bluuebunny/crossref_metadata_2025_split huggingface dataset:
Gemischte und nicht-standard Finite-Elemente-Methoden mit Anwendungen
Authors: Braess Dietrich , Carstensen Carsten , Hackl Klaus 
Published: December 2009

Mixed finite element methods (MFEM) form a general mathematical framework for the spatial discretisation of partial differential equations, mainly applied to elliptic equations of second order. They become increasingly important for the solution of nonlinear problems. In contrast to standard finite element schemes the mixed finite element discretisation of problems in divergence form, i.e.

f +{\rm div} \, \sigma = 0

where

\sigma = A (\nabla \, u)

,

\sigma \in L

and

u \in H

, allows more flexibility in the design of the discrete approximation spaces contained in

L

and

H

, i.e. in the spaces for the direct variables and the Lagrange multipliers.

The workshop focuses on new developments in the field of mixed and non-standard finite element methods. The main points are
The workshop aimed at bridging the gap between the computational engineering community and applied mathematicians and in consequence to unify the scientific language and foster later collaboration. Nonlinear mixed schemes were of particular concern for problems in elasticity and plasticity, but electromagnetics and mathematically related topics were also included.
Mixed finite element methods for elliptic problems are based on a variational description in saddle-point form. Side conditions such as divergence free velocity fields in incompressible fluid dynamics are usually treated in this framework. The appearance of ‘soft’ side conditions is typical for structural mechanics as is the case with nearly incompressible materials or plates and shells with small thickness parameters. We also mention materials which almost satisfy the Kirchhoff condition, i.e. problems with a high but finite shear stiffness. In such cases, which are by no means ‘soft’ from the mathematical point of view, mixed methods lead to a more robust discretisation.
The arising stability conditions and computational techniques cannot be understood fully by intuitive mechanical principles; however, from the mathematician's point of view their reasoning is natural, clear and insightful.
Mixed and non-standard finite element methods gain increasing prominence in the prevention of locking phenomena. We highlight a topic which is currently actively investigated: the development of stable and efficient plate and shell elements with regard to shear locking, which is more intricate than volume locking. Here it is important to understand how techniques based on heuristic ideas are consistent with more modern mathematical methods.
Availability of fast solvers is decisive for the competitiveness of numerical techniques. For a variety of applications, multigrid methods are crucial for the efficiency of the implementation.
Methods have been proposed which do not appear plausible if one wants to deduce the algorithms directly from the physical model. The advanced methods depend on rigorous error estimators in order to guarantee that the numerical solutions represent the exact solutions of the physical model.
Mini-Workshop: Analytical and Numerical Methods in Image and Surface Processing
Authors: Dziuk Gerhard , Reif Ulrich , Rumpf Martin , Schröder Peter 
Published: December 2009

In different areas of mathematics and computer science significant progress has been achieved with respect to geometric modeling, the processing of detailed surface models and geometric methods in morphological image processing. In contrast to classical tools in approximation theory such as spline curves and surfaces these new approaches rely on methods from the calculus of variations and geometric evolution problems. As examples we would like to mention here:

Recently it has become clear that even though the applications areas differ significantly the methodological overlap is enormous. E.g. currently Willmore flow is one of the hot topics in geometric analysis and at the same time it turned out to be an adequate approach for the restoration of destroyed surfaces. Simultaneously the Willmore energy is a fundamental building block in variational methods of morphological image processing. Indeed as for surfaces it can be used for restoration and the disocclusion of images. Furthermore, it enters important fairness criteria in geometry processing and the generation of

C^1

smooth surface models.

The progress in this new, active and already very successful research field has its roots in originally completely different disciplines of mathematics and computer science. Hence, an exchange of different methodologies and a joining of activities is very promising. This was the intention of the mini workshop. To achieve this goal researchers from completely different areas of mathematics and mathematically oriented computer science were invited to share their point of view and their recent achievements with others. The intention of this mini-workshop was to initiate stronger exchange, to encourage the exploration of synergies and the building of bridges between the different disciplines and the different types of approaches for closely related research fields. In particular we intended to bring together people from
Let us unroll one of these synergy fields:
The organizers contributed introductory talks into the different fields to stimulate discussion and interaction.
In parallel to this mini workshop another mini workshop on fluid interfaces took place. Together with the organizers of the latter workshop we arranged for several joined talks on the common interest field of geometric evolution problems by participants of both workshops. This interplay turned out to be very fruitful and inspiring.
One Thursday evening we had a longer evening talk on a new approach for the discretization of fluid motion given by Peter Schröder.
Enveloping Algebras and Geometric Representation Theory
Authors: Kumar Shrawan , Littelmann Peter , Soergel Wolfgang 
Published: December 2009

Since its inception in the early seventies, the study of Enveloping Algebras has undergone a significant and continuous evolution and moreover has inspired a wide variety of developments in many areas of mathematics including Ring Theory, Differential Operators, Invariant Theory, Quantum Groups and Hecke Algebras.
As indicated above, one of the main goals behind this meeting was to bring together a group of participants with a wide range of interests in and around the geometric and the combinatorial side of the representation theory of Lie groups and algebras. We strongly believe that such an approach to representation theory, in particular interaction between geometry and representation theory, will open up new avenues of thought and lead to progress in a number of areas.
This diversity was well reflected in the expertize represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under the following three headings:

(1) The study of the structure and representation theory through
a) Slodowy slices and the Joseph ideal,
b) Equivalence of categories of representations and certain geometric counterparts,
c) Hall algebras, Quotient schemes and canonical bases, d) Richardson elements and birationality questions,
e) free Lie algebras and current algebras,
f) Dirac cohomology,
h) Gelfand-Zeitlin theory

(2) Combinatorial Aspects of Lie Systems particularly through
a) Saturation problems and Buildings,
b) Affine and double Affine Hecke algebras, Cherednik algebras,
c) Solvable lattice models

(3) Geometric Structures including
a) Representation theoretic methods in enumerative geometry,
b) Multiplicity free symplectic reduction,
c) Monodromy actions of braid groups,
d) Associated varieties for Lie super algebras,
e) affine Grassmannians,
f) Schubert varieties and Demazure modules
The single most important development reported in this conference seems to us to be the generalization of the categorical Satake isomorphism to quantum groups at roots of unity, which allows to bring in a whole lot of new geometry into representation theory. It seems likely that it will lead to the proofs of some open conjectures of Lusztig. We were particularly happy to have quite a few very strong younger participants and some highly promising postdocs. We tried to accomodate young postdocs, as much as feasible, to present their work by devoting one afternoon to their (shorter) talks.
Free Probability Theory
Authors: Biane Philippe , Speicher Roland , Voiculescu Dan-Virgil 
Published: December 2009

The workshop
Free Probability Theory
, organised by Philippe Biane (Paris), Roland Speicher (Kingston), and Dan Voiculescu (Berkeley) was held March 27th–April 2nd, 2005. This meeting was well attended with over 50 participants with broad geographic representation from Austria, Canada, Denmark, France, Germany, Hungary, Japan, Netherlands, Poland, USA.

This workshop was sponsored by a project of the European Union which allowed us to invite in addition to established researchers also a couple of young people who were interested in learning about free probability.
Free probability theory is a line of research which parallels aspects of classical probability, in a non-commutative context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. It grew out from attempts to solve some longstanding problems about von Neumann algebras of free groups. In the almost twenty years since its creation, free probability has become a subject in its own right, with connections to several other parts of mathematics: operator algebras, the theory of random matrices, classical probability and the theory of large deviations, algebraic combinatorics, topology. Free probability also has connections with applied mathematics (wireless communication) and some mathematical models in theoretical physics.
The Oberwolfach workshop on free probability brought together a very strong group of mathematicians representing the current directions of development in the area. The diversity of the particants and the ample free time left in the programme stimulated a lot of fruitful discussions, laying the seed for many new collaborations.
The programme consisted of 13 lectures of 50 minutes, supplemented by 13 shorter contributions of 30 minutes. Because of the various backgrounds of the participants much emphasis was put on making the lectures accessible to a broad audience; most of them provided a survey on the background as well as highlighting some recent developments in connection with free probability. Instead of trying to summarize all these developments we will let the following abstracts speak for themselves.
Kommutative Algebra
Authors: Bruns Winfried , Flenner Hubert , Huneke Craig 
Published: December 2009

The workshop on Commutative Algebra was very well attended by the important senior researchers in the field and many promising young mathematicians. A major subgroup of the participants was formed by researchers in affine geometry, a neighboring field that has strong interactions with commutative algebra.
The NSF Oberwolfach program made it possible to increase the number of young participants considerably, and the organizers are very grateful to the NSF for its support.
The conference took place in a very lively atmosphere, made possible by the excellent facilities of the institute. There were 53 participants and 19 talks with a considerable number of lectures given by young researchers. The program left plenty of time for cooperation and discussion among the participants. We highlight the areas in which new results were presented by the lecturers:

(a)

Characteristic

p

-methods

Starting with the landmark results of Peskine and Szpiro, characteristic

p

methods have had a extremely strong influence on the development of commutative algebra. They were crystalized by Hochster and Huneke in the notion of tight closure, and have led to remarkable results in ideal and module theory.

(b)
Combinatorial commutative algebra and effective methods
The spectacular applications of commutative algebra to enumerative combinatorics two decades ago have developed into a subfield of commutative algebra that is very active now. The main objects are algebraic structures defined by monomials, in particular face rings of simplicial complexes and affine monoid algebras. Another driving force of this development are the powerful effective methods based on Gröbner bases since they ultimately rely on monomial computations.

(c)
Homological methods and invariants
This area started from the fundamental theorems of Auslander, Buchsbaum and Serre on regular rings. Homological properties are used in the major classification of commutative rings and their modules. In the last decade considerable progress has been made through the use of differential graded algebras, derived categories and the duality between the polynomial ring and the exterior algebra.

(d)
Multiplier ideals
The multiplier ideals of an ideal

I\subset \mathbb C[X_1,\dots,X_n]

constitute subtle invariants of the singularity defined by

I

. Originally defined in analytic terms, they have now been applied to algebraic problems and have led to surprising results in ideal theory, especially on uniform bounds for the asymptotic behavior of various kinds of powers of an ideal.

(e)
Affine geometry
Affine geometry deals with algebro-geometric questions of affine varieties. The talks concerned actions of the additive and multiplicative groups

{\mathbf G_a}

and

{\mathbf G_m}

on affine varieties and, in particular, the affine space. Moreover there were lectures on the famous Jacobian problem and possible generalizations to other varieties. In many of the contributions the close ties to commutative algebra became apparent and led to fruitful discussions.
Mini-Workshop: Aspects of Ricci-Flow
Authors: Ecker Klaus , Huisken Gerhard , Ilmanen Thomas 
Published: December 2009

The motion of a Riemannian metric along its Ricci curvature,

\tfrac{d}{dt}g_{ij} = - 2R_{ij} (g(t))

was proposed in 1982 by Richard S. Hamilton as a geometric version of the heat equation suitable for uniformizing and smoothing the geometry of a given initial Riemannian manifold

(M^3, g_0)

. Hamilton's work has opened up the whole area of geometric evolution equations, leading to the discovery of new phenomena in these equations and to topological applications such as the classification of 3-manifolds of positive Ricci-curvature and certain 4-manifolds. Recent work of G. Perelman has indicated how to approach a proof of the Poincaé-conjecture and the Thurston Geometrization conjecture for 3-manifolds using Hamilton–Ricciflow.

The mini workshop has concentrated on a thorough technical investigation of that part of the work of Perelman that is related to Hamilton–Ricciflow with surgeries on a finite time interval. Together with further work of Perelman and also Colding–Minicozzi this part of Perelman's work implies a proof of the Poincaré conjecture when confirmed. The efforts of the workshop were greatly helped by previous work of other mathematicians on Perelman's work., e.g. the notes of B. Kleiner and J. Lott.
The workshop was able to confirm major sections in the two papers of G. Perelman including the entropy- and reduced volume estimates, the compactness properties of ancient solutions to the flow and the surgery construction. It was also able to reinterpret several arguments involving Alexandrov spaces from the viewpoint of smooth Differential Geometry.
When the workshop had to come to an end, its participants agreed that it would be very desirable to establish self-contained expositions of the following points:

(1) The boundedness of the curvature

\sup_{B_\rho(x)} R \leq c(\rho)

in the proof of the approximation theorem I.12.1 of Perelman's first paper.

(2) The approximation of mini-max surfaces in the varifold distance for immersed surfaces in the paper of Colding–Minicozzi.
(3) The survival of the reduced volume estimate past surgeries.

(4) The uniform control of a fixed scale

\rho > 0

past all surgeries on a finite time interval, below which the approximation theorem applies.

It seems that detailed self-contained expositions of (3) and in particular (4) require more effort than (1) and (2).
Nonlinear Evolution Problems
Authors: Ecker Klaus , Shatah Jalal , Struwe Michael 
Published: December 2009

The workshop ‘Nonlinear Evolution Problems’ focussed on three types of nonlinear evolution equations:
(1) Geometric evolution equations (essentially of parabolic type)
(2) Nonlinear hyperbolic equations
(3) Dispersive equations
The programme consisted of 22 talks presented by international specialists from Australia, France, Germany, Italy, Sweden, Switzerland and the United States.Generally, three lectures were delivered in the morning sessions and two in the late afternoon which left ample time for individual discussions.
In the first group of equations, in particular the Ricci flow and conformal flows such as the Yamabe flow and the Q-curvature flow were considered. A further focus was on curvature flows for hypersurfaces such as the Gauss and harmonic mean curvature flow and on geometric flows of higher order. Julie Clutterbuck presented a direct approach to certain fundamental estimates for a general class of parabolic equations. Mete Soner pointed out important relations between fully nonlinear parabolic equations and backward stochastic differential equations.
The class of hyperbolic equations was represented by wave maps, Einstein's equations of gravitation and nonlinear wave equations in waveguides and cones. Among the dispersive equations the nonlinear Schroedinger equation and the KdV equation were considered.
The individual discussions covered connections between approaches to the different equations in each group but also on common techniques used across the three classes of equations. Among such techniques are for instance the careful examination of the algebraic and geometric structure of the nonlinear terms. Another common theme appears to be the phenomenon of blow-up (singularity formation) for solutions and methods to describe the solution near these. Selfsimilar solutions and blow-up rates play an important role here.
The organizers decided to particularly encourage the young researchers among theparticipants to present their work, including a PhD student and several recent post-doctoral fellows.
Real Analysis, Harmonic Analysis and Applications to PDE
Authors: Müller Detlef , Stein Elias M. 
Published: December 2009

Major areas and results represented at the workshop are:

I. Methods in harmonic analysis
(a) Multilinear analysis: this is an outgrowth of the method of “tile decomposition” which has been so successful in solving the problems of the bilinear Hilbert transform. Recent progress involves the control of maximal trilinear operators and an extension of the Carleson–Hunt theorem.

(b) Geometry of sets in

\R^d

: This includes recent progress on the interaction of Fourier analysis and geometric combinatorics related to the Falconer distance problem.

(c) Singular integrals: A break trough has been obtained on singular integrals on solvable Iwasawa

AN

- groups, which require a new type of Calderón–Zygmund decomposition, since the underlying spaces have exponential volume growth. Further significant progress involves the theory of operator-valued Calderón–Zygmund operators and their connection to maximal regularity of evolution equations.

(d) Oscillatory integrals, Fourierintegral operators and Maximal operators: This includes estimates of maximal operators related to polynomial polyhedra and their relations with higher dimensional complex analysis, sharp estimates for maximal operators associated to hypersurfaces in

\R^3,

estimates for degenerate Radon transforms and linear and bilinear estimates for oscillatory integral operators, as well as optimal Sobolev regularity for Fourier integral operators.

II. Applications to P.D.E.

(a) Dispersive linear and non-linear equations: Far reaching new approaches to dispersive estimates for Schrödinger equations via coherent state decompositions and a related new phase space transform adapted to the wave operator were introduced. Further significant progress includes

L^p

- estimates respectively blow up rates for eigenfunctions and quasimodes of elliptic operators on compact manifolds with and without boundary, and related problems for globally elliptic pseudodifferential operators, and well-posedness of the periodic KP-I equations. All these results are based in part on important ideas in harmonic analysis (such as I(d) above).

(b) Schrödinger operators with rough potentials: This includes quantitative unique continuation theorems and their relations with spectral properties and the study of embedded eigenvalues of Schrödinger operators.
The meeting was attended by 50 participants. The official program consisted of 23 lectures, and left sufficiant room for further activities, such as self-organised sessions and discussions among groups of participants. The organisers made an effort to include young mathematicians, and greatly appreciate the new joint program of the Oberwolfach Institute and the american NSF, which allowed to invite several outstanding young scientists from the United States.
Explicit Methods in Number Theory
Authors: Cohen Henri , Lenstra, Jr. Hendrik W. , Zagier Don B. 
Published: December 2009

The workshop
Explicit Methods in Number Theory
was organised by Henri Cohen (Talence), Hendrik W. Lenstra (Leiden), and Don B. Zagier (Bonn) and was held July 17–23, 2005. Three previous workshops on the topic had been held in 1999, 2001, and 2003. The goal of this meeting was to present new methods and results on concrete aspects of number theory. In many cases, this included computational and experimental work, but the primary emphasis was placed on the implications for number theory rather than on the computational methods employed.

There was a ‘mini-series’ of five

1

-hour morning talks given by Bas Edixhoven, Johan Bosman, Robin de Jong, and Jean-Marc Couveignes on the topic of computing the coefficients of modular forms. Let

\Delta=q \prod_{n \geq 1} (1-q^n)^{24}=\sum_{n \geq 1} \tau(n) q^n

be Ramanujan's tau function, a newform of weight

12

for

SL_2(\mathbb Z)

. The speakers exhibited a method to compute

\tau(p)

for

p

prime in time polynomial in

\log p

.

Some of the other main themes included:
As always in Oberwolfach, the atmosphere was lively and active, providing an ideal environment for the exchange of ideas and productive discussions. This meeting was well-attended – with over 50 participants from a variety of backgrounds and with broad geographic representation from all continents, including a number of younger researchers. There were 30 talks of various lengths, and ample time was allotted for informal collaboration.
Differentialgeometrie im Grossen
Authors: Leeb Bernhard , Seidel Paul , Tian Gang 
Published: December 2009

The meeting continued the biannual conference series
Differentialgeometrie im Großen
at the MFO which was established in the 60's by Klingenberg and Chern. While global Riemannian geometry with its connections to geometric analysis and topology remained an important focus of the conference, this time special emphasis was given to complex and symplectic geometry.

There were 50 participants from 8 countries, more specifically, 22 participants from Germany, 9 from other European countries, 1 from China, 19 from North-America. 10% of the participants were women. More than half of the participants (about 29) were young researchers (less than 10 years after diploma or B.A.), both on doctoral and postdoctoral level.
The official scientific program consisted of 22 one hour talks leaving plenty of time for informal discussions. The organizers would like to thank the institute staff for their great hospitality and support before and during the conference.