Análise funcional geométrica e aç˜oes de grupos

Análise funcional geométrica e ações de grupos
Universidade Federal de Santa Catarina
Florianópolis, SC, Brazil
19–21 de Fevereiro 2014
Conferência é apoiada em parte pelo projeto “Pesquisador Visitante Especial” do programa
“Ciência sem Fronteiras” (Linha 1 - CAPES), processo 085/2012
i
Sumário
Horário das palestras . . . . . . . . .
Resumos das palestras . . . . . . . .
Abdelmoubine Amar Henni . .
Alcides Buss . . . . . . . . . . .
Arkady Leiderman . . . . . . .
Artur Hideyuki Tomita . . . .
Bernhard Burgstaller . . . . . .
Brice Mbombo . . . . . . . . .
Camila Fabre Sehnem . . . . .
Christina Brech . . . . . . . . .
Cleon S. Barroso . . . . . . . .
Dana Bartosova . . . . . . . . .
Daniel Pellegrino . . . . . . . .
Eduardo Scarparo . . . . . . .
Eliezer Batista . . . . . . . . .
Giuliano Boava . . . . . . . . .
Ivan Pontual Costa e Silva . . .
Johan Öinert . . . . . . . . . .
Maria Inez Cardoso Gonçalves
Martin Weilandt . . . . . . . .
Nilson C. Bernardes Jr. . . . .
Ruy Exel . . . . . . . . . . . .
Tristan Bice . . . . . . . . . . .
Valentin Burcea . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ii
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. iii
. 1
. 1
. 1
. 2
. 2
. 3
. 3
. 3
. 4
. 4
. 4
. 5
. 6
. 6
. 6
. 7
. 7
. 7
. 8
. 8
. 9
. 9
. 10
Horário das palestras
quarta
19 fev.
Nilson C. Bernardes Jr.
9:15-9:45
Generic Dynamics of
the Cantor space
Giuliano Boava
10:00-10:30
10:45-11:15
quinta
20 fev.
Cleon Barroso
On the approximate fixed
point property in abstract spaces
and applications
Alcides Buss
12:00-2:00
2:00-2:30
2:45-3:15
Eduardo Scarparo
Quasi-lattice ordered groups
and C ∗ -algebras
Abdelmoubine Amar Henni
Daniel Pellegrino
Camila Fabre Sehnem
Instanton bundles on
projective spaces
As desigualdades de
Bohnenblust-Hille
Uma classificação de
fibrados de Fell estáveis
Descrição das Álgebras de Cuntz-Li
como Produtos Cruzados Parciais
The universal property of
inverse semigroup equivariant
KK-theory
Tristan Bice
Annihilators in C ∗ -Algebras
Pausa para o almoço
Ivan Pontual Costa e Silva
Marginally outer trapped surfaces
and geodesic incompleteness of
spacetimes
Isometries of a generalized
numerical radius
Ruy Exel
Martin Weilandt
Christina Brech
Isospectral Alexandrov spaces
Universal Banach spaces
Open G-bases and compact
resolutions in topological groups
and locally convex spaces
Maria Inez Cardoso Gonçalves
K-theory of
Cuntz-Pimsner algebras
Arkady Leiderman
3:30-4:00
Artur Tomita
Countably compact groups
with convergent sequences
Weakly proper actions,
imprimitivity theorems and
duality
Bernhard Burgstaller
11:30-12:00
sexta
21 fev.
Dana Bartosova
Lelek fan and
projective Fraı̈ssé theory
Eliezer Batista
4:15-4:45
5:00-5:30
Partial actions and
partial representations of
Hopf algebras
Johan Öinert
Outerness of partial actions
Brice Mbombo Dempowo
Valentin Burcea
Amenability test spaces
for Polish groups
CR Singularities in
Codimension 2: Normal Forms
A conferência terá lugar na sala do LAED do Departamento de Matemática (a do Colóquio, no
andar térreo).
Organizadores:
Daniel Gonçalves (daniel.goncalves arroba ufsc ponto br),
Ivan Pontual Costa e Silva (pontual.ivan arroba ufsc ponto br),
Vladimir Pestov (vpest283 arroba uottawa ponto ca).
iii
Resumos das palestras
Instanton bundles on projective spaces
Abdelmoubine Amar Henni
MTM-UFSC
[email protected]
I’ll present an introduction to the study of instanton bundles using complexes called Monads
and which allow us to study families of instantons and in particular construct a ”moduli space”.
Weakly proper actions, imprimitivity theorems and duality
Alcides Buss
Universidade Federal de Santa Catarina
[email protected]
Coauthors: Siegfried Echterhoff
Abstract: In this talk I will give an introduction to the theory of weakly proper group
actions on C ∗ -algebras and show how this class of actions provides a unified approach to prove
(and in many cases improve and/or generalize) all known imprimitivity and duality theorems
involving groups (co)actions and their crossed products, like for instance Green’s Imprimitivity
Theorem, or more generally the Symmetric Imprimitivity Theorem, Mansfield’s Imprimitivity
Theorem, and the Quigg-Landstad Duality Theorem for group coactions. This is joint work
with Siegfried Echterhoff.
1
Open G-bases and compact resolutions in topological groups and locally convex
spaces
Arkady Leiderman
Ben-Gurion University of the Negev, Israel
[email protected]
Coauthors: S. Gabriyelyan (Beer Sheva), J. Kakol (Poznan)
A Hausdorff topological group is said to have G-base if it admits a base of neighbourhoods
of the unit {Uα }, where indices α belong to the partially ordered set NN , and the condition
Uα ⊂ Uβ , whenever β < α, holds.
Each metrizable group has a G-base. One of the main results is
Theorem 1. A topological group is metrizable iff it is Frechet-Urysohn and has a G-base.
Characterizations of metrizability of locally convex spaces, in particular the spaces of continuous
real functions with compact-open topology Cc (X), are given using G-bases.
Theorem 2. If X is a metrizable kω -space, then the free abelian topological group A(X) and
the free locally convex topological space L(X) have a G-base.
It will be explained that G-bases are related naturally to the compact resolutions swallowing
the compact sets Another link is established with sequential properties.
We show that any topological group with a G-base which is a k-space has the strong Pytkeev
property.
Theorem 3. For a Cech-complete X the space Cc (X) has the strong Pytkeev property if and
only if is Lindelof.
The strong Pytkeev property has been investigated for several well known classes of locally
convex spaces including (DF)-spaces and strict (LM)-spaces. We pose a dozen open questions.
Note that recently T. Banakh proved that the space Cc (X) has the strong Pytkeev property
for any separable metrizable X.
Countably compact groups with convergent sequences
Artur Hideyuki Tomita
Department of Mathematics - Universidade de São Paulo
[email protected]
Coauthors: Ana Carolina Boero (UFABC), Irene Castro Pereira (UFPA)
The construction of countably compact groups without non-trivial convergent sequences was
obtained using some set-theoretic assumption and it is still unknown if there is such an example
in ZFC.
Dikranjan asked whether every abelian group that admits a compatible countably compact
group topology admits also a countably compact group topology with a non-trivial convergent
sequence. For the torsion case it is not difficult to prove this is true. For the non-torsion case,
Boero, Garcia Ferreira and Tomita (Topology Appl., 2012) showed that free abelian group
of cardinality continuum admits such a group topology using selective ultrafilters. We will
discuss this example and sketch the more general result that answer Dikranjan’s question for
non-torsion abelian groups of cardinality at most continuum.
2
The universal property of inverse semigroup equivariant KK-theory
Bernhard Burgstaller
UFSC
[email protected]
After prework of Cuntz, Higson proved that every homotopy invariant, stable and splitexact functor from the category of separable C ∗ -algebras to abelian groups factors through
Kasparov’s KK-theory. Thomson generalized this result to the G-equivariant setting, where G
denotes a locally compact group. It seems to us that at a slight adaption of Thomson’s proof
gives Higson’s universality result also in the inverse semigroup equivariant setting.
Amenability test spaces for Polish groups
Brice Mbombo
Universidade de São Paulo
[email protected]
Coauthors: Yousef Al-Gadid and Vladimir Pestov
A topological group G is amenable if every continuous action of G on a compact space X
admits an invariant Borel probability measure.
A compact space K is an amenability test space for a class C of topological groups if a group
G ∈ C is amenable if and only if every continuous action of G on K admits an invariant Borel
probability measure.
De la Harpe and Giordano (C. R. Acad. Sci. Paris 324 (1997), 1255–1258), answering a question
of Grigorchuk, had proved that the Cantor space 2ℵ0 is an amenability test space for discrete
countable groups. Bogatyi and Fedorchuk (Topol. Methods Nonlinear Anal. 29 (2007), 383–
401) had obtained the same conclusion for the Hilbert cube [0, 1]ℵ0 .
In this talk, we will eventualy (If time permits) provide a brief historic introduction to the
notion of amenability and discuss the problem of the existence of test spaces for amenability
and extreme amenability for all polish groups.
Uma classificação de fibrados de Fell estáveis
Camila Fabre Sehnem
Universidade Federal de Santa Catarina
[email protected]
Coauthors: Ruy Exel
Dada uma ação parcial α de um grupo discreto G sobre uma C ∗ -álgebra A, um fibrado
de Fell é associado ao C ∗ -sistema dinámico parcial (A, G, α). No entanto, nem todo fibrado
de Fell sobre um grupo discreto pode ser obtido a partir de um C ∗ -sistema dinámico parcial.
Quando B é um fibrado de Fell estável sobre um grupo enumerável e a álgebra da fibra unidade
é separável, R. Exel (Proc. London Math. Soc. 74 (1997), 417-443) provou que existe uma
açõ parcial torcida do grupo base na álgebra da fibra unidade cujo fibrado de Fell associado
coincide com B. Nesta palestra, apresentaremos um progresso neste resultado, mostrando que
B pode ser obtido a partir de uma ação parcial não torcida do grupo base na álgebra da fibra
unidade.
3
Universal Banach spaces
Christina Brech
Universidade de São Paulo
[email protected]
Coauthors: Piotr Koszmider
The classical result of Banach and Mazur states that C[0, 1] is a universal separable Banach
space. We consider the question on the existence of a universal space for the class of Banach
spaces of some fixed larger density. The version which concerns isometric universality was
proved to be undecidable by results of Parovicenko and Shelah-Usvyatsov. Our main result is
to prove the consistency of the nonexistence of a universal Banach space of density continuum.
On the approximate fixed point property in abstract spaces and applications
Cleon S. Barroso
Universidade Federal do Ceará
[email protected]
Coauthors: O. F. K. Kalenda and P.-K. Lin
In this talk we will present a research line known as approximate fixed point property in
abstract spaces. With views to the structure of Banach spaces, some related problems and
contributions will be presented. We will also present some interesting applications related to
the fixed point theory, and also to the theory of ordinary differential equations in Banach spaces.
Lelek fan and projective Fraı̈ssé theory
Dana Bartosova
Universidade de São Paulo
[email protected]
Coauthors: Aleksandra Kwiatkowska
We construct the Lelek fan (the unique subfan of the Cantor fan with a dense set of endpoints) as a natural quotient of a projective Fraı̈ssé family of finite directed trees. This allows us
to show a homogeneity and projectivity property of the Lelek fan in the class of all smooth fans.
We further explore the group of homeomorphisms of the Lelek fan, for instance its simplicity,
dense conjugacy classes or connectivity properties. We also mention connections with the Dual
Ramsey theory.
The theory of projective Fraı̈ssé classes and their limit was originated by Irwin and Solecki
in their model-theoretic and combinatorial treatment of the pseudoarc.
4
As desigualdades de Bohnenblust–Hille
Daniel Pellegrino
Universidade Federal da Paraiba
[email protected]
Resumo. A desigualdade multilinear de Bohnenblust–Hille, demonstrada em 1931 no Annals
of Mathematics, garante que para cada inteiro positivo m existe uma constante Cm ≥ 1 tal que
 m+1

2m
N
X
2m
m+1
T (ei1 , ..., eim )


≤ Cm kT k
i1 ,...,im =1
N
para todos inteiros positivos N e todas formas m-lineares T definidas em `N
∞ × · · · × `∞ . A
desigualdade polinomial de Bohnenblust–Hille é um resultado similar, que controla a norma
dos coeficientes dos polinômios homogêneos. Embora tenham sido concebidas como ferramentas para o estudo de problemas relacionados a séries de Dirichlet, atualmente as desigualdades
de Bohnenblust–Hille têm aplicações em diferentes reas da matemtica e até mesmo em Teoria
da Informação Quântica. Curiosamente, em tais aplicações, o controle das constantes Cm tem
papel central. Apresentaremos resultados recentes que mostram que, em forte contraste com
as previsões dos últimos 80 anos, as constantes Cm têm um crescimento muito lento. Também
mostramos que o crescimento das constantes da desigualdade polinomial de Bohnenblust–Hille,
no caso de escalares complexos, é subexponencial. Aplicamos esse resultado para obter o comportamento assintótico exato do raio de Bohr do polidisco n-dimensional.
References
1. N. Albuquerque, F. Bayart, D. Pellegrino and J. B. Seoane-Sepúlveda, Sharp generalizations of the multilinear Bohnenblust–Hille inequality, J. Funct. Anal.,
DOI: 10.1016/j.jfa.2013.08.013 (to appear in print).
2. F. Bayart, D. Pellegrino and
p J. B. Seoane-Sepúlveda, The Bohr Radius of the n-dimensional
polydisk is equivalent to (log n)/n, arXiv:1310.2834v2 [math.FA], 15 Oct 2013.
3. D. Diniz, G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane-Sepúlveda, The asymptotic
growth of the constants in the Bohnenblust-Hille inequality is optimal. J. Funct. Anal.
263 (2012), no. 2, 415–428.
4. D. Nuñez-Alarcón, D. Pellegrino, and J.B. Seoane-Sepúlveda. On the Bohnenblust-Hille
inequality and a variant of Littlewood’s 4/3 inequality. J. Funct. Anal. 264 (2013), no.
1, 326–336.
5. D. Nuñez-Alarcón, D. Pellegrino, J.B. Seoane-Sepúlveda, D.M. Serrano-Rodrı́guez. There
exist multilinear Bohnenblust-Hille constants (Cn )∞
n=1 with limn→∞ (Cn+1 − Cn ) = 0. J.
Funct. Anal. 264 (2013), no. 2, 429–463.
6. D. Pellegrino, J.B. Seoane-Sepúlveda, New upper bounds for the constants in the BohnenblustHille inequality. J. Math. Anal. Appl. 386 (2012), 300–307.
5
Quasi-lattice ordered groups and C ∗ -algebras
Eduardo Scarparo
Universidade Federal de Santa Catarina
[email protected]
Quasi-lattice ordered groups were introduced by A. Nica, who associated C ∗ -algebras to
these structures and defined amenability in a natural way. We will present some examples and
results due to Nica, Laca and Raeburn about these C ∗ -algebras and also discuss a possible
generalization of quasi-lattice ordered groups.
Partial actions and partial representations of Hopf algebras
Eliezer Batista
UFSC
[email protected]
Partial actions is a subject of interest for operator algebras, dynamical systems and pure
algebra as well. The intent of this talk is to present a generalization of the concept of partial
actions and partial representations to the context of Hopf algebras. Partial representations of
Hopf algebras are directly related to Hopf algebroids and this important connection allows one
to describe the monoidal category of partial modules. We present some nontrivial examples, as
the category of partially G graded modules and partially G graded algebras.
Descrição das álgebras de Cuntz-Li como Produtos Cruzados Parciais
Giuliano Boava
UFSC, Florianópolis, Brasil.
[email protected]
Há poucos anos, Cuntz e Li introduziram a C ∗ -álgebra A[R] associada a um domı́nio de
integridade R com quocientes finitos por ideais principais. Nesta apresentação, mostramos que
A[R] é uma álgebra parcial de grupo com um conjunto apropriado de relações e que o espectro de
tais relações é homeomorfo ao completamento profinito do anel R. A partir desta representação,
exibimos uma caracterização de A[R] como produto cruzado parcial. Por fim, mostramos que
tais caracterizações podem ser usadas para recuperar resultados provados por Cuntz e Li como,
por exemplo, que A[R] é simples (concluindo que a ação do sistema é topologicamente livre e
minimal).
6
Marginally outer trapped surfaces and geodesic incompleteness of spacetimes
Ivan Pontual Costa e Silva
Universidade Federal de Santa Catarina
[email protected]
In the context of Lorentzian geometry and mathematical Relativity, there has been a great
surge of interest in (marginally) outer trapped surfaces, whose importance stems from the
fact that they are both quasi-local versions of black hole horizons and natural generalizations of
minimal surfaces in initial data sets. In this lecture we review recent work relating the existence
of such objects in a spacetime with causal geodesic incompleteness therein.
Outerness of partial actions
Johan Öinert
Lund University, Sweden
[email protected]
Coauthors: Patrik Nystedt (University West, Sweden)
In this talk, I will explain how one can extend the classicial notion of an outer action α of a
group G on a unital ring A, to the case when α is a partial action on ideals, all of which have
local units. Using a very general result from the theory of graded rings, we shall see that if α
is an outer partial action of an abelian group G, then its associated partial skew group ring
A ?α G is simple if and only if A is G-simple. Finally, I will explain how this result can be
applied in order to obtain generalizations of recent results by D. Gonçalves (UFSC) on partial
skew group algebras associated with certain types of partial dynamical systems.
Isometries of a generalized numerical radius
Maria Inez Cardoso Gonçalves
UFSC
[email protected]
Coauthors: A. R. Sourour, V. Pestov
For 0 < q < 1, the q-numerical radius is defined on the algebra Mn of all n × n complex
matrices by
rq (A) = max{|x∗ Ay| : x, y ∈ Cn , kxk = kyk = 1, hy, xi = q}.
This definition can easily be extended to infinite dimension.
During this talk I’ll characterize all the isometries of the q-numerical radius both on Mn an
on the space K(H) of compact operators on an infinite-dimensional Hilbert space H.
While the strategy of the proof is broadly similar to the finite dimension case, the infinite
dimensional case poses numerous difficulties and requires new techniques. The main challenge
was the usage of extreme points, which was the key component of the proofs in the finite
dimension case.
7
Isospectral Alexandrov spaces
Martin Weilandt
Universidade Federal de Santa Catarina
[email protected]
Coauthors: Alexander Engel (Universitt Augsburg)
The quotient of a compact Riemannian manifold by a compact group of isometries can be
seen as an Alexandrov space. Generalizing Schth’s version of the so-called torus method to this
setting, we construct the first non-trivial examples of compact non-isometric Alexandrov spaces
which are isospectral with respect to the Laplacian (but do not fall into the simpler category
of Riemannian orbifolds).
Generic Dynamics of the Cantor Space
Nilson C. Bernardes Jr.
Instituto de Matemtica - Universidade Federal do Rio de Janeiro
[email protected]
Coauthors: Udayan B. Darji
We develop unifying graph theoretic techniques to study the dynamics and the structure
of spaces H(C) and C(C), the space of homeomorphisms and the space of self-maps of the
Cantor space, respectively. Using our methods, we give characterizations which determine
when two homeomorphisms of the Cantor space are conjugate to each other. We also give a
new characterization of the comeager conjugacy class of the space H(C). The existence of this
class was established by Kechris and Rosendal and a specific element of this class was described
concretely by Akin, Glasner and Weiss. Our characterization readily implies many old and
new dynamical properties of elements of this class. For example, we show that no element of
this class has a Li-Yorke pair, implying the well known Glasner-Weiss result that there is a
comeager subset of H(C) each element of which has topological entropy zero. Our analogous
investigation in C(C) yields a surprising result: there is a comeager subset of C(C) such that
any two elements of this set are conjugate to each other by an element of H(C). Our description
of this class also yields many old and new results concerning dynamics of a comeager subset of
C(C). This is a joint work with Udayan B. Darji.
8
K-theory of Cuntz-Pimsner algebras
Ruy Exel
UFSC
[email protected]
Given a C ∗ -algebra A and a corespondence M over A, one may construct an associated
C -algebra, usually denoted by OM , called the Cuntz-Pimsner algebra of M . Choosing the
apropriate correspondence, one gets lots of interesting outcomes such as all Graph C ∗ -algebras,
all Kirchberg algebras, and in fact many C ∗ -algebras that have been studied in the literature
in the last 35 years.
In the original 1997 paper where Pimsner introduced this construction a method was described for the computation of the K-theory groups of OM . However the application of the method
in concrete examples presents some daunting difficulties, sometimes preventing a satisfactory
final answer.
In this talk I will briefly describe the construction of the Cuntz-Pimsner algebras and a
more pedestrian way to compute K-theory groups under certain (quite often verifiable) special
hypothesis. Time permiting I will show how these ideas may be used to study certain C ∗ algebras associated to integer dilation matrices.
This is joint work with Astrid an Huef and Iain Raeburn, from the University of Otago,
New Zealand.
∗
Annihilators in C ∗ -Algebras
Tristan Bice
UFSC
[email protected]
I outline my recent research regarding a quite basic C ∗ -algebraic structure which has, up
till now, received little attention from the operator algebra community. Specifically, I look
at C ∗ -algebra annihilators, the hereditary C ∗ -subalgebras coming from left/right annihilator
ideals. It turns out that these are, in many ways, the best C ∗ -algebra analogs of projections
in von Neumann algebras and, indeed, every annihilator in a von Neumann algebra A is of the
form pAp for some p in A. Unlike projections, however, annihilators still exist in abundance
and form a complete lattice in an arbitrary C ∗ -algebra. They also admit a simple notion of
equivalence which is completely consistent with the classical notion of Murray-von Neumann
equivalence. Moreover, we can use annihilators to decompose an arbitrary C ∗ -algebra in a way
that is again completely consistent with the classical von Neumann algebra type decomposition.
9
CR Singularities in Codimension 2: Normal Forms
Valentin Burcea
U.F.S.C.
[email protected]
I will survey some results about the real submanifolds in the complex space and the normal
forms constructions near a CR singularity. In particularly, I shall focus on the case when the
codimension of M is 2. Depending on the available time, I will present also some of my very
recent work in this direction.
10