ICMS

Sep 3, 2007 - Sep 7, 2007

Heriot Watt University, Edinburgh

Organisers

Name	Institution
Böttcher, Albrecht	Techical University of Chemnitz
Deift, Percy	New York University
Shargorodsky, Eugene	King's College London

Short Report

Applications of the Riemann-Hilbert problem to integrable systems, orthogonal polynomials and random matrices is one of the most vigorously developing branches of mathematics at present. The recent research in this area has resulted in spectacular progress in many directions, especially those related to the study of asymptotic behaviour of integrable models, and in creation of new powerful methods such as the nonlinear steepest descent method for the Riemann-Hilbert problem.

The spectral theory of Toeplitz operators has also seen significant recent progress in several directions including operators with piece-wise continuous symbols on general Carleson curves with arbitrary Muckenhoupt weights and operators with matrix semi-almost periodic symbols. There is also quite a number of recent striking, often counterintuitive, results on Toeplitz matrices. The latter is a subject attracting a great deal of interest from researchers involved with numerical linear algebra, pseudospectra, C*-algebras, orthogonal polynomials, random matrices, representation theory, stochastic processes and other fields.

The main aim of the workshop was to bring together the top researchers who apply the Riemann-Hilbert problem to integrable systems, orthogonal polynomials and random matrices, and those who work in the spectral theory of Toeplitz operators, to review the current state of the field and, most importantly, to discuss plans for the future.

The workshop was attended by 34 researches representing a wide variety of backgrounds and interests. The programme consisted of six 50 minutes long talks and twenty 30 minutes long talks. A most important feature of the timetable were very long lunch breaks from 12:30–12:45 to 15:50. These breaks provided the participants with plenty of time for discussions and much mathematics was done during these periods.

Participants list and links to available presentations are further down this page.

Download the pdf file of the full report

Photos from the workshop

Arrangements

The workshop will begin on Monday 3 September and finish on Friday 7 September 2007. Participation is by invitation only.

Venue
The Workshop will be held in Scott Russell Building Room 112 at Heriot-Watt University. The campus is on the western edge of the city, near the airport, with excellent public transport to and from Edinburgh city centre. The easiest way to get there from the airport is by taxi at a cost of about £10.00. There is no direct bus from the airport to HW. If you arrive by train or coach in central Edinburgh, there are a number of buses that will take you directly to the campus.

The university's website has a map showing the location of the university and instructions on how to reach it by various forms of transport, http://www.hw.ac.uk/home/dir/40/edinburgh-campus then choose ‘Edinburgh Campus Maps and Directions’.

Registration
There will be a combined Registration and Welcome Buffet from 19.00 to 21.00 on Sunday 2 September, in the College Lounge, Heriot-Watt University. Food will be available between 19.00 and 20.30. Please note you may register at anytime between 19.00 and 21.00. Those who cannot register on Sunday evening may do so on Monday 3 September prior to the start of the lectures or during the first coffee break, at a Registration desk which will be situated in the Scott Russell Room 114.

Audio/Visual Facilities
A data projector, overhead projector, laptop, two blackboards and flipchart will be available.

Accommodation and Facilities
Single study-bedrooms have been reserved at Heriot-Watt University. All rooms have a private shower and toilet and a telephone. The telephone will access incoming calls. You need to buy a telephone card from Reception in order to make outgoing calls. Each room also has an internet point should you wish to bring your laptop, although there is a charge for this service. We have also arranged free email and internet access in some of the central computer labs and the Library. Please see http://www.hw.ac.uk/uics/images/pc_cal_lab_map_print_out.pdf. WIFI access areas are located in some parts of the university. Please see http://www.hw.ac.uk/uics/Help_FAQs/WiFi_Locations.html

On arrival go to the main University and Conference Centre Reception in the James Watt Centre which is manned 24 hours. You may check into your room from 14.00 on the day of arrival. Your room must be vacated by 10.00 on the day of departure. There is a £20.00 charge for any bedroom key not returned to Reception by 11.00 on the morning of departure. Please note that if you do not check out by noon, you will be charged for a full night of accommodation. If you are not leaving immediately, luggage may be left temporarily in the baggage room at James Watt Centre Reception.

Meals and Refreshments
Breakfast will be provided for Heriot-Watt University residents for the period of their stay in the Middle Floor Dining Room. Lunch will be provided on weekdays only (Monday 3 September to Friday 7 September) 11.45 to 14.00 also in the Middle Floor Dining Room.
Evening meals will be provided on campus in the Middle Floor Dining Room on Monday, Tuesday, Wednesday and Friday from 18.00 to 20.00.

There will be a buffet supper (free of charge to delegates) in the College Lounge on the evening of Sunday 2 September between 19.00 and 20.30. Participants may register for the workshop during this buffet. There will also be an opportunity to register on Monday morning prior to the first talk.

A Workshop Dinner will be held on Thursday 6 September in Scholars Restaurant at Heriot-Watt.

Financial Matters
The workshop grant will cover the cost of single en-suite bed and breakfast accommodation at Heriot-Watt. Lunches and evening meals Mon-Fri inclusive on campus and the Sunday evening buffet supper will also be provided. Please note that we cannot reimburse any meals taken off campus.

Under the terms of our EPSRC funding we are required to charge a 30.00 GBP registration fee to cover costs not admissable under the grant. Please contact ICMS if this fee will be difficult for you to pay, as we may be able to waive this cost for some participants e.g. graduate students. Otherwise, you will be advised about payment methods at a later date.

 

Programme

Sunday 2 September

19.00 - 21.00

Registration & buffet, College Lounge (Food available until 20.30)

Monday 3 September

08.30 - 09.30	Registration
09.30 - 10.00	Estelle Basor (California Polytechnic State University) Dimer Models and Toeplitz Determinants
10.05 - 10.35	Igor Krasovsky (Brunel University) Hankel determinants with singular symbols
10.40 - 11.10	Yuri Karlovich (Universidad Autonoma del Estado de Morelos, Mexico) Algebras of pseudodifferential operators and their applications
11.10 - 11.40	Tea/Coffee
11.40 - 12.30	Harold Widom (Univeristy of California, Santa Cruz) On the inverse and determinant of certain truncated Wiener-Hopf operators
12.30 - 15.50	Lunch (Middle floor dining room)
15.50 - 16.40	Jinho Baik (University of Michighan) Total Integrals of Painleve II Solutions
16.40 - 17.10	Tea/Coffee
17.10 - 17.40	Elias Wegert (TU Bergakademie Freiberg) Blaschke representation of functions on the circle
18.00	Dinner (Middle floor dining room)

Tuesday 4 September

09.10 - 09.40	Marko Lindner (University of Reading) Fredholmness of operators in the Wiener algebra
09.45 - 10.15	Nikolai Vasilevski (Centro de Investigacion y Estudios Avanzados, Mexico) On the structure of the C^*-algebra generated by Toeplitz operators with piecewise continuous symbols
10.20 - 10.50	Miroslav Englis (Academy of Sciences, Prague) Toeplitz operators from various viewpoints
10.50 - 11.20	Tea/Coffee
11.20 - 12.10	Torsten Ehrhardt (University of California, Santa Cruz) Gap probabilities in Random Matrix Theory and Determinants of Wiener-Hopf Hankel operators
12.15 - 12.45	David Wenzel (Technische Universitat Chemnitz) A probability argument in favor of ignoring small singular values of Toeplitz operators
12.45 - 15.50	Lunch (Middle floor dining room)
15.50 - 16.40	Maurice Duits (Katholieke Universiteit Leuven) An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices
16.40 - 17.10	Tea/Coffee
17.10 - 17.40	Sergei Grudsky (Centro de Investigacion y Estudios Avanzados, Mexico) Uniform boundedness of Toeplitz matrices with variable coefficients
18.00	Dinner (Middle floor dining room)

Wednesday 5 September

09.30 - 10.00	Mo Man Yue (University of Bristol) Entanglement in the generalized XY model and block Toeplitz determinant
10.05 - 10.35	Francesco Mezzadri (University of Bristol) Derivative of Riemann-zeta function, Toeplitz determinants
10.40 - 11.10	Yang Chen (Imperial College London) A singular linear statistics and Painleve III
11.10 - 11.40	Tea/Coffee
11.40 - 12.10	Beatrice Pelloni (University of Reading) Riemann-Hilbert and d-bar problems and moving boundaries
12.10	Lunch then free afternoon
18.00	Dinner (Middle floor dining room)

Thursday 6 September

09.30 - 10.00	Arno Kuijlaars (Katholieke Universiteit Leuven) Riemann-Hilbert steepest descent analysis for non-intersecting squared Bessel paths
10.05 - 10.35	Jani Virtanen (University of Helsinki) Toeplitz and Hankel operators on the Bergman space A^1
10.40 - 11.10	Eugene Shargorodsky (King's College London) Toeplitz operators with bounded measurable coefficients
11.10 - 11.40	Tea/Coffee
11.40 - 12.30	Ilya Spitkovsky (College of William and Mary) On the almost periodic factorization
12.30 - 15.50	Lunch (Middle floor dining room)
15.50 - 16.40	Alexander Its (Indiana University-Purdue University Indianapolis) On the Riemann-Hilbert approach in the theory of Toeplitz and Hankel determinants
16.40 - 17.10	Tea/Coffee
17.10 - 17.40	Thanasis Fokas (Cambridge University) Riemann-Hilbert and Dbar formalism, imaging and integrability
19.30	Workshop Dinner at Scholar's Restaurant

Friday 7 September

09.30 - 10.00	Sandra Pott (University of Glasgow) Tangential interpolation in vector-valued H^p spaces, Carleson measures and applicationd to controlability
10.05 - 10.35	Vladimir Peller (Michighan State University) Approximation by analytic matirx functions in L^p
10.40 - 11.10	Jonathan Partington (University of Leeds) Laplace's equation and approximation in Hardy spaces in 2 and 3 dimensions
11.10 - 11.40	Tea/Coffee
11.40 - 12.30	Discussion
12.30 - 15.50	Lunch (Middle floor dining room)
15.50 - 16.40	Discussion
16.40 - 17.10	Tea/Coffee
17.10 - 17.40	Discussion
18.00	Dinner (Middle floor dining room)

Presentations:

Presentation Details
Baik, Jinho
Total Integrals of Painleve II Solutions
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We compute the total integrals of real and purely imaginary solutions to Painleve II equations on the real line. The motivation comes from the Tracy-Widom distribution functions which arise in random matrix theory, random permutations and random tilings. This is a joint work with Buckingham, DiFranco and Its.
Basor, Estelle
Dimer Models and Toeplitz Determinants
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The talk will describe the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice ($t=0$) and the triangular lattice ($t=1$), and we obtain the asymptotics for $0<tle 1$. For $0<t<1$ we apply the Szeg"o Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form. This is joint work with Torsten Ehrhardt.
Chen, Yang
A singular linear statistics and Painleve III
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A particular singular linear statistics motivated by problems in integrable quantum field gives rise to a class of orthogonal polynomials where the weight has an infinitely strong zero. It is shown the recurrence coefficients are related a particular PIII.
Duits, Maurice
An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices
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I will report on recent joint work with Arno Kuijlaars on the limiting eigenvalue distribution of $ntimes n$ banded Toeplitz matrices as $nto infty$. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as $ntoinfty$. Our main result is a characterization of the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimizes an energy functional defined on admissible vectors of measures. If time permits, I will also illustrate our results by an example originating from a particular coupled random matrix model.
Ehrhardt, Torsten
Gap probabilities in Random Matrix Theory and determinants of Wiener-Hopf Hankel operators
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The gap probabilities is the probability that no eigenvalue of a random matrix (taken from an appropriate ensemble of random matrices) lies in an interval of a certain length. It is by now well known that for various ensembles these probabilities can be expressed in terms of Fredholm determinants of corresponding integral operators. For instant, in the bulk scaling limit for the three classical Gaussian ensembles this leads to determinants involving integral operators with sine-kernels $$ frac{sin(x-y)}{pi(x-y)}, quad mbox{ and } quadfrac{sin(x-y)}{pi(x-y)}pm frac{sin(x+y)}{pi(x+y)}$$ on the finite interval $[0,alpha]$. Furthermore, the hard egde scaling limit for the Jacobi and the Laguerre ensembles involves integral operators of Bessel type. A question of interest is the asymptotics of the corresponding Fredholm determinants as the parameter $alpha$ (which corresponds to the size of the gap) tends to infinity. Positive results are now available for the sine-kernel cases. There the asymptotics can be reduced to the asymptotics of Wiener-Hopf plus Hankel determinants with Fisher-Hartwig symbols. The reduction is based on precise relations between Hankel and Toeplitz plus Hankel determinants. I am planning to talk about these results and tentatively also about the Bessel cases.
Englis, Miroslav
Toeplitz operators from various viewpoints
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The purpose of the talk is to highlight a selection of topics, ranging from quantization to group representations and function theory in several complex variables, in which Toeplitz operators, sometimes surprisingly, have turned out to play an important role.
Fokas, Thanasis
Riemann-Hilbert and Dbar formalism, imaging and integrability
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A new method for inverting analyticaly certain integrals will be reviewed. This method involves the spectral analysis of an appropriate linear eigenvalue problem and the formulation of a local d-bar problem. Aplications include imaging and the solution of initial boundary value problems for PDEs involving a moving boundary. In addition, the long standing open problem of constructing integrable multidimensional nonlinear PDEs will be discussed. It will be shown that the Cauchy problem for such PDEs leads to a nonlocal d-bar problem. Integrable nonlinear PDEs in 3+1 compatible with the Laplace equation will be presented.
Grudsky, Sergei
Uniform boundedness of Toeplitz matrices with variable coefficients (A. Böttcher and S. M. Grudsky)
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Uniform boundedness of sequences of variable-coeffcient Toeplitz matrices is a surprisingly delicate problem.We showe that if the generating function of the sequence belongs to a smoothness scale of the Holder type and if alfa is the smoothness parameter, then the sequence may be unbounded for alfa< 1/2 while it is always bounded for alfa> 1/2.
Its, Alexander
On the Riemann-Hilbert approach in the theory of Toeplitz and Hankel determinants
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We will outline the basic features of the asymptotic analysis of Toeplitz and Hankel determinants via the Riemann-Hilbert method including the fundamental connections to the theory of Painlevé equations. We will also review some of the most recent results obtained in the field and discuss some of its open questions.
Karlovich, Yuri
Algebras of pseudodifferential operators and their applications
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Using the boundedness of the maximal singular integral operator related to the Carleson-Hunt theorem, we study the boundedness and compactness of pseudodifferential operators with bounded measurable $V(R)$-valued symbols and Lipschitz $V_d(R)$-valued symbols on weighted Lebesgue spaces, where $V(R)$ and $V_d(R)$ are the Banach algebras of all functions of bounded variation, respectively, on $R$ and on dyadic shells of $R$. This allows one to study pseudodifferential operators with symbols admitting discontinuities of the first kind with respect to the spatial and dual variables. Appearance of discontinuous symbols leads to non-commutative algebras of Fredholm symbols. Several Banach algebras of pseudodifferential operators with discontinuous symbols are investigated. We construct Fredholm symbol calculi for these algebras and establish Fredholm criteria for the operators in these algebras in terms of their Fredholm symbols. Applications to algebras of singular integral operators with/without shifts are considered.
Krasovsky, Igor
Hankel determinants with singular symbols
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Recent progress will be reported on calculation of large-n asymptotics for n-dimensional Hankel determinants whose symbol possesses Fisher-Hartwig singularities.
Kuijlaars, Arno
Riemann-Hilbert steepest descent analysis for non-intersecting squared Bessel paths
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I will consider non-intersecting squared Bessel paths that are conditioned to start at 1 and end at 0. At any intermediate time the positions of the paths give a determinantal point process with a kernel that can be expressed in terms of the 3x3 matrix valued Riemann-Hilbert problem for multiple orthogonal polynomials with respect to modified Bessel functions. As the number of paths increases and under appropriate scaling, the paths fill out a region. A steepest descent analysis for the Riemann-Hilbert problems shows that the correlation kernel has the usual scaling limits from random matrix theory, except at the critical time when the paths just hit the hard edge at 0. There we find new scaling limits.
Lindner, Marko
Fredholmness of Operators in the Wiener algebra
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Toeplitz operators are polynomials $sum a_i V^i$ (or limits of such) of the shift operator $V$ with constant coefficients $a_i$. If we allow the coefficients $a_i$ to be non-constant bounded functions and impose the summability condition $sum ||a_i||_infty < infty$, then we get what is called an operator in the Wiener algebra. We discuss the Fredholm property of such operators acting on an $ell^p$ space, show that this property does not depend on $p in [1,infty]$ and show new results obtained by so-called limit operator techniques. We will illustrate these results by looking at classes of discrete Schrödinger operators on the axis.
Man Yue, Mo
Entanglement in the generalized XY model and block Toeplitz determinant
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In this joint work with Its and Mezzadri, we have studied the asymptotics of the entropy a class of spinchain that generalizes the XY spinchains. The entropy of these spinchains are expressible in terms of a block Toeplitz determinant whose asymptotics can be computed by using the Riemann-Hilbert method. We were able to express the entropy in terms of hyperelliptic theta functions and have studied the limiting entropy when a phase transition occurs. The results generalizes the previous work by Its, Jin and Korepin.
Mezzadri, Francesco
Derivative of the Riemann-zeta function, Toeplitz determinants
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The properties of the zeros of the derivative of the Riemann-zeta function are closely connected with those of the zeros of zeta(s) itself. In particular, the knowledge of the distribution of the zeros of zeta'(s) provide information on the fraction of zeros of zeta(s) on the critical line Re(s)=1/2. It is well known that the local statistical properties of zeta(s) can be modeled by the characteristic polynomials of random unitary matrices with probability distribution given by Haar measure. The idea behind this work is to compute the distribution of the roots of the derivatives of such polynomials, which numerical experiments show to be in excellent agreement with the distribution of the zeros of zeta'(s). This distribution can be expressed as a Toeplitz determinant whose symbol has essential singularities in proximity of the unit circle. In order to study the asymptotics in the limit as such essential singularities approach the unit circle, we adopt the techniques used in studying the double scaling limits that appear in Random Matrix Theory. However, in this case the limiting distribution is described by an isomonodromic problem with three essential singularities, instead of one, as in the usual double scaling limit.
Partington, Jonathan
Laplace's equation, and approximation in Hardy spaces in 2 and 3 dimensions
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This talk describes joint work with B. Atfeh, L. Baratchart and J. Leblond. We describe an approach to inverse problems for the Laplacian by means of an approximation problem in Hardy spaces, for which the solution is expressible in terms of Toeplitz operators. We explain the two-dimensional situation in detail, and describe an extension to three dimensions using the Hardy spaces defined by E. Stein, which consist of gradients of real harmonic functions.
Peller, Vladimir
Perturbation of functions of contractions
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Hankel operators are used to obtain results on approximation by analytoc matrix functions in L^p. This is a joint work with L. Baratchart.
Pelloni, Beatrice
Riemann-Hilbert and d-bar problems and moving boundaries
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We describe the recent explicit solution of moving boundary value problem for linear evolution PDEs, obtained by casting the problems as d-bar equations in the complex plane. This solution depends on the inversion formula for a generalised Fourier transform. The proof of the relevant inversion theorem follows the line of a general approach for interpreting integral transforms that has recently been studied by Fokas, Pelloni and al.
Pott, Sandra
Tangential interpolation in vector-valued H^p spaces, Carleson measures and applications to controllability
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We obtain norm estimates for the problem of minimal-norm weighted tangential interpolation by functions in vector-valued $H^p$ spaces, expressed in terms of the Carleson constants of related scalar and vector measures. Applications are given to the controllability properties of linear semigroup systems with a Riesz basis of eigenvectors. This is joint work with Birgit Jacob (Delft University of Technology, Netherlands) and Jonathan R. Partington (University of Leeds,UK).
Shargorodsky, Eugene
Toeplitz operators with bounded measurable coefficients
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The talk is intended as a nontechnical review of some recent progress and open problems in the study of the essential spectra of Toeplitz operators with bounded measurable coefficients.
Spitkovsky, Ilya
On the almost periodic factorization
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We will give an overview of currently known result on the existence and construction of the almost periodic factorization, and will also discuss related open problems.
Vasilevski, Nikolai
On the structure of the C^*-algebra generated by Toeplitz operators with piece-wise continuous symbols
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It is well known that the $C^*$-algebra generated by Toeplitz operators with piece-wise continuous symbols has quite a complicated structure. Apart of its initial generators, it contains all finite sums of finite products of the initial generators as well as the uniform closure of the set of such sums of products. We clarify the structure of this algebra presenting a certain canonical form of any element from the algebra and characterizing all Toeplitz operators which belong to this algebra.
Virtanen, Jani
Toeplitz and Hankel operators on the Bergman space A^1
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We recall some recent results on Toeplitz operators acting on the Hardy space $H^1$ with continuous and $C+H^infty$ symbols that are of logarithmic bounded mean oscillation. In particular we concentrate on boundedness, compactness and Fredholmness. Similar questions are considered when Toeplitz and Hankel operators act on the Bergman space $A^1$ and their symbols belong to the logarithmic $BMO$ in the Bergman metric.
Wegert, Elias
Blaschke representation of functions on the circle
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There are several representations of a complex valued function $f$ on the unit circle $mathbb{T}$ involving functions $f_+$ and $f_-$ which extend holomorphically into the interior $mathbb{D}$ and the exterior $mathbb{E}:=mathbb{C}setminusoverline{mathbb{D}}$ of the circle, respectively. The two most familiar representations are the additive Riesz decomposition $f = f_+ + f_-$ and the multiplicative Wiener-Hopf factorization $f(t) = f_+(t) cdot t^kappa cdot f_-(t)$. We show that every unimodularly bounded measurable function on $mathbb{T}$ admits a (essentially unique) ``Blaschke representation'' [f=frac{f_++f_-}{1+overline{f}_-f_+}.]
Wenzel, David
A probability argument in favor of ignoring small singular values of Toeplitz operators
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If the matrix of a square linear system is nonsingular but has very small singular values, then tiny perturbations of the right-hand side may cause drastic changes in the solution. We show that the probability for this to happen is very close to zero if sufficiently many singular values of the matrix are bounded away from zero. The results are illustrated by normal and multi-level Toeplitz matrices.
Widom, Harold
On the inverse and determinant of certain truncated Wiener-Hopf Operators
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The solution of a problem arising in integrable systems requires sharp asymptotics for the inverses and determinants of truncated Wiener-Hopf operators, both in the regular case (where the non-truncated Wiener-Hopf operator is invertible) and in singular cases. We consider the cases where the symbol has one double zero or two simple zeros and find formulas for the inverse and determinant with very small error. The application is to the cylindrical Toda equations.

Participants

Name	Institution
Baik, Jinho	University of Michigan
Basor, Estelle	California Polytechnic State University
Borodin, Alexei	California Institute of Technology
Böttcher, Albrecht	Techical University of Chemnitz
Cafasso, Mattia	SISSA
Chen, Yang	Imperial College London
Deift, Percy	New York University
Dritschel, Michael	University of Newcastle
Duits, Maurice	Katholieke Universiteit Leuven
Ehrhardt, Torsten	University of California Santa Cruz
Englis, Miroslav	Academy of Sciences, Prague
Fokas, Thanasis	Cambridge University
Grudsky, Sergei	CINVESTAV
Its, Alexander	Indiana University-Purdue University Indianapolis
Karlovich, Yuri	Universidad Autonoma del Estado de Morelos
Krasovsky, Igor	Brunel University
Kuijlaars, Arno	Katholieke Universiteit Leuven
Lasarow, Andreas	Katholieke Universiteit Leuven
Lindner, Marko	University of Reading
Man Yue, Mo	University of Bristol
McGuinness, Bronagh	University of Ulster
Mezzadri, Francesco	University of Bristol
Partington, Jonathan	University of Leeds
Peller, Vladimir	Michigan State University
Pelloni, Beatrice	University of Reading
Pott, Sandra	University of Paderborn and University of Glasgow
Shargorodsky, Eugene	King's College London
Spitkovsky, Ilya	College of William and Mary
Toland, John	University of Bath
Vasilevski, Nikolai	Cinvestav
Virtanen, Jani	University of Helsinki
Wegert, Elias	Tech Univ Bergakademie Freiberg
Wenzel, David	Technical University of Chemnitz
Widom, Harold	University of California, Santa Cruz