SCIENTIFIC PROGRAMS AND ACTIVITIES

May 8, 2014
2.10pm FI210

Claire Shelly
Skein Theory for D^(3n) Planar Algebras

In this talk we will review a construction of the D^(3n) subfactors and give a presentation of their (A_2) subfactor planar algebra in terms of generators and relations.

May 1, 2014
2.10pm FI210

Hannes Thiel
The generator problem for C*-algebras

The generator problem asks to determine for a given C*-algebra the minimal number of generators, i.e., elements that are not contained in a proper C*-subalgebra. It is conjectured that every separable, simple C*-algebra is generated by a single element. The generator problem was originally asked for von Neumann algebras, and Kadison included it as Nr. 14 of his famous list of 20 "Problems on von Neumann algebras". The general problem is still open, most notably for the free group factors.

With Wilhelm Winter, we proved that every a unital, separable C*-algebra is generated by a single element if it tensorially absorbs the Jian-Su algebra. This generalized most previous results about the generator problem for C*-algebra.

In a different approach to the generator problem, we define a notion of `generator rank', in analogy to the real rank. Instead of asking if a certain C*-algebra A is generated by k elements, the generator rank records whether the generating k-tuples of A are dense. It turns out that this invariant has good permanence properties, for instance it passes to inductive limits. It follows that every AF-algebra is singly generated, and even more the set of generators is generic (a dense G_delta-set).

April 22, 2014
2.10pm FI210

Dave Penneys
Frobenius algebras in rigid C*-tensor categories

Frobenius algebras in unitary fusion categories give
subfactors by work of many people, including Longo-Rehren and Mueger,
which show this result for subfactors of type III factors. We will
give a straightforward proof for type II_1 factors.

April 17, 2014
2:10 pm FI210,

Eusebio Gardella
Classification of circle actions on Kirchberg algebras.

In this talk we will outline the classification of circle actions with the Rokhlin property on Kirchberg algebras in terms of their fixed point algebra together with the KK-class of its predual automorphism. We will also consider a continuous analog of the Rokhlin property, asking for a continuous path of unitaries instead of a sequence, and show that circle actions with the continuous Rokhlin property on Kirchberg algebras are classified by their fixed point algebra, and in the presence of the UCT, by their equivariant K- theory. We moreover characterize the K-theoretical invariants that arise from circle actions with the continuous Rokhlin property on Kirchberg algebras.

April 15, 2014
3.30pm in FI210,

I will present the result, obtained in joint work with Eusebio Gardella, that the relations of conjugacy and cocycle conjugacy of automorphisms of the Cuntz algebra O_2 are not Borel. I will focus on the motivations and implications of such result, and I will provide the main ideas of the proof. No previous knowledge of descriptive set theory will be assumed.

April 4
Time: 3:30 p.m.

Location: BA1160

Narutaka Ozawa
Noncommutative real algebraic geometry of Kazhdan's property (T)

I will start with a gentle introduction to the emerging (?) subject of "noncommutative real algebraic geometry," a subject which deals with equations and inequalities in noncommutative algebra over the reals, with the help of analytic tools such as representation theory and operator algebras. I will mention some results toward Connes's
Embedding Conjecture, and then present a surprisingly simple proof that a finitely generated group has Kazhdan's property (T) if and only if a certain equation in the group algebra is solvable. This suggests the possibility of proving property (T) for a given group by computers. arXiv:1312.5431

Rui Okayasu
Haagerup approximation property for arbitrary von Neumann algebras

We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property. This is based on a joint work with Reiji Tomatsu.

James Lutley
Finite dimensional approximations of product systems

Product systems have been the subject of recent study as a generalization of the Pimsner construction which contains the algebras of higher rank graphs as well as crossed products by certain partially ordered groups and a large class of reduced semigroup C*- algebras. We will discuss a particularly well-behaved class of such algebras with built in representations of a remarkable form. We then look at when these algebras are QD, when they are AF and when they are nuclear.

March 25

3.30pm in FI210

Sherry Gong
Finite Part of Operator $K$-Theory and Traces on Reduced $C^*$ Algebras for Groups with Rapid Decay

This talk is about the part of the operator $K$ theory of groups arising from torsion elements in the group. We will see how idempotents arise from torsion elements in a group, and discuss the part of $K$ theory they generate, and in particular, how to detect such idempotents using traces. We conclude with a condition for when such elements can be detected in the case of groups of rapid decay. We further analyse traces on the reduced $C^*$ algebras of hyperbolic groups and in doing so completely classify such traces.

Guihua Gong
Classification of AH algebras with ideal property, Elliott invariant and Stevens Invariant

In this talk, I will present the classification of AH algebra with ideal property with no dimension growth. The talk is based on three joint papers, two papers for reduction theorem which are joint with Jiang-Li-Pasnicu, and one for isomorphism theorem which is joint with Jiang-Li. Also I would like to discuss Kun Wang's work about the equivalence between Elliott invariant and Stevens invariant, which can be used to give two different descriptions of the invariants for the classification of our class.

March 18

Let G be a compact, simply connected Lie group and \A is a Dixmier-Douady bundle over G. All the sections of \A vanishing at infinity forms a G-C*-algebra A. The K-homology of A is defined to be the twisted K-homology. Freed-Hopkins-Teleman shows that twisted K- homology is isomorphic to the Verlinde ring R_{k}(G). In this talk, I will try to generalize their result to the crossed product case and prove that the K-homology of the crossed product of A is isomorphic sort of “formal Verlinde module”.

I will speak about the recent result of Sato, White, and Winter. Namely, Z-stability implies finite nuclear dimension for the class of simple, separable, unital, nuclear C*-algebras with a unique tracial state.

Is there a way of constructing separable, nuclear C*-algebras that radically differs from the classical constructions? I will present some preliminary results on this problem, subsuming some recent projects and work in progress with a number of logicians and operator algebraists.

Zhuang Niu
The classification of rationally tracially approximately point-line algebras

I’m going to briefly describe a classification result on the rationally tracially approximately point-line algebras. Then I’ll discuss the range of the invariant for this class of C*-algebras. This is based on a joint work with Guihua Gong and Huaxin Lin.

Shuhei Masumoto, University of Tokyo
A Definition of CCC for C*-Algebras

In this talk, I will define the countable chain condition (CCC) for C*-algebras. In case of von Neumann algebras this is equivalent to $\sigma$-finiteness of the center. Then I will investigate the relation between this condition and minimal tensor products by using a set theoretic principle, Martin's Axiom.

Qingyun Wang
Tracial Rokhlin property for actions of discrete amenable groups on C*-algebras

In this talk, I'll define a version of the (weak) tracial Rokhlin property for actions of discrete amenable groups acting on a unital simple separable C*-algebra. It is a generalization of the tracial Rokhlin property defined for actions of finite groups and the integer group. I'll then show that several known structural results about the crossed product could be generalized to our case. Then I will give some examples of amenable group actions on the Jiang-Su algebra \mathcal{Z} with the tracial Rokhlin property, and use it to show that actions with tracial Rokhlin property are generic for \mathcal{Z}-stable C*-algebras.

Sutanu Roy
Quantum group-twisted tensor product of C*-algebras

We put two C*-algebras together in a noncommutative tensor product using quantum group actions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively.
This is a joint work with Ralf Meyer and Stanisław Lech Woronowcz.

 

Please note the Mini-courses on Group Structure, Group Actions and Ergodic Theory will be on, so the next seminar will be on February 20, 2014

N. Christopher Phillips (University of Oregon)
A survey of $L^p$ operator algebras

In this talk, I will give a general survey of what is known about several classes of examples of operator algebras on $L^p$ spaces. I will also give some open questions (but there are many more than there is time for in the talk).

I will describe results on:
Spatial $L^p$ analogs of UHF algebras (simplicity and K-theoretic classification).
A more general class of $L^p$ analogs of UHF algebras, in which Banach algebra amenability is equivalent to being isomorphic to a Spatial $L^p$ UHF algebra.
Spatial $L^p$ analogs of Cuntz algebras (simplicity, pure infiniteness, uniqueness, and K-theory).
Reduced $L^p$ operator transformation group algebras for free minimal actions of discrete groups (simplicity and traces).
Reduced $L^p$ operator group algebras for discrete groups (simplicity for Powers groups [due to Pooya] and $L^p$ nuclearity for amenable
groups [due to An, Lee, and Ruan]).

Eusebio Gardella
Circle actions on \mathcal{O}_2-absorbing C*-algebras with the Rokhlin property

We de fine a Rokhlin property for circle actions on unital C*-algebras, and show that any circle action on a separable \mathcal{O}_2-absorbing C*-algebra can be norm-pointwise approximated by actions with the Rokhlin property. We also show that if A absorbs \mathcal{O}_2 and \alpha is a circle action on A with the Rokhlin property, then the restriction of to any closed subgroup also has the Rokhlin property. As an application, we classify circle actions with the Rokhlin property on separable nuclear \mathcal{O}_2- absorbing C-algebras up to conjugacy by an approximately inner automorphism of the algebra. We also provide examples of how most of these results fail if the algebra on which the circle acts is assumed to be \mathcal{O}_\infty-absorbing (or more speci cally, a Kirchberg algebra) instead of \mathcal{O}_2-absorbing. If time permits, we will explain how these results could potentially be used to classify certain not necessarily outer automorphisms of \mathcal{O}_2.

James Lutley
The Nuclear Dimension of UCT Kirchberg Algebras

It was recently shown by Enders that the nuclear dimension of any UCT Kirchberg algebra with torsion-free K_1 is one. This class exactly corresponds to those which occur as graph algebras. Here we construct a family of outstanding examples using higher rank graphs and describe a surprisingly general type of CPC approximation that approximates a unital inclusion of Toeplitz-type extension of said algebra into a somewhat larger enveloping algebra. We discuss how this range defect was corrected for in the O_n and O_inf cases and how it might be overcome in the more general setting.

Hannes Thiel
Recasting the Cuntz category

(joint work with Ramon Antoine and Francesc Perera)
The Cuntz semigroup W(A) of a C*-algebra A plays an important role in the structure theory of C*-algebras and the related Elliott classification program. It is defined analogously to the Murray-von Neumann semigroup V(A) by using equivalence classes of positive elements instead of projections.
Coward, Elliott and Ivanescu introduced the category Cu of (completed) Cuntz semigroups. They showed that the Cuntz semigroup of the stabilized C*-algebra is an object in Cu and that this assignment extends to a continuous functor.
We introduce a category W of (pre-completed) Cuntz semigroups such that the original definition of Cuntz semigroups defines a continuous functor from C*-algebras to W. There is a completion functor from W to Cu such that the functor Cu is naturally isomorphic to the completion of the functor W.
If time permits, we will apply this to construct tensor products in W and Cu.

Max Lein
Analysis of Pseudodifferential Operators by Combining Algebraic and Analytic Techniques

This talk will focus on a link between pseudodifferential theory and the theory of C*-algebras, so-called ${\psi}$*-algebras. Viewing pseudodifferential operators (${\psi}$DOs) as elements of ${\psi}$*-algebras, one sees that they are affiliated to twisted crossed product C*-algebras, and thus, algebraic tools can be used to investigate properties of ${\psi}$DOs. The talk concludes with an application, the decomposition of the essential spectrum of a ${\psi}$DO in terms of the spectra of a family of asymptotic ${\psi}$DOs. This makes the intuition that »the essential spectrum is determined by the operator's behavior at infinity« rigorous.

Alessandro Vignati
An amenable operator algebra that is not a C*-algebra

Recently Farah-Choi-Ozawa constructed a (nonseparable) amenable operator algebra that is not isomorphic to a C*-algebra, using a particular gap discovered by Luzin. After a brief introduction of the objects, we will explain how to generalize their construction, in order to construct an amenable operator algebra A such that every nonseparable amenable subalgebra of A is not isomorphic to a C*-algebra.

Dave Penneys
The operator-valued Fock space of a planar algebra

In joint work with Hartglass, we find the operator-valued Fock space associated to a planar algebra. We get natural analogs of the Toeplitz, Cuntz, and semicircular algebras, as well as a $C^*$-dynamics. These tools allow for the computation of the K-theory of these algebras. Certain (inductive limits of) compressions recover Cuntz-Krieger, Doplicher-Roberts, and Guionnet-Jones-Shlyakhtenko algebras.

Hongliang Yao (Nanjing University of Science and Technology)
Extensions of Stably Finite C*-algebras

I will show that for any C*-algebra A with an approximate unit consisting of projections, there is a smallest ideal I of A such that the quotient A/I is stably finite. I will give a necessary and sufficient condition for a given ideal to be equal to this ideal, in terms of K-theory. I will introduce an outline of the proof.
This talk will start at 3:30 p.m.

Dave Penneys
1-supertransitive subfactors with index at most 6.2

I will begin with a brief introduction to the subfactor classification program, which has two main focuses: restricting the list of possible principal graphs, and constructing examples when the graphs survive known obstructions. I will discuss recent joint work with Liu and Morrison which classifies 1-supertransitive subfactors without intermediates with index in $(3+\sqrt{5},6.2)$. We show there are exactly 3 examples corresponding to a BMW algebra and two "twisted" variations.

Yanli Song
An introduction to Baum-Connes conjecture for the case when G is a countable discrete group

I will outline how the Higson-Kasparov C*- algebra plays a role in the proof to that conjecture.

James Lutley
C*-algebras of Higher Rank Graphs

We will introduce the notion of a higher rank graph and discuss how one generates a C*-algebra from such an object. Whereas the Cuntz algebras can be though of as being generated by a free semi-group, graph algebras place restrictions on multiplication, yielding a free semi-groupoid construction. Higher rank graphs generalize this by allowing paths to admit distinct factorizations, thus introducing relations into the semi-groupoid. We will also introduce the infinite path representation these algebras carry. The infinite path space admits a natural locally compact Hausdorff topology with a natural inclusion into the k-torus. We will ask if this is might be used to compute properties of the algebra. Emphasis will be placed on examples of well-known algebras which occur as higher-rank graph algebras but not as conventional graph algebras.

Qingyun Wang
mathcal{Z}-stability of crossed product by actions with certain tracial Rokhlin type property

In this talk I will present a recent paper by Ilan Hirshberg and Joav Orovitz, where they defined a tracial notion of \mathcal{Z}-absorbing. They showed that tracially Z-absorbing coincide with \mathcal{Z}- absorbing in the simple nuclear case. With the help of this notion, they proved that, if $A$ is a simple nuclear \mathcal{Z}-absorbing C*-algebra, then the crossed products by actions (of finite group or integer group) satisfying certain tracial Rokhlin type property is again \mathcal{Z}-absorbing. I will then discuss some related questions regarding nuclear dimension and strict comparison.

James Lutley
Explicit Constructions of Kirchberg Algebras and their Applications

After discussing the obstructions to exhibiting an arbitrary Kirchberg algebra as a graph algebra, we will describe two explicit generalizations which yield additional examples. First, we will give an overview of a construction of Katsura using an integer action with a co-cycle to give all UCT Kirchberg algebras with countable K-groups. Katsura used this construction to prove a remarkable theorem on the lifting of group actions from an algebra's K-groups to the algebra itself. Then, for those who are interested, we can discuss k-graphs and their algebras, and describe how to compute the nuclear dimension of certain specific examples.

Ilijas Farah
Event InformationTitle: An amenable operator algebra not isomorphic to a C*-algebra

The algebra is a subalgebra of a finite von Neumann algebra, 2-subhomogeneous, and for any e>0 we have an example whose amenability constant is at most 1+e. The algebra is nonseparable and I will prove that the methods used in the proof cannot give a separable example. This is a joint work with Y. Choi and N. Ozawa.

Martino Lupini
The algebraic eigenvalues conjecture for sofic groups

If G is a group, then the integral group ring of G is the linear span over the integers of G inside its group von Neumann algebra. The algebraic eigenvalues conjecture asserts that any element of the integral group ring of G has only algebraic integers as eigenvalues. This conjecture is still open in general, but it has been verified by Andreas Thom when G is sofic. I will present a short proof of Thom's theorem in the framework of model theory for operator algebras. No previous knowledge of model theory will be assumed.

Dave Penneys
Free graph algebras and GJS C^*-algebras (part 2)

I will discuss ongoing joint work with Hartglass on the C^*- algebras arising from Guionnet-Jones-Shlyakhtenko's diagrammatic proof of Popa's reconstruction theorem for subfactor planar algebras. I will discuss the Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras associated to a planar algebra, and I will explain how we think they fit together with the GJS C^*-algebra of the planar algebra. We also think there should be a nice story with the graph loop algebras (due to many, including Evans, Izumi, Kawahigashi, Ocneanu, and Sunder)
arising from connections on principal graphs.

Dave Penneys
Free graph algebras and GJS C^*-algebras

I will discuss ongoing joint work with Hartglass on the C^*- algebras arising from Guionnet-Jones-Shlyakhtenko's diagrammatic proof of Popa's reconstruction theorem for subfactor planar algebras. I will discuss the Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras associated to a planar algebra, and I will explain how we think they fit together with the GJS C^*-algebra of the planar algebra. We also think there should be a nice story with the graph loop algebras (due to many, including Evans, Izumi, Kawahigashi, Ocneanu, and Sunder) arising from connections on principal graphs.

Nicola Watson
Discrete order zero maps and nuclearity of C^*-algebras with real rank zero

Order zero maps are an integral part of the recent advances made in the study of the structure of nuclear C*-algebras. Discrete order zero maps are a particularly nice special case, and are, in some sense, "dense" amongst those order zero maps with finite dimensional domain and real rank zero codomain. Consequently, they are of particular interest when studying both the nuclear dimension of C*-algebras with real rank zero, and more generally, when these algebras are nuclear. As a direct consequence of the structure of these maps we will prove a couple of results in these situations.

Claire Shelly
Planar Algebras and Type III Subfactors (Part 2)

I will begin by reviewing some basic ideas about planar algebras and type III subfactors. Using graph algebra techniques I will show how a type III subfactor can be used to define a planar algebra. Finally I will discuss a simple example, showing how planar algebras can be used to construct C^* algebras, type III factors and subfactors.

Claire Shelly
Planar Algebras and Type III Subfactors

I will begin by reviewing some basic ideas about planar algebras and type III subfactors. Using graph algebra techniques I will show how a type III subfactor can be used to define a planar algebra. Finally I will discuss a simple example, showing how planar algebras can be used to construct C^* algebras, type III factors and subfactors.

James Lutley
Toeplitz-Cuntz-Krieger algebras, Cuntz-Toeplitz algebras and permutations of words

We will describe a method originally due to Evans which relates the natural Fock space representations of these two classes of algebras, where the former is a projective cutdown of the latter. Whereas Winter and Zacharias used the Cuntz-Toeplitz algebras to compute the nuclear dimension of the Cuntz algebras, we will show progress towards computation of that of more general Cuntz-Krieger algebras.

James Lutley
Toeplitz-Cuntz-Krieger algebras, Cuntz-Toeplitz algebras and permutations of words

We will describe a method originally due to Evans which relates the natural Fock space representations of these two classes of algebras, where the former is a projective cutdown of the latter. Whereas Winter and Zacharias used the Cuntz-Toeplitz algebras to compute the nuclear dimension of the Cuntz algebras, we will show progress towards computation of that of more general Cuntz-Krieger algebras.

The quantization commutes with reduction problem for Hamiltonian actions of compact Lie groups was solved by Meinrenken in the mid-1990s, and solved again afterwards by many other people using different methods. In this talk, I will consider a generalization of [Q, R]=0 theorem when the manifold is noncompact. In this case, the main issue is that how to quantize a non-compact manifold. I will adopt some ideas from geometric K-homology introduced by Baum and Douglas in 1980s and examine this problem from a topological perspective. One of the applications is that it provides a geometric model for the Kasparov KK group KK(C*(G, X), C).

David Barmherzig after tea, a continuation of last week (report on operator algebra techniques in signal processing).

Grazia Viola
Tracially central sequences

A central sequence in a C*-algebra is a sequence that asympotically commute in norm with every element in the algebra. The reduced C*- algebra of the free group on two generators have an abundance of central sequences, while the group von Neumann algebra of the group on two generators have only trivial central sequences (where convergence is in L^2-norm). To solve this dichotomy we introduce a new notion of central sequences, the tracially central sequences. We show that if A is a simple, stably finite, unital, separable C*-algebra, which has strict comparison of positive elements and a unique tracial state, and if assume also some other condition, then the tracially central algebra of A coincide with the central algebra of the von Neumann algebra associated to the Gelfand-Naimark-Segal representation of A.

Martino Lupini
The automorphisms of a Jiang-Su stable C*-algebra are not classifiable up to conjugacy

After surveying various classification results for automorphisms of C*- algebras, I will explain how one can obtain negative results about classification using tools from descriptive set theory and, in particular, Hjorth's theory of turbulence. As an application I will show that the automorphisms of any Jiang-Su stable C*-algebra are not classifiable up to conjugacy using countable structures as invariants (joint work with David Kerr, Chris Phillips, and Wilhelm Winter).

Danny Hay
A classification result for recursive subhomogeneous algebras

Lin has shown that a C*-algebra is classifiable whenever it is tracially approximated by interval algebras (TAI) and satisfies the Universal Coefficients Theorem. In a recent paper of Strung & Winter, a class of recursive subhomogeneous algebras is introduced, and classified by showing its members are TAI. We will look at the main result and corollaries of this paper, and discuss some of the techniques developed—the excision of large interval algebras and finding tracially large intervals therein.
See http://arxiv.org/abs/1307.1342

David Barmherzig
Mathematical Signal Processing and Operator Algebras

The classical theory of signal processing was formalized during the last century by Nyquist, Shannon, etc. and studies how to process, transmit, and encode information signals. It draws heavily on techniques from Fourier theory, harmonic analysis, complex analysis, finite field theory, and differential equations. As well, more modern techniques have recently been introduced such as wavelets, frame theory, compressive sensing, and spectral graph theory methods. In recent years, many applications of operator algebras to signal
processing have also been developed. This talk will give an introduction and overview of these topics.

James Lutley
Constructing Kirchberg Algebras from Cuntz-Toeplitz Algebras

We will review a construction of Evans which allows us to construct Cuntz-Krieger algebras from Cuntz-Toeplitz algebras. This construction allows us to compute the nuclear dimension of those algebras as an extension of the method Winter and Zacharias used to compute that of the Cuntz algebras. Subsequently we will discuss multiple methods of
constructing algebras from infinite graphs as limits of algebras from finite graphs such as the Cuntz-Krieger algebras. Finally, we will introduce a construction of Katsura which was recently recontextualized by Exel and Pardo that can produce any UCT Kirchberg algebra through the implementation of an integer action on a graph algebra.

Dave Penneys
Part 3: A new obstruction (July 11th)

In this talk, we will use Liu's relation to derive a strong triple point obstruction. We will then recover all known obstructions discussed in Part 1 of the talk. As an example, we will determine the chirality of all subfactors of index at most 4, and we will show that D_{odd} and E_7 are not principal graphs of subfactors.

Dave Penneys
Part 2: Wenzl's relation (July 9th)

In this talk, we will focus on skein theory in a planar algebra. The main goal of this talk will be to discuss two strong quadratic relations in a subfactor planar algebra. The first is Wenzl's relation, which is the recursive formula for obtaining the Jones-Wenzl idempotents in the Temperley-Lieb planar algebra. We will talk about a variation of this relation which holds in a general planar algebra. We will then derive what I call Liu's relation, a clever variant of Wenzl's relation due to Zhengwei Liu.

Nicola Watson
Discrete order zero maps

Order zero maps are an integral part of the recent advances made in the study of the structure of nuclear C*-algebras. Discrete order zero maps are a particularly nice special case, and this talk will focus on just how nice they are. Discrete order zero maps are, in some sense, "dense" amongst those order zero maps with finite dimensional domain and real rank zero codomain, and so they are of particular interest when studying the nuclear dimension of C*-algebras with real rank zero. As a direct consequence of the "niceness" of these maps we will prove a couple of results in this situation.

Dave Penneys
Triple point obstructions, a 3 part talk

Overall abstract:
There has been recent success in the classification program of subfactors of small index. The classification program has two main objectives: restricting the list of possible principal graphs, and constructing examples of subfactors for the remaining graphs. The former task relies on principal graph obstructions, which rule out many possibilities by either combinatorial constraints, or by some rigidity phenomenon which relates the local structure of the principal graph to constants in the standard invariant of the subfactor. A triple point obstruction is an obstruction for possible principal graphs with an initial triple point. I will talk about a new triple point obstruction which is strictly stronger than all known triple point obstructions.

Part 1: Triple point obstructions (July 4th)

After a brief review of the definition of the principal graphs of a subfactor, we will discuss the former state of the art of triple point obstructions, including Ocneanu's triple point obstruction, Jones' quadratic tangles obstruction, the triple-single obstruction of Morrison-Penneys-Peters-Snyder (probably known to Haagerup), and Snyders singly valent obstruction.

We study solvability of certain linear nonhomogeneous elliptic problems and show that under reasonable technical conditions the convergence in L^2(R^d) of their right sides implies the existence and the convergence in H^2(R^d) of the solutions. The equations involve second order differential operators without Fredholm property and we use the methods of spectral and scattering theory for Schroedinger type operators.