationary models are not well suited to cover the situation in which a time series may experience a sudden change in its dynamics. That is, there may be changes in the mean, variance, dependence structure, or some other feature that occur at random times. Often it is reasonable to assume that the time series is stationary between these changepoints . This assumption is tacitly made in the applications of classical changepoint analysis techniques to stochastic processes of dependent data which have been proposed recently. It also forms the basis for segmentation techniques which partition the time axis such that the data in each subinterval may be modeled by a stationary process. Another approach explicitly takes the piecewise stationarity into account in the modeling step. There, the sudden changes in the dynamic structure of the data-generating process are controlled by a hidden Markov chain with finitely many states resulting in hidden Markov models, or, more generally, in Markov switching models. Such models are becoming increasingly popular in the analysis of economic, environmental and other data. Some progress has been recently made in the theory regarding the stochastic structure of such piecewise stationary time series models as well as regarding the properties of parametric and nonparametric statistical procedures for estimating the system characteristics, of estimating locations of changepoints and of reconstructing the unobservable state process, i.e. of finding an appropriate segmentation of the data. Much of this work also requires the development of algorithms that are numerical feasible for carrying out these procedures. Nevertheless, there are still many open questions, and one of the main goals of the workshop beyond the exchange of current results between different statistical communitites working on those problems, was to join forces to discuss ideas on how to attack various open problems and to investigate the relations between different approaches. A combination of review talks as well as presentations of recent progress in theory and of challenging applications formed the basis for the exchange of ideas. Altogether, there were 16 talks which invariably were accompanied by a very lively discussion. The presented results and the raised questions served as inspiration for future research and as stimulation for joint projects. An open problem session also contributed to the fruitful communication between the different research communities. At the end of the conference, the participants decided to continue the efforts started at the workshop and to further communication by creating a website which serves as a platform for researchers working on time series with change points.

Minor summation formula and a proof of Stanley’s open problem

In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley’s open problem.