Non-commutative Geometry

When studying problems in difficult algebras one often tries to reduce
the problem to a simpler problem in a simpler algebra. For example when
studying pseudodifferential operators, problems are often reduced to
studying the commutative algebra of symbols. This can be formalized by
studying extensions of algebras. For C*-algebras this theory is
welldeveloped and is studied via an algebraic invariant. This algebraic
invariant has relations with KK-theory and index theory. I will present
a generalization of this algebraic invariant to a class of *-algebras
and indicate how this can be used to derive explicit index formulaes such
as Atiyah-Singers index theorem.

Time: 13.15-14.30, Wednesday, April 9

Place: MH:333, Centre for Mathematical Sciences, Lund

Speaker: Abdenacer Makhlouf, Université de Haute Alsace, Mulhouse, France

Title: Deforming binary and ternary algebras: cohomological tool and some structure results.

OBS! Colloquium in Mathematics

Abstract:

In the this talk, I describe the deformation theory
introduced by Gerstenhaber for associative algebras and by
Nijenhuis-Richardson for  Lie algebras.  Then  I extend it to ternary
algebras.  These deformation theories are an algebraic point of view
based on the cohomology which lead to several structure results on rigid
algebras and the global study of the algebraic varieties of a given
finite dimensional algebras.

Time: 15.30-16.30, Wednesday, February 27

Place: MH:333, Centre for Mathematical Sciences, Lund

Speaker: Christian Svensson, Matematikcentrum, Lunds universitet

Title: Crossed Product Structures Associated With Discrete Dynamical Systems

OBS! Licentiate seminarium

Abstract:

The origin of the research done in the papers included in this thesis is the theory of crossed product C*-algebras. In our context, discrete dynamical systems generated by a homeomorphism of a topological space correspond to crossed product C*-algebras obtained from the actions of the group of integers on the C*-algebra of complex-valued continuous functions on the space, induced by the homeomorphism. There exist many well-known striking equivalences between topological properties of such dynamical systems and their associated C*-crossed products, one of the most famous ones being the equivalence between minimality of a system and simplicity of its C*-algebra.

In the papers included in this thesis we introduce an algebraic crossed product structure induced by an automorphism of a commutative Banach algebra, and investigate its connections with a naturally associated topological dynamical system on the character space of the Banach algebra. Many analogues of the known equivalences from the theory of crossed product C*-algebras hold true in this context, as well as results that appear to lack correspondents in the C*-algebra case. We consider e.g. the ideal structure and maximal abelian subalgebras of the crossed product and how these algebraic properties can be related to topological features of the system. We also start the investigation of a Banach algebra crossed product whose C*-envelope is the usual crossed product C*-algebra.

Time: 15.30-16.30, Tuesday, February 26

Place: MH:333, Centre for Mathematical Sciences, Lund

Speaker: Sten Kaijser, Department of Mathematics, Uppsala University

Title: The Mellin transform and some well-known inequalities.

Mellin-transformen och några välkända olikheter

Abstract:

The starting point for this seminar is the observation that
L^p spaces are invariant under a change of density. This implies that
a change of density is always possible and this can often lead to new
knowledge. An important example of this is replacing the additively
invariant Lebesgue measure by the multiplicatively invariant
Euler-Lebesgue measure.

Sammanfattning: Utgångspunkten för detta föredrag är iakttagelsen att
L^p-rum är invarianta vid byta av täthet. Detta innebär att ett enkelt
byte av täthet kan göras problemfritt, något som ofta leder till helt ny
insikter. Ett viktigt exempel på detta är övergången från det additivt
invarianta Lebesgue-måttet till det multiplikativt invarianta
Euler-Lebesgue-måttet.

Time: 16.15-17.15, Friday, February 22

Place: MH:333, Centre for Mathematical Sciences, Lund

Speaker: Klas Modin, Numerisk analys, Matematikcentrum, Lunds universitet

Title: Explicit adaptive geometric integration based on splitting

and its connection to Nambu–Poisson brackets

Abstract:

Geometric integrators are numerical integration methods where the discrete phase flow
preserve some underlying geometric structure. The two most common examples are symplectic methods and reversible
methods. For problems in Hamiltonian dynamics (or Poisson dynamics),
geometric integrators are preferable over standard methods, because they
usually conserve, or ``almost conserve'', first integrals over very long time intervals.

Implementation of adaptive step size control for geometric numerical integration schemes
is non-trivial, since conventional strategies destroy the structure preserving properties.
The common approach is to introduce a dynamic time transformation of the original
system, and then to utilize a geometric (structure preserving) discretization of the transformed system.

An explicit fully reversible integrating step size control algorithm for reversible systems is suggested in [HS]
The idea is to consider an augmented dynamical system, with an additional first integral that corresponds to the step
size control objective, and then to carry out explicit reversible discretization of this system.
Theoretical justification is given of the good long time behaviour of the algorithm when applied to integrable
reversible systems. However, when applied to Hamiltonian systems, the resulting adaptive integrator
is not symplectic, which for many problems (e.g. non-reversible and/or non-integrable problems)
is a drawback.

In the first part of the talk I will sketch the basic principle behind
splitting methods (for construction of symplectic methods) and show
how the Baker--Campbell--Hausdorff (BCH) formula
is used to obtain theoretical results on the long time energy behaviour.

In the second part I will show that the time transformed augmented system used as a basis in [HS]
admits a generalized Poisson structure, which is some cases
correspond to a Nambu--Poisson structure. Furthermore, it turns out that the adaptive numerical scheme
suggested in [HS], can be viewed as a splitting method in the framework of Nambu--Poisson brackets.
Possible theoretical implications and open problems connected to this will also be discussed.

Bibliography

[HS] E.Hairer and G. Söderlind. Explicit, time reversible, adaptive step size control.
SIAM J. Sci. Comput., 26:1838--1851, 2005.

PAST SEMINARS and EVENTS

list is incomplete, more web info on past seminars and events will be added

info on activities during 2005 is under construction and will be added soon

AUTUMN 2007

Time: 10.15-11.00, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Michael Cwikel, Technion – Israel Institute of Technology, Haifa, Israel:

Title: Complex interpolation of compact operators. Creeping towards an answer?

Abstract:

The problem that we are struggling with goes back to Alberto Calderon's celebrated and beautiful work on complex interpolation spaces. Thus it has been open since August 15, 1963.

Suppose that T is a linear operator which acts compactly on both of the Banach spaces X and Y. We still do not know whether, in general, T is also compact on Calderon's space [X,Y]_θ.

Partial answers to this question have been given by Calderon himself, by Lions and Peetre, Arne Persson, and many others.

I will report on some recent partial answers which at least give the impression (illusion?) that we might be creeping closer to a full answer.

Unless the audience prefers otherwise, I will begin by recalling some of the history and applications of interpolation theory and the definitions and some of the basic relevant facts about Calderon's spaces. I will also indicate various connections with Fourier series. Indeed Fourier series are apparently a significant part of the arsenal we have for attacking this problem. (As some of you may recall, Svante Janson has characterised Calderon's spaces via sequence spaces of Fourier coefficients, and Fedor Nazarov has used Fourier series to give an "almost counterexample" to a closely related question.)

Some background about these things can be found at

http://www.math.technion.ac.il/~mcwikel/compact

Time: 11.10-11.55, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Lars-Erik Persson, Luleå University of Technology, Luleå, Sweden:

Title: Hardy-type inequalities and convexity- historical remarks and some

recent results and questions

Abstract:

I will first shortly present some remarkably facts from the period 1915-1925, which finally led Hardy to present and prove his inequality in his famous 1925 paper (in particular, the Swedish mathematician Riesz was important for this development !), see [1]. After that I will present the easiest (and, in my opinion, most natural) proof of this inequality I know. This proof depends only on a convexity argument and this way of thinking is also related to interpolation theory. Finally, I present some facts from the books [2] and [3] but also some even more recent results and open questions.

[1]A. Kufner, L. Maligranda and L.E. Persson, The prehistory of the Hardy
inequality, Amer. Math. Monthly 113 (2006), No. 8, 715-732.

[2]C.P. Niculescu and L.E. Persson, CONVEX FUNCTIONS AND THEIR APPLICATIONS-A CONTEMPORARY APPROACH, Canad. Math. Series Books in Mathematics, Springer, 2006 (252 pages).

[3] A. Kufner, L. Maligranda and L.E. Persson, THE HARDY INEQUALITY. ABOUT ITS HISTORY AND SOME RELATED RESULTS, Vydavatelsky Servis Publishing House, Pilsen, 2007 (161 pages).

Time: 13.15-13:45, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Jaak Peetre, Lund University, Lund, Sweden:

Title: Neoclassical Invariant theory

Abstract:

Some topics and ideas related to this interesting neoclassical subject will be presented.

Time: 13.50-14.10, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Christian Svensson: Lund University, Lund, Sweden

Title: Crossed product algebras and Banach algebras associated to dynamical systems.

Abstract:

Given a discrete dynamical system, one may construct an associative (non-commutative) complex algebra with multiplication determined via the action defining the system, being an example of a crossed product. It turns out that for large classes of systems, one obtains striking equivalences between, in particular, topological properties of the system and algebraic properties of the crossed product. In this talk we will discuss the ideal structure of crossed products and ideal intersection properties of (commutative) subalgebras between the canonical "coefficient algebra" and its commutant.

Time: 14.10-14.30, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Johan Oinert, Lund University, Lund, Sweden

Title: Crystalline graded rings and generalized crossed products

Abstract:

In this talk we will focus on a certain 'intersection property' of ideals in certain kinds of rings and algebras. We start by looking at C*-crossed product algebras associated with dynamical systems, which is our main motivating example. We will then show that the 'intersection property' also holds in a purely algebraic framework, namely for algebraic crossed products. Thereafter we generalize this result to a more general class of rings, the so called 'crystalline graded rings'. Both of these classes of rings are group graded, but we will define a new class of rings (with motivation coming from irreversible dynamical systems) which is only assumed to be monoid graded. After giving the definition, we shall give some concrete examples of such rings and show that in general the 'intersection property' need not hold for these rings.

Time: 14.35-14.55, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Erik Darpö, Uppsala University, Lund, Sweden

Title: On the classification of finite-dimensional division algebras

Abstract:

Let k be a field. A division algebra over k is a vector space A over k endowed with a bilinear multiplication map AxA --> A; (x,y) --> xy such that left and right multiplication with any non-zero element in A are invertible linear operators on A. In general, a division algebra is not assumed to be associative or commutative, nor to have an identity element.

We shall give a survey of the classification problem for finite-dimensional division algebras, with focus on algebras over the real ground field. This includes the construction of the four classical real
division algebras, theorems by Frobenius and Zorn on associative and alternative division algebras respectively, and the celebrated (1,2,4,8)-theorem, which asserts that every finite-dimensional division algebra over R has dimension either 1, 2, 4 or 8.

We will also touch upon some recent development in the field, originating mainly from Osborns work on quadratic division algebras in the sixties.

Time: 15.30-15.50, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Olivier Verdier, Lund University, Lund, Sweden

Title: Application of interpolation between anisotropic Sobolev spaces to PDEs

Abstract:

The interpolation between anisotropic Sobolev spaces with p=2 is elementary but turns out to be an effective tool, coupled with the vectorial Sobolev inequalities, to obtain estimates of nonlinear terms. I will show some applications to the Burgers equation with highly irregular source terms.

Time: 15.50-16.35. Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Svante Janson, Uppsala University, Sweden

Title: Interpolation of subcouples and quotient couples

Abstract:

A subcouple (Y₀,Y₁) of a Banach couple (X₀,X₁) is K-closed if, for elements in Y₀+Y₁, the K-functional evaluated in the smaller couple is equivalent to the K-functional evaluated in the larger couple.

This is not true in general, but it is a useful property when it holds. This property was used and investigated by Pisier, in connection with interpolation of H^p-spaces. I will discuss this and general results concerning interpolation by the real method of subcouples and quotient couples. It seems that very little is known about corresponding results for the complex method.

This is a survey based on my paper with the same title in Ark. Mat. 31 (1993), no. 2, 307--338.

Time: 16.40-17.10, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Sten Kaijser, Uppsala University, Uppsala, Sweden:

Title: Pseudodeterminants

Abstract:

This talk will be devoted to Pseudodeterminants.

Time: 17.15-18.00, Monday, October 8

Place: MH:332A, Centre for Mathematical Sciences, Lund

Speaker: Frank Hansen, Copenhagen University, Copenhagen, Denmark

Title: The application of operator monotone functions in economics

Abstract:

Two notions in microeconomics, decreasing relative risk premium and risk vulnerability, are connected to matrix or operator monotonicity. We show that a decision maker with an increasing utility function has preferences representing decreasing relative risk premium, if and only if his utility function is matrix monotone of order two. We then show that this property may be equivalently formulated in terms of preferences on binary lotteries. We finally explain the notion of risk vulnerability and show that an
operator monotone function necessarily is risk vulnerable.

SPRING 2007

STINT intensive advanced mini-course by Palle E. T. Jorgensen

Time: kl 13:15-15:00,

Monday, June 11,

Tuesday June 12,

Wednesday, June 13

Place: MH:333, Centre for Mathematical Sciences, Lund

Speaker: Palle E. T. Jorgensen, The University of Iowa , USA

Title: Wavelets, fractals, iterated function systems and operator representations.

(http://www.maths.lth.se/matematiklth/personal/ssilvest/PalleJorgensenWaveletsLundJune2007.htm )

Abstract:

These lectures, combining analysis and tools from mathematical probability, give a systematic presentation of recent trends in three fields, wavelets, signals and fractals. The unity of basis constructions and their expansions is emphasized as the starting point for the development of bases that are computationally efficient for use in several areas from wavelets to fractals.

From operator algebra theory, a key topic in the talks will be representations of the Cuntz algebras.

They aim to bring together tools from engineering and math, especially from signal- and image processing, and from harmonic analysis and operator theory. The presentation is aimed at graduate students.

Special:

• motivation;

• glossary of terms, their use in mathematics and in engineering;

• graphic renditions of algorithms, and separate illustrations;

• explaining engineering terms to mathematicians, and operator theory to engineers;

• guide to the literature.

Palle E.T. Jorgensen is a Professor of Mathematics at the University of Iowa.

He has taught some of this in courses over the last several years.

He is the author of a GTM Springer book v 234 in 2006.

Analysis and Probability: Wavelets, Signals, Fractals, Graduate Texts in Mathematics, Vol. 234, Springer.

And his other most recent book was written jointly with Ola Bratteli and is entitled

Wavelets through a Looking Glass, Applied and Numerical Harmonic Analysis, Birkhäuser, 2002.

Time: kl 10.15-11:05, Wednesday, June 13,
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Örjan Stenflo, Uppsala University:

Title: Random V-variable fractals.

Workshop on wavelet analysis, fractal geometry, iterated function systems and applications, Lund, June 13, 2007

Abstract:

A standard way to generate probability measures supported on fractal sets is to regard them as limiting probability distributions for processes obtained from random iterations of functions. It is typically not possible to generate natural random fractals like e.g. Brownian motion in a similar way due to the complexity of these objects. This has restricted applications in fractal modeling. In joint work with Michael Barnsley and John Hutchinson we introduced V-variable fractals as a way of resolving this. Random V-variable fractals can be generated quickly as "points" along trajectories of a fractal-valued random iteration process. Simultaneously they can be used to approximate "standard" random fractals.

Time: kl 11.10-12:00, Wednesday, June 13,
Place: MH:309B, Centre for Mathematical Sciences, Lund
Speaker: Ole Christensen, Danmarks Tekniske Universitet:

Title: Gabor systems and frames.

Workshop on wavelet analysis, fractal geometry, iterated function systems and applications, Lund, June 13, 2007

Abstract:

In signal processing and pure mathematics it is useful to represent complicated signals or functions f in terms of linear combinations of simpler building blocks ek; the classical case, where ek is assumed to be an orthonormal basis for a Hilbert space H to which the relevant signals belong, leads to the representation f= Σ< f,ek> ek. However, the assumption that ek forms an ONB is quite restrictive, and it limits the other properties one can expect from ek. We discuss concrete cases from wavelet analysis and Gabor theory, where desirable properties of ek can not be combined with ek being an ONB. It turns out that much more freedom can be attained by assuming that ek is a frame rather than an ONB. We will discuss recent results concerning frames, with particular focus on Gabor frames.

Time: kl 13:15-14:00, Tuesday, May 8,
Place: MH:309B, Centre for Mathematical Sciences, Lund
Speaker: Sergei Silvestrov, Lund University

Title: Twisted Derivations and Introduction to Quasi Lie Algebras.

Workshop on homological devices and non-commutative algebra, Lund, May 8, 2007

Abstract:

In this talk, I will present a review on quasi-Lie algebras, quas-hom-Lie algebrasand hom-Lie algebras generalizing color Lie algebras, Lie superalgebras and Lie algebras and providing a unifying framework for quasi-Lie quasi-deformations via twisted derivations of infinite-dimensional Lie algebras of vector fields of Witt and Virasoro type.

Time: kl 14:05-15:00, Tuesday, May 8,
Place: MH:309B, Centre for Mathematical Sciences, Lund
Speaker: Lars Hellström, Umeå university

Title: A Generic Diamond Lemma with applications for nonassociative algebra, operads,
Hopf algebras, and beyond

Workshop on homological devices and non-commutative algebra, Lund, May 8, 2007

Abstract:

When studying algebraic structures defined by generators and relations, one often relies on a diamond lemma (or some more specialised counterpart, such as may be found in e.g. Gröbner basis theory) to obtain an effective model for the structure studied. A problem is however that diamond lemmas tend to be stated only for a specific family of algebraic structures, so that each new kind of algebraic structures (with different axioms, a different set of operations, or whatever) requires its own diamond lemma. In this talk, I will present my Generic Diamond Lemma and explain how one can apply it for a large variety of algebraic structures. Focus will be on (i) how a single result can handle many different kinds of multiplicative structures (commutative, associative, free non-associative, path algebra, etc.) and (ii) how to handle many-sorted algebraic structures (operads, PROPs, symocats, etc.).

Time: kl 15:30-16:30, Tuesday, May 8,
Place: MH:309B, Centre for Mathematical Sciences, Lund
Speaker: Goro Kato, California Polytechnic State University, San Luis Obispo

Title: N-Complexes and Precohomologies, [NCPC]

Workshop on homological devices and non-commutative algebra, Lund, May 8, 2007

Abstract:

For an abelian category A, we can from the category Se(A) of sequences and morphisms of A. Recall that in order to define cohomologies we need the category of complexes. There is a new invariant, generalizing the notion of a cohomology, defined from Se (A) to A. This invariant is said to be a precohomology. There exist two functors from Se(A) to the category of N-complexes, where N>1. Those functors are said to be complexifying functors. Then we can define a cohomology on the complexified objects in the category of N-complexes, which is the precohomology of (i,k)-type. In order to define the Derived Category on N-complexes, we need several projects including the notions of quasi-isomorphism of N-complexes, homotopy of N-complexes, spectral sequences based on N-complexes and precohomologies of composite functors, hyperprecohomologies. Also one can define a notion of inverse limits connecting to precohomologies. We will outline those projects and main statements.

Time: kl 16:30-17:30, Tuesday, May 8,
Place: MH:309B, Centre for Mathematical Sciences, Lund
Speaker: Daniel Larsson, Institute Mittag-Leffler, Stockholm

Title: Some Applications and Origins of N-complexes.

Workshop on homological devices and non-commutative algebra, Lund, May 8, 2007

Abstract:

In this talk I will discuss definitions, motivations and applications, present and future, of so-called N-complexes. These generalize ordinary complexes in homological algebra in that one do not assume the differentials to be square-zero. Instead they are nilpotent of some higher order.

Time: kl 11:00-12:00, Tuesday, February 27,
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Tomo Matsusita, Ritsumeikan University, Japan.

Title: Static Hedging in Supermarkets.

Joint with Mathematical Statistics seminar

Abstract:

We introduce a notion of supermarkets and show that in the supermarkets there are plenty of assets and "static hedging " is always possible. Typical examples are concerning Cox-Ross-Rubinstein's binary markets and Black-Sholes markets. Mathematically, expansions with respect to exponential martingales play central roles.

Time: kl 15:15-16.15, Tuesday, February 27,
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Jun Tomiyama, Tokyo Metropolitan University, Japan Women’s University.

Title: Matrix Monotone, Convex Functions and Truncated Moment Problem.

Abstract:

For a nontrivial interval I in the real line, a real valued continuous function f defined in I is said to be n-matrix monotone if the function calculus of f on the selfadjoint part of n by n matrix algebra M(n) keeps the order in M(n), and n-matrix convex when the calculus keeps the convexity. Notions of operator monotonicity and operator convexity of f are similarly defined replacing M(n) by the algebra of all bounded linear operators on an infinite-dimensional Hilbert space. Write the sets of those functions as P_n(I) and K_n(I) and the latter classes as P_{\infty}(I) and K_{\infty}(I).
In this talk, I shall show a deep relationship between the existence of polynomials in P_n(I) as well as in K_n(I) and truncated moment problems for a finite interval I, from which we can see an important aspect of structure of pilings of those P_n(I)'s and K_n(I)'s down to P_{\infty} and to K_{\infty}. This also brings an insight of the pilings for an infinite interval.

Time: kl 16:20-17:20, Tuesday, February 27,
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Tamotsu Teruya, Ritsumeikan University, Japan.

Title: Equivalence bimodules arising from index two inclusions of C*-algebras.

Abstract:

Let A be a unital C*-algebra. We shall introduce involutive

A-A-equivalence bimodules and prove that any C*-algebra

containing $A$ with Watatani index 2 is constructed by an involutive

A-A-equivalence bimodule.

AUTUMN 2006

Lund 2006, October 12- October 14, 2006, “Algebra, Geometry and Mathematical Physics” conference organized by AGMF: “Algebra, Geometry and Mathematical Physics” Baltic Nordic network

SPRING 2006

Time: kl. 13.15-15.00, Wednesday, May 24,
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Jun Tomiyama, Tokyo Metropolitan University, Japan Women’s University, Japan

Title: On the structure of the spaces of matrix monotone functions and matrix convex functions.

Abstract:

Let I be a nontrivial interval in the real line, and denote by P_n(I) and K_n(I) the spaces of monotone functions and convex functions defined on the matrix algebra M_n and the interval I respectively. We investigate what kinds of polynomials are contained in those spaces and how rich they are besides those so-called operator monotone functions and operator convex functions contained in P_n(I) and K_n(I) for every n. Discussions may contain the characterizations of 2-convex functions as well as (if possible) the consideration of convex functions on C*-algebras.

Time: kl. 15.15-16.15, Wednesday, May 24,

Place: MH:333, Centre for Mathematical Sciences, Lund

Speaker: Yacin Ameur, Kalmar Högskolan:

Title: On the structure of the spaces of matrix monotone functions and matrix convex functions.

Abstract:

The talk will concern some results related to a joint work with Sergei Silvestrov and Sten Kaijser. I will also discuss generalizations of interpolation functions, for example to Hilbertian operator spaces.

Time: kl. 16.30-17.30, Wednesday, May 24,

Place: MH:333, Centre for Mathematical Sciences, Lund

Speaker: Hiroyuki Osaka, Ritsumeikan University, Kusatsu, Japan

Title: Stable rank of inclusion of C*-algebras of depth 2

Abstract:

The notion of topological stable rank for a C*-algebra $A$, denoted by $\tsr(A)$, was introduced by Rieffel, which generalizes the concept of dimension of a topological space.

He presented the basic properties and stability theorem related to K-Theory for

C*-algebras. He also proved that $\tsr(A \rtimes_\alpha {\mathbb Z}) \leq \tsr(A) + 1$, and asked if an irrational rotation algebra $A_\theta$ has topological stable rank two. I. Putnum (\cite{pu}) gave a complete answer to this question, that is, $\tsr(A_\theta) = 1$.

Time: kl 13.15-14.15, Friday, March 31,
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Nacer Makhlouf, Université de Haute Alsace, Mulhouse, France

Title: Hopf Algebras and Hom-Hopf Algebras.

Abstract:

The discovery of quantum groups gives impulse to a strong development in the theory of Hopf algebras. In my talk I will summarize the recent developments and give some results on rigidity of Hopf algebras and irreducible components of Hopf algebraic varieties. This geometric and local study uses the algebraic formal deformations introduced by Gerstenber. Then, I will introduce the basic concepts of Hom-Hopf algebras. This structure is based on a modified associative algebra structure by a homomorphism. Using the reversing arrow ansatz, one defines the Hom-comultiplication, Hom-bialgebras and Hom-Hopf algebras.

Intensive advanced mini-course in algebraic geometry with applications

Detailed information is at the course home page:

Minikurs i algebraisk geometri och tillämpningar

Nacer Makhlouf, Université de Haute Alsace,Mulhouse, France,

Exploring Algebraic Sets with Computer Algebra Systems.

Time:

Monday 27 mars, 15.15-17.00,

Wednesday 29 March, 13.15-15.00,

Thursday 30 March, 10.15-12.00

Place: MH:228, Centre for Mathematical Sciences, Lund

Time: kl 15.30-16.30, Tuesday, March 21,
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Kjetil Røysland, University of Oslo, Norway

Title: Symmetric MRA Wavelets in L2(R^n).

Abstract:

Given various group actions on L2(R^n) , we will consider MRA's generated by invariant scaling functions.
Inspired by a paper of Packer and Rieffel, we will give conditions for
the existence of invariant MRA wavlets.

Packer, Judith A. and Rieffel, Marc A. (2003). Wavelet filter functions,
the matrix completion problem, and projective modules over C( T^n). J.
Fourier Anal. Appl. 9 No.2, 101-116.

PhD thesis defence of Daniel Larsson, Lund University, Sweden

Time: kl 15.30-16.30, Friday, February 24.
Place: MH:C, Centre for Mathematical Sciences, Lund

Title of the PhD Thesis: Quasi-Lie Algebras and Quasi-Deformations. Algebraic Structures Associated with Twisted Derivations

Opponent: Professor Jürgen Fuchs, Institutionen för ingenjörsvetenskap, fysik och matematik, Karlstad Universitet

Abstract:
This thesis introduces a new deformation scheme for Lie algebras, which we refer to as “quasi-deformations” to clearly distinguish it from the classical Grothendieck-Schlessinger and Gerstenhaber deformation schemes. The main difference is that quasi-deformations are not in general category-preserving, i.e., quasi-deforming a Lie algebra gives an object in the larger category of “quasi-Lie algebras”, a notion which is also introduced in this thesis. The quasi-deformation scheme can be loosely described as follows: represent a Lie algebra by derivations acting on a commutative, associative algebra with unity and replace these derivations with twisted versions. An algebra structure is then imposed, thus arriving at the quasi-deformed algebra. Therefore the quasi-deformation takes place on the level of representations: we “deform” the representation, which is then “pulled-back” to an algebra structure.

The different Chapters of this thesis is concerned with different aspects of this quasi-deformation scheme, for instance: Burchnall-Chaundy theory for the q-deformed Heisenberg algebra (Chapter II), the (quasi-Lie) algebraic structure on the vector space of twisted derivations (Chapter III), deformed Witt, Virasoro and loop algebras (Chapter III and IV), Central extension theory (Chapter III and IV), the Lie algebra sl(2) and some associated quadratic algebras.

Time: kl 15.30-16.30, Friday, February 17.
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Lionel Richard, Antwerp University, Belgium
Title: On the Gelfand-Kirillov conjecture, its quantum version and a mixed version.

Abstract:
Initiated in their paper of 1966, the Gelfand-Kirillov problem is dealing with the

rational equivalence (isomorphism of skew-fields of fractions) of universal enveloping algebras of algebraic Lie algebras and Weyl algebras (differential operators on the affine space). In the 90's appeared an analogue of this problem for quantum groups.

In this talk I will first present the Gelfand-Kirillov conjecture and its quantum version. In the

quantum case, I will present a theorem of G. Cauchon (2003), dealing with a large class of Ore extensions (non-commutative polynomial algebras). I will then introduce some 'mixed' (classical and quantum) Gelfand-Kirillov problem, initiated by a question of Corwin-Gelfand-Goodman (1994). After looking at some examples in 3 variables, I will introduce some general classes of skew-fields of noncommutative rational functions and give some results of classification upon these.

AUTUMN 2005

Time: kl. 11.00-12.00, Tuesday, December 6.

Place: MH:333, Centre for Mathematical Sciences, Lund

Examensarbete. Master Thesis.

Speaker: Johan Öinert, F00, Lund University (LTH)

Title: Connections between Topological Dynamical Systems and Non-Commutative C*-algebras.

SPRING 2005

Time: kl. 15.30-16.30, Monday, 21 February.

Place: MH:332:B, Centre for Mathematical Sciences, Lund

Speaker: Jun Tomiyama, Tokyo Metropolitan University:

Title: Non-Linear Calculus for Monotone Operator Functions on C*-Algebras.

Time: kl. 15.30-16.30, Monday, 17 February.

Place: MH:C, Centre for Mathematical Sciences, Lund

Matematiska Kollokviet. Colloquium.

Speaker: Jun Tomiyama, Tokyo Metropolitan University:

Title: Interplay between Topological Dynamical Systems and C*-Algebras (Case of Recurrent Sets).

Time: kl. 10.15-11.00, Friday, 4 February

Place: MH:333, Centre for Mathematical Sciences, Lund

Examensarbete. Master Thesis.

Speaker: Christian Svensson, F99, Lund University (LTH)

Title: Commuting Differential Operators and Commuting Elements in Non-commutative Algebras.

Time: kl. 10.15-11.00, Friday, 28 January.

Place: MH:333, Centre for Mathematical Sciences, Lund

Examensarbete. Master Thesis.

Speaker: Roland Mezöfi, E00, Lund University (LTH)

Title: On the q-Deformed Gelfand-Dickey Hierarchy, the q-Deformed Volterra Equation and Their Generalizations.

AUTUMN 2004

Time: kl 15.30-16.30, Friday, August 27.
Place: MH:333, Centre for Mathematical Sciences, Lund

Speaker: Yacin Ameur, Kalmar Högskolan,
Title: Intepolation functions on unital C*-algebras

Abstract: (English)
Let $a$ be a strictly positive element of a unital C*-algebra $A$, and define
the '$a$-norm' on $A$ by $\|x\|_a=\|a^{1/2}xa^{-1/2}\|$, $x\in A$. A
positive function $h$ defined on $\sigma(a)$ is called an interpolation function with
respect to $a$ if the inequality $\|x\|_h(a)\le \max(\|x\|,\|x\|_a)$ holds for
all $x\in A$. We shall give a necessary and sufficient condition for $h$ to be an
interpolation function, and also discuss the relations between interpolation functions and the recently studied classes of monotone functions defined on the positive elements of a C*-algebra.

Abstract: (Swedish)

Låt $a$ vara ett strikt positivt
element i en C*-algebra $A$ med enhet. Sätt
$\|x\|_a=\|a^{1/2}xa^{-1/2}\|$. En positiv
funktion $h$ definierad på $\sigma(a)$ säges
vara en interpolationsfunktion m.a.p. $A$ om
olikheten
$\|x\|_{h(a)}\le \max(\|x\|,\|x\|_a)$ är
uppfylld för alla $x\in A$. Vi skall ge ett
nödvändigt och tillräckligt villkor på $h$ för
att vara en interpolationsfunktion. Vi skall
även diskutera relationerna mellan
interpolationsfunktioner och de nyligen
studerade klasserna av monotona funktioner
definierade på de positiva elementen av en
C*-algebra.

4'nd Öresund Symposium on Non-commutative Geometry and Non-commutative Analysis

Lund, November 25, 2004,

Centre for Mathematical Sciences, Lund University,

Organizers:
Sergei Silvestrov, (e-mail: sergei.silvestrov@math.lth.se), Centre for mathematical Sciences, Lund University,
Soeren Eulers (e-mail: eilers@math.ku.dk)

Program and abstracts can be downloaded at

Öresund symposiums on non-commutative geometry, non-commutative analysis and applications

or as separate file here: PDF

Program and abstracts of the past Öresund symposiums can be downloaded here

Öresund symposiums on non-commutative geometry, non-commutative analysis and applications

AUTUMN 2003:

Time: kl 11.15-12.00, Torsdag, December 11.
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Jakob Willborg, student (F96), Lund
Title: Non-commutative operator method for equations of difference and
q-difference type

OBS! Master thesis presentation

Abstract: (English)
Non-commuting operators are of great interest in many different disciplines
e.g. applied physics and applied economics besides the pure mathematics. One
needs only to point to the fact that some very fundamental theorems in
quantum mechanics involve the aspect of non-commuting operators to see the
use of this theory. Different generators and defining relations result in
different quantum mechanic interpretations. This thesis centers on the
application of operator methods for solving differential, difference,
q-difference and q,H-difference equations. Methods for solving these
equations are developed by considering the commutation relations satisfied
by the operators. First some necessary notions are introduced and then a
general and effective operator method is presented to solve all of those
different types of equations. This operator method is expanded to all such
operator equations of order n with constant coefficients. The nature of the
operator commutation relations are studied and some results e.g. reordering
formulas, covariance commutation relations and more or less explicit
formulas for the powers of the operators are presented. Examples are given
of solutions to different equations. Both homogenous equations and
particular solutions of the non-homogeneous equations are studied. The
convergence of the q-deformed difference equation is investigated in some
detail with the help of visualizations done numerically with a Matlab
program. Finally, D{q,H}-equations are investigated.

Time: 15:30-16:30, Friday, November 28, 2003.
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Ralf Fröberg, Department of Mathematics, Stockholm University
e-mail: ralff@math.su.se
Title: Noncommutative Groebner bases, Koszulity, and coherence

Abstract: (English)
I am going to show how one with algebra can calculate the number
of steps of length n in a digraph. This leads naturally to the definition
of Koszul algebras, which I am going to review.

If someone wants to prepare oneself for Koszul algebras, then they can
read the review article which I have written:

Ralf Fröberg, Koszul algebras, Proc. Fez Conf. 1997, eds. Dobbs,
Fontana, Kabbaj, lect. Notes Pure Appl. Math. 205, Marcel Dekker.

Abstract: (Swedish)
Jag kommer att visa hur man med algebra kan räkna ut antalet stigar av
längd $n$ i en digraf. Detta leder naturligt till definitionen av
Koszulalgebror, som jag kommer att tala översiktligt om.

Om någon vill förbereda sig på Koszulalgebror kan de läsa en
översiktsartikel jag har skrivit:

Ralf Fröberg, Koszul algebras, Proc. Fez Conf. 1997, eds. Dobbs,
Fontana, Kabbaj, lect. Notes Pure Appl. Math. 205, Marcel Dekker.

Time: 15:30-16:30, Friday, November 14, 2003.
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker:Dmitri Piontkovski
Central Institute of Economics and Mathematics
Nakhimovsky prosp. 47, Moscow 117418, Russia
e-mail: piont@mccme.ru
Title: Noncommutative Groebner bases, Koszulity, and coherence
Abstract:
We will consider a class of associative graded algebras (so-called algebras
with non-commutative generalized Koszul filtrations) which includes
coherent algebras (in partivular, Noetherian algebras) and many important Koszul
rings. Our main examples are algebras where every right-sided finitely generated
ideal has a finite Groebner basis. If time will allow, also the commutative examples
arising from algebraic geometry will be under consideration.

Time: 13:30-14:30, Monday, November 17, 2003.
Place: MH:309C, Centre for Mathematical Sciences, Lund
Speaker: V.Manuilov, Moscow State University, manuilov@higeom.math.msu.su
Title:
Operators under non-rigid relations and almost representations of groups.
Abstract:
Let r₁,...,r_m be a set of relations on elements a₁,...,a_n.
An $\varepsilon$-almost representation of this set of relations is a set
b₁,...,b_n of operators on a Hilbert space (often finite-dimensional)
such that ||r_j(b₁,...,b_n)||< or =\varepsilon. A set of continuous
families b₁(t),...,b_n(t), t \in [0,\infty), of operators is an
asymptotic representation if it is an $\varepsilon(t)$-representation for
each t and if lim_t_{->\infty\varepsilon}(t)=0. Some properties of
almost representations can be seen already in the case of the commutation
relation r(a₁,a₂)=a₁a₂-a₂a₁. In the case of finite-dimensional Hilbert
spaces (with increasing dimension) there are topologically nontrivial
almost representations and the corresponding topological invariant (which is
the Chern character) is stable under small deformations and distinguishes
also unbounded representations. In the special case when the relations define
a group almost representations, we define almost flat vector bundles over
classifying spaces of groups and we discuss the relation to the Novikov
conjecture. We also discuss a possibility to extend an almost representation
to an asymptotic one.

SPRING 2003:

Time: 15:30-16:30, Friday, May 23, 2003.
Place: MH:332a, Centre for Mathematical Sciences, Lund
Speaker: Pär Kjellberg, Department of Chemical Physics, Lund University
Title:''Molecular Symmetry.''
Abstract: The mission of the natural sciences is to find relations or rules,
which can be used in order to understand nature.
The world is a complex place and it is often not
an easy task to describe it in a simple way.
When exploring an object in this our diverse universe
it is a well-known fact that if one finds some general symmetrical
properties of it, what ever it may be, this symmetry will
simplify the problem.
The talk will be about symmetry and group theory in the field of
Molecular Physics. The classification of molecules according to their
symmetries will be explained. An overview of the major point groups
will be given, as well as their application to vibrational modes,
chemical bonding and electronic transitions. Examples of more
complicated problems will be given in relation to the work at the
department of Chemical Physics at Lund University.

2'nd Öresund Symposium on Non-commutative Analysis and Non-commutative Geometry

Copenhagen, May 2, 2003,

Department of Mathematics, University of Copenhagen,

Time: 11.00 -- 17.20

Organizers:
Sergei Silvestrov, Lund,
Soeren Eulers, Gert Pedersen, Copenhagen

Want the program ? Click here

PDF PS file

Time: 15.30-16.30, Wednesday, April 2, 2003.
Place: MH:C, Centre for Mathematical Sciences, Lund
Speaker: Professor Erik Alfsen, Department of Mathematics, Oslo University, Norway

Colloquium talk !

Title:''Jordan algebras versus associative algebras as models
of quantum mechanics.''
Abstract: Non-commutative associative algebras were
introduced as a model of quantum mechanis by Heisenberg and Born in the
mid-twenties. Shortly afterwards P. Jordan proposed an alternative model based on the commutative non-associative algebras named after him. The
measurement theory of quantum mechanical observables can be dealt with
in both approaches. However, it is an imortant feature of quantum
mechanics that the physical variables play a dual role, as observables,
and as generators of of one-parameter groups of transformations which
describe the time development of the quantum system. An associative
product can be decomposed in a Jordan part and a Lie part. In the
associative models of quantum mechanics, the measurement theory relates
to the Jordan part and the time development relates to the Lie part.
Thus both aspects of a physical variable can be accounted for in an
associative model. Not so in the Jordan model, where more structure is
necessary to describe the time development. What is needed, is the
existence of a "dynamical correspondence", defined in terms of
one-parameter groups of order preserving transformations generated by
"order derivations". Such a dynamical correspondence will give each
element of the Jordan algebra the "double identity" required for the
application to physics. Mathematically, the theory of dynamical
correspondences carracterizes the relevant
associative algebras (C*-algebras) among the corresponding Jordan
algebras (JB-algebras). In the talk this will be explained
with precise statements of definitions and results together with some
brief comments on the ideas of the proofs.

Time: 13.15-15.00, Friday, April 4, 2003.
Place: MH:140, Centre for Mathematical Sciences, Lund
Speaker: Professor Erik Alfsen, Department of Mathematics, Oslo University, Norway
Title:''On the state space geometry of Jordan and C*-algebras.''
Abstract: For a JB-algebra, as well as a C*-algebra, one
can define the concept of a "state space", wich models the "mixed states"
of quantum theory. In both cases, the state space is a compact convex set.
In the JB-algebra case, this compact convex set completely determines the
algebra. But in the C*-algebra case, more structure is necessary to recover
the non-commutative product. It turns out that what is needed, is a "global
orientation" which "locally distinguishes multiplication from the left from
multiplication from the right". In fact, the existence of such an
orientation characterizes the state spaces of C*-algebras among those of
JB-algebras. In the talk this will be explained with precise statements of
definitions and results together with some brief comments on the ideas of
the proofs.

Time: 17.15-18.00, Thursday, March 6, 2003
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Jakob Wilborg, student, Lund

Title:"Operator method for solution of differential, difference and q-difference equations"
Abstract: This is a well prepared introductory lecture
about the famous operator method for obtaining solutions for
differential, difference and q-difference equations.
The central role of non-commutativity will be explained.

Time: 15.30-16.30, Thursday, February 27, 2003
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Stefan Heimgård, Student, Lund

Title:"Transfer operators, wavelets and Cuntz operator algebras"
Abstract: This is a well prepared introductory lecture,
based on a section from the recent excellent book by Ola Bratteli and Palle Jorgensen:
Ola Bratteli, Palle E. T. Jorgensen,
Wavelets through a Looking Glass. The World of the Spectrum,
Series "Applied and Numerical Harmonic Analysis",
Birkhäuser, Boston, 2002. (ISBN: 0-8176-3962-4)

AUTUMN 2002:

Time: 13.30-14.30, Friday, December 13, 2002.
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Professor Jun Tomiyama, Tokyo, Japan

Title:"Projection theorem revisited. "
Abstract: In a basic interplay between topological dynamics and C*-theory,
the Projection theorem plays a central role. With a simple proof recently found,
I will discuss important consequences of this result together with further recent
development.

Time: 13.30-14.30, Thursday, December 12, 2002.
Place: MH:309B, Centre for Mathematical Sciences, Lund
Speaker: Professor Jun Tomiyama, Tokyo, Japan

Colloquium talk !

Title:"Interplay between topological dynamics and theory of C*-algebras. "
Abstract: Here I will discus, with brief history, general background and
basic relations between two theories:

how noncommutative C*-algebras naturally appear in relation
with topological dynamical systems ?

Time: 13.30-14.30, Wendesday, November 6, 2002.
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Niels Jakob Laustsen, Department of Mathematics, University of Copenhagen, Denmark,
http://www.math.ku.dk/~laustsen/

Title:"Commutators of operators on Banach spaces "
Abstract: The commutator of two operators S and T is given by
[S,T] := ST - TS. The original motivation for studying commutators of
operators comes from quantum mechanics, in which case the operators S
and T are unbounded and act on a Hilbert space. Commutators are also
useful from a purely mathematical point of view when studying
non-commutativity. Indeed, the theme of this talk is that we can gain
insight into the non-commutative structure of the Banach
algebra B(X) consisting of all (bounded, linear) operators on
a Banach space X by studying the commutators of operators on X and the
subspace that they span.

First I shall review some classical results about commutators of operators
on a Hilbert space, in particular Halmos's theorem that each operator on
an infinite-dimensional Hilbert space is the sum of two commutators. Then
I shall explain how this result can be generalized to operators on many
other Banach spaces, including c₀, C([0,1]), l_p, and L_p([0,1])
for 1<=p<=infinity. The proof has a strong algebraic flavour
and so may appeal to algebraists as well as analysts.

Halmos's theorem does not extend to all Banach spaces: there are Banach
spaces X such that B(X) admits a non-zero trace, and so not
all operators on such a Banach space X can be a sum of commutators. One
family of Banach spaces with this property is the quasi-reflexive James
space J_p. I shall introduce this important Banach space and outline a
proof of the fact that there is a unique normalized trace
on B(J).

Time: 13.15-14.15, Friday, October 11, 2002.
Place: MH:333, Centre for Mathematical Sciences, Lund
Speaker: Magdalene Grantson, Department of Computer Science, Lund
Title:"An Introduction to Sphere Packing , Kissing Numbers and Symmetry "
Abstract:
Sphere packing is a problem where mathematicians try to determine how tightly
spheres can be packed in a box. The problem asks for the densest packing of equal
spheres in n-dimensional space. A typical example is a grocer trying to stack
apples in his store or pack them in a box. Arranging the apples in any random way
may result in an inefficient miss-use of space. Even though there exists several
forms of packing, I will be looking at lattice periodic packing
for identical spheres in 1,2, 3-dimensions. In lattice packing the centers of the spheres
form a lattice in Rⁿ. Actually, because of their regularity lattice packings
are easier to study than non-lattice ones.

In three dimensions ,the Kissing number problem ask how many spheres
can be arranged so that they all just touch or "kiss" another sphere of the same size.
Note that for a lattice packing this is simply the number of balls touching any given one,
since every ball touches the same number of others.

Mathematicians define symmetry in terms of reflections, rotations, and
inversions that produce an object that is indistinguishable from the original.

There is a relation between symmetry and lattice packing of spheres, since
Lattices give rise to symmetry groups

SPRING 2002 and AUTUMN 2001

SPRING 2002

Time: Friday, May 31, 2002, 14.00-15.00.
Place: Room: MH 333, Centre for Mathematical Sciences, Lund
Speaker: Jonas Hartwig, Centre for Mathematical Sciences, Lund
Title: "Generalized Derivations on Algebras"
Abstract:
We consider linear operators on associative algebras
which satisfy a generalized version of the Leibniz rule for differentiating the product of
two functions. Examples of such operators include coloured (or graded) derivations,
in particular superderivations, and varius kinds of difference and shift operators.

Some necessary equations for generalized derivations on the quantum plane is stated.
We define a module structure on spaces of derivations, and
then state a theorem which determines the structure of some generalized
derivations on associative algebras which are unique factorization domains
as rings, in particular on polynomial algebras. Finally we generalize the Witt algebra
to a skew-symmetric algebra of generalized derivations on a commutative associative algebra.

Titel på svenska: "Generaliserade Deriveringar på Algebror"
Sammanfattning på svenska:
Vi studerar lineära operatorer på associativa algebror
som uppfyller en generaliserad version av
Leibniz regel för derivatan av produkten av två funktioner.
Exempel på sådana operatorer omfattar färgare (eller graderade)
deriveringar, speciellt superderiveringar,
och olika typer av differens- och skiftoperatorer.

Några nödvändiga ekvationer för generaliserade deriveringar
på kvantumplanet formuleras. Vi definierar en modul-struktur på rum av deriveringar,
och formulerar en sats som bestämmer strukturen på några
generaliserade deriveringar på associativa algebror med
unik faktorisering, speciellt på polynomalgebror.
Slutligen generaliserar vi Witt algebran till en skev-symmetrisk
algebra av generaliserade derinveringar på en kommutativ associativ algebra.

Time: Wednesday, May 15, 2002, 14.00-15.00.
Place: Room: MH 333, Centre for Mathematical Sciences, Lund
Speaker: Lars Hellström, Department of Mathematics, Umeå Universitet.
Title: "On semigroups for filtered structures."
Abstract:
In Mora's unified theory for Gröbner and standard bases using filtered
structures, the term order serves two purposes. On one hand it is (as
usual) discerning the head term of a polynomial, and on the other it
defines the topology of the algebra. A consequence of the latter is that
the elements in the neighbourhood basis of zero will be indexed by the
elements from a pretty arbitrary semigroup. In topologies that are defined
by a real-valued norm, elements in the neighbourhood basis of zero would
instead be indexed by positive real numbers. Since semigroups are more
general, one might expect that filtered structures can define topologies
that real-valued norms cannot, but in fact the opposite is true: every
topology that is induced by a filtered structure that fulfills all
conditions in Mora's theory can alternatively be defined using a norm that
merely takes rational numbers as values. The reason for this is that the
semigroup of any such filtered structure must have an order-preserving
homomorphism to the integers.

Titel på svenska: "Om semigrupper hos filtrerade strukturer"
Sammanfattning på svenska:
I Moras förenade teori för Gröbner/standard-baser med hjälp av
filtrerade strukturer så används termordningen för två olika saker:
dels för att som vanligt urskilja den ledande termen i ett polynom,
men även för att definiera den topologi som används. En följd av det senare är att
elementen i topologins omgivningsbas till punkten noll kan vara indexerade med
element ur en i stort sett godtycklig semigrupp; som en jämförelse kan
nämnas att en topologi som istället definieras av en reellvärd norm har
en omgivningsbas där elementen är indexerade med positiva reella tal. Man
skulle därför kunna förvänta sig att man utifrån filtrerade strukturer
kan definiera topologier som inte hör ihop med någon reellvärd norm, men
det visar sig faktiskt vara tvärt om; en filtrerad struktur som uppfyller
alla villkor i Moras teori inducerar en topologi som lika gärna kan
definieras av en rationellvärd norm. Orsaken till detta är att varje sådan
semigrupp har en ordningsbevarande homomorfi till heltalen.

Time: Tuesday, May 14, 2002, 15.15-16.15.
Place: Room: MH 333, Centre for Mathematical Sciences, Lund
Speaker: Samuel Rydh, Department of Mathematical Physics, KTH, Stockholm.
Title: "NCG field theory models and anomalies."
Abstract:
I will give a brief introduction to NCG field theory
models in the Hamiltonian framework with the focus on the
quantization of chiral fermions. I will discuss the cocyles
describing the chiral symmetry breaking and address the issue
of locality.

Titel på svenska:

Sammanfattning på svenska:
Jag kommer att ge en kort introduktion till
icke kommutativa kvantfältmodeller med fokus pa kvantiseringen
av chirala fermioner i den hamiltonska formuleringen. Jag
kommer att diskutera de cocykler som beskriver det chirala
symmetribrottet och ta upp fragan om lokalitet.

Time: Friday, April 12, 2002, 13.15-14.15
Place: Room: MH 333, Centre for Mathematical Sciences, Lund
Speaker: Soeren Eilers, Department of Mathematics,University of Copenhagen.
Title: "Invariants for substitutional subshifts."

Abstract:
Based on a substitution on a finite alphabet, such as
T(0)=01 T(1)=10
one may define a dynamical system, a substitution subshift,
where the points are doubly infinite
sequences such as

..... 10100110010110100101100110100110010110011010010110 .....

where each finite subword is a subword of Tⁿ(a) for suitable n and a,
and where the automorphism is the forward shift.

Such dynamical systems are of interest not only as tractable examples
in the theory of dynamical systems, but also as models for complex
behaviour in other branches of mathematics and applications. Work in
such diverse areas as formal languages, number theory, tiling geometry
and mathematical physics employs substitutions and their subshifts.

One basic problem is to determine whether or not two such systems are
equal is the sense of conjugacy. Attacking this problems
requires a rich and diverse supply of computable invariants.

In the talks, after a careful introduction to the subject, it will be
my goal to demonstrate how the seemingly unrelated non-commutative
discipline of
operator algebras becomes an important source of invariants
here. The focus in the first talk will be on seminal work by Durand,
Host and Skau. In the second, I hope to cover recent work by myself
and T. M. Carlsen.

Titel på svenska:

Sammanfattning på svenska:

Time: Wednesday, April 3, 2002, 15.15-16.30
Place: Room: MH 333, Centre for Mathematical Sciences, Lund
Speaker: Prof. Dmitrii Silvestrov
Department of Mathematics and Physics,
Mälardalen University
Title:Non-linearly Perturbed Non-negative Matrices
and Ergodic Type Theorems

Abstract:
The lecture presents results of asymptotical analysis for the powers P(t)ⁿ
of non-linearly perturbed non-negative matrices
P(t) = P₁ t + P₂ t² + ... + P_k t^k + o(t^k ).
The explicit algorithm is described for calculations of the corresponding
expansions for invariant and quasi-invariant projectors and other functionals.
The method is based on techniques of perturbed renewal equations.
It turns out that the balance between perturbation parameter t and n has a delicate
influence upon the form of the expansions. This balance can be of
the form t^r n --> const as t --> 0 and n --> infinity, where 0 < r < k+1.
Applications to analysis of ergodic and quasi-stationary phenomena for non-linearly perturbed
stochastic systems are described. Prospective generalisations for presented results are discussed.

Titel på svenska:

Sammanfattning på svenska:

Time: Friday, March 22, 13.15-15.00
Place: MH:333, Matematikcentrum
Speaker: Nadia S. Larsen, Department of Mathematics,
University of Copenhagen
Title:"Hecke algebras and their C^*-algebra completions"
Abstract:
A group G together with a subgroup G₀ form a
Hecke-pair if for each element g in G, the double coset
G₀g G₀ is a finite union of left cosets g_iG₀.
A trivial example is that when the subgroup G₀ is
normal, in which case every double coset is a single left coset. One
associates to a Hecke pair a *-algebra, and then a natural question is
whether a *-algebra completion may be constructed in a manner similar
to the completion of the group algebra to the group *-algebra.
Following Bost and Connes's construction of a Hecke *-algebra arising
in number theory, several authors have recently considered the
problem of constructing completions of more general Hecke algebras. The
aim of this talk is to introduce the necessary terminology and to review
some of these approaches.

Titel på svenska:

Sammanfattning på svenska:

Time: Thursday, March 7, 15.30-17.00
Place: MH:333, Matematikcentrum
Speaker: Jaak Peetre (Centre for Mathematical Sciences, Lund)
Title:"Loewner theory and interpolation functions"
Abstract:
Loewner theory is about matrix monotone functions.
It has many important applications, e.g. in mathematical physics
and in net work analysis (Duffin and his school).
In this talk I discuss Loewner theory in connection with
the theory of interpolation functions.
In fact, interpolation functions involve an additional
parameter p so in the latter theory may be viewed
as a kind of generalized Loewner theory. This is this
aspect which will be emphasized in the talk.
No previous knowledge of interpolation is supposed.

Titel på svenska: Operatormonotona funktioner och Löwners sats

Sammanfattning på svenska:

Time: Tuesday, Fabruary 19, 16.00-17.00
Place: MH:333, Matematikcentrum
Speaker: Nail Ibragimov (Centre for Mathematical Sciences, Lund)
Title: "Non-linear superposition principles and
the Vessiot-Guldberg-Lie algebras"

(Obs ! Nail Ibragimov will give also lecture on Dynamical Systems
Seminar starting at 17.15. Topics are related but not the same.)

Abstract:
The talk is aimed at discussing the theory of nonlinear
superposition principles for nonlinear differential equations on Lie
algebras. The problem of identifying all systems of first-order ordinary
differential equations whose general solution can be expressed
as a function of a finite number of particular solutions
(they are termed a fundamental system of solutions)
was considered by E. Vessiot, A. Guldberg and S. Lie in 1893,
and I call the underlying Lie algebra the Vessiot-Guldberg-Lie algebra.
In the first lecture, we will discuss definitions and the main theorem
due to Lie and illustrate it by several examples.
A new application will be considered in Lecture 2.
Different criterias for existence of weak solutions and three different
existence theorems will be discussed.

Titel på svenska:
Sammanfattning på svenska:

Time: Wednesday, February 13, 14.00-15.00 (OBS! time)
Place: MH:333, Matematikcentrum, Lund
Speaker: Daniel Bundzik (Malmö Högskola)

Title:"Quantum physics, Quantum mechanics and Applications"

Abstract:
A historical review of quantum physics
will be presented. The fundament
of quantum mechanics is discussed with Schrödinger equation as
reference. If time admits, modern quantum
mechanical applications are illustrated.

Titel på svenska: "Kvantfysik, Kvantmekanik och Tillämpningar"
Sammanfattning på svenska:

En historisk överblick av kvantfysikens frågeställningar och fenomen
presenteras. Kvantmekanikens fundament diskuteras med Schrödingers
ekvation som utgångpunkt. I mån av tid illustreras moderna
kvantmekaniska tilläpningar.

Time: Monday, January 21, 13.15-14.15
Place: MH: 332A, Centre for Mathematical Sciences
Speaker: Oskar Hagberg (Mathematical Statistics, Centre for Mathematical Sciences, Lund)
Title:"Hilbert space techniques for discrete time series"
Abstract: (This talk is my examination of the course Spectral Theory for Hilbert Spaces.)

The spectral representation theorems for unitary operators in Hilbert spaces will
be shown to have the spectral representation theorems of random processes as
special cases. For simplicity and brevity, I confine myself to the case of
discrete time, as described e.g. in chapter VI of Shiryaev's ``Probability''.
(The theory for the continuous case is analoguous.) I will make a short
summary of the needed theory of random processes, so no previous
knowledge of this subject will be necessary to understand the talk.

Titel på svenska:Hilbertrumsteknik for diskreta tidsserier

Sammanfattning på svenska:
(Detta föredrag är min examination av kursen Spektralteori för hilbertrum.)

Spektralrepresentationssatserna for unitära operatorer i hilbertrum kommer att visas
ha satserna om spektralrepresentation av stokastiska processer som
specialfall. För enkelhels skull kommer jag att begränsa mig till fallet med
diskret tid, som det beskriv i t.ex. kapitel VI i Shirjaevs "Probability". (Fallet
med kontinuerlig tid är analogt) Jag kommer helt kort att gå igenom den
nödvändiga teorin för stokastiska processer; därför behövs inga speciella
förkunskaper i sannolikhetsteori för att förstå det som sägs.

AUTUMN 2001

Time: Thursday, October 18, 16.00--17.00
Place: MH: 333, Centre for Mathematical Sciences
Speaker: Tomas Persson (Centre for Mathematical Sciences, Lund)
Title:" Operator Monotone functions and Löwner's theorem"
Abstract: The notion and some basic properties
of operator monotone functions will be discussed.
The famous Löwner theorem with two different proofs
by Gunnar Sparr and by A. Koranyi will be presented.

Titel på svenska: Operatormonotona funktioner och Löwners sats

Sammanfattning på svenska: Begrepp och några grundläggande egenskaper
av operatormonotona funktioner och Löwners sats
kommer att diskuteras. Den berömda satsen av Löwner med två olika bevis
av Gunnar Sparr och A. Koranyi kommer att presenteras.

Time: Thursday, October 25, 14.00-15.00
Place: MH: 309B, Centre for Mathematical Sciences
Speaker: Mattias Nilsson (Centre for Mathematical Sciences, Lund)
Title:"Non-commutative operators of wavelet analysis"
Abstract: Introduction to non-commutative operator methods
in wavelet analysis will be given.

It will be shown that Wavelet analysis is based on properties
of several important non-commuting operators.
The significance of these operators in Wavelet theory
originates in the fact that they satisfy some commutation relations
(Cuntz relations) and consequently generate operator representations
of the so called Cuntz C^*-algebras.

Some basic constructions describing this interplay will be presented.

Background on wavelets will be given.

The lecture will be of introductory nature, and no special knowledge will be needed.

Titel på svenska: Icke-kommutativa operatorer inom waveletanalys.

Sammanfattning på svenska:
En introduktion till icke-kommutativa operatormetoder
inom waveletanalysen kommer att ges.

Det visar sig att waveletanalysen grundas på flera viktiga egenskaper hos vissa icke-kommuterande operatorer. Betydelsen för dessa operatorer i waveletteorin
härstammar från det faktum att de uppfyller vissa kommuteringsrelationer
(Cuntz-relationer) som således genererar operatorrepresentationer
av så kallade Cuntz C*-algebror.

Grundläggande konstruktioner som beskriver dessa samband kommer att presenteras.
En bakgrund till waveletanalysen kommer även att ges.

Det här kommer att vara en introducerande föreläsning.
Inga särskilda förkunskaper krävs !!!

Time: Wednesday, October 31, 14.00-15.00
Place: MH: 333, Centre for Mathematical Sciences
Speaker: Anders Dahlner (Centre for Mathematical Sciences, Lund)
Title:"Commutative Banach algebras, The Gelfand Transform and Wiener's Lemma"
Abstract: We give a brief introduction to commutative Banach algebras and to
the Gelfand transform (a generalization of Fourier transform and other transforms).
Applications to certain problems in classical analysis will also be discussed.

Titel på svenska:"Kommutativa Banachalgebror, Gelfandtransformen och Wieners Lemma"

Sammanfattning på svenska:
Vi ger en kortfattad introduktion till kommutativa Banachalgebror
och till Gelfandtransformen (en generalisering av Fouriertransformen och andra transformer). Tillämpningar till vissa problem inom klassisk analys kommer också att diskuteras.

Time: Wednesday, November 14, 15.30-16.30
Place: MH: 333, Centre for Mathematical Sciences
Speaker: Serguei Shimorin (Centre for Mathematical Sciences, Lund)
Title: Completely bounded maps and similarity to a contraction
Abstract:
We shall discuss the concepts of completely bounded
and completely positive maps between operator algebras.
As an application of the theory of completely bounded maps,
we shall consider Paulsen's criterion:
an operator in a Hilbert space is similar to a contraction if and only if
it is completely polynomially bounded.

Titel på svenska: "Fullständigt begränsade avbildningar och likformighet med en kontraktion."

Sammanfattning på svenska:

Vi kommer att diskutera begreppen fullständigt begränsade
och fullständigt positiva avbildningar mellan operator algebror.
Som en tillämpning av teorin kommer vi att studera Paulsens kriterium:
En operator i ett Hilbert rum är likformig med en kontraktion om och
endast om den är fullständigt polynomiellt begränsad.

Time: Tuesday, December 4 at 14.00-15.00
Place: MH: 331, Centre for Mathematical Sciences
Speaker: Per Östborn (Mathematical Physics, Lund)
Title:"Introduction to quantum computing - an application
of the theory of linear operators in Hilbert space"
Abstract:
At this seminar you will learn what a quantum computer is,
how it works, and what makes it different from an ordinary, classical computer.
Properties of quantum mechanics make it possible for
the quantum computer to solve some problems more efficiently
than an ordinary computer. We will describe some of this properties.
We finish by describing a quantum algorithm which factorizes
large integers more efficiently than any known classical algorithm.

The seminar will stay at a basic level. No previous knowledge
of the subject is required.

Titel på svenska:"Inledning till kvantberäkningar -- en tillämpning
av teorin för operatorer på Hilbertrum."

Sammanfattning på svenska:

På detta seminarium får du lära dig vad en kvantdator är,
principerna för hur den fungerar, och vad som skiljer
den från en vanlig, klassisk dator.
Egenskaper hos kvantmekaniken gör att kvantdatorn kan lösa vissa problem
effektivare än en vanlig dator.
Vi kommer att beskriva vilka egenskaper det rör sig om.
Vi avslutar med att beskriva en kvantalgoritm som faktoriserar stora heltal
effektivare än alla kända klassiska algoritmer.

Seminariet kommer att hållas på en grundläggande nivå.
Inga förkunskaper i ämnet krävs.

Time: Tuesday, December 21 at 14.00-15.00
Place: MH: 332A, Centre for Mathematical Sciences
Speaker: Kristian Sandberg
(Dept. of Applied Mathematics, University of Colorado at Boulder, USA)

Title:"Representations of dynamical systems via commutation relations."

Abstract:

Operator algebras are of fundamental importance in quantum mechanics.
An observable is represented by a self-adjoint operator A and
according to the Heisenberg picture the time evolution F_t of a
physical system is related to a unitary transformation U such that
F_t(A)=U^*AU.

The mathematical framework for the dynamics of operators is based on
so-called C*-dynamical systems. The representations of such systems
are closely connected to commutations relations of operators. However,
such representations are often limited to cases where the underlying maps of the
dynamical system are homeomorphisms.

In this talk we will study the matrix equation AB=BF(A) where F is a
polynomial and see how the pair of matrices (A,B) contain significant
information about the dynamical system generated by the polynomial map
acting on the complex plane. Our study is not limited to self-adjoint
matrices A and we present solutions for maps that are not bijective. We
will also see how the solutions include information about the
stablity of the dynamics.

Titel på svenska:"Representationer of dynamiska system via kommuteringsrelationer."

Sammanfattning på svenska:

Operator algebror är av fundemental betydelse i kvantmekanik.
Ett observerbart objekt representeras av en sjalv-adjungerande operator A
och enligt Heisenbergs formulering är systemts tidsberoende F_t relaterat
till en unitar transformation U så att F_t(A)=U^*AU.

Matematiken bakom dynamik av operatorer kan studeras med C*-dynamiska
system. Representationerna av sådana system är nära relaterat till
kommuteringsrelationer av operatorer. Sådana representationer är dock
ofta begränsade till fall då den underliggande avbildningen för den dynamiska
systemet ar en homeomorfi.

I detta föredrag studerar vi matrisekvationen AB=BF(A) dar F är ett
polynom och ser hur matrisparet (A,B) innehåller viktig information of
det dynamiska systemet genererat av polynomet F i det komplexa
talplanet. Våra studier är inte begränsade till själv-adjungerade matriser A
och vi presenterar lösningar for avbildningar som inte är bijektiva.
Vi kommer också att se hur lösningarna innehåller information om det
dynamiska systemets stabilitet.

Principles of the NANG seminar

NANG seminar has a broad scope thanks to universal appearance and important role of non-commutativity.

We hope that the activities of the seminar will be useful for researchers, teachers and especially
for graduate and undergraduate students.

One of the main goals of NANG seminar is to highlight fundamental and unifying
role of non-commutativity in exploration of the borders (and bridges) between different parts of mathematics and its various applications in Physics, Chemistry, Engineering, Economics, Computer Science, Informatics, Biology, Medicine and other subjects.

Analysis, algebra, geometry, operator algebras and operator theory, algebraic geometry,
combinatorics and graph theory, stochastic processes, probability theory and statistics,
dynamical systems and ergodic theory, numerical analysis,
and other areas of Mathematics are connected in many ways via non-commutativity
to its applications outside mathematics.

Thus the NANG seminar is important member of seminars in Lund.

Please note the following principles which are followed by the NANG seminar:

1. Talks at the NANG seminar should have some direct or indirect relation to non-commutativity.
There is NO other restrictions on the subjects or directions of the talks. The seminar has equally high respect to all research directions and cherishes scientific freedom.

2. We are very grateful and meet with highest respect everybody
who wish to give a talk or contribute in financial or other ways to seminars activities.

3. The seminar encourages cooperation between all research groups
in The Centre for Mathematical Sciences,

in other departments and institutions in Lund
and in other places in Sweden and abroad.

This page is created and maintained by Sergei Silvestrov