SCIENTIFIC PROGRAMS AND ACTIVITIES

December 10, 2013

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
 Archive of talks 2011-2012 Archive of talks 2012-2013 Map to Fields
 Upcoming Seminars: every Tuesday and Thursday at 2 pm Room 210 Ongoing Seminars Past Seminars Nov. 28 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. Nov. 26 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. Nov. 21 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. Nov. 19 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. Nov. 14 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. Nov. 12 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. Nov. 7 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. Nov. 5 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. Oct. 29 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. Oct. 31 Luis Santiago Oct. 22 Luis Santiago Oct. 24 Yanli Song Oct. 17 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. Oct. 15 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. Oct. 8 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. Tues. Oct. 3 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. Tues. Oct. 1 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. Thurs. Sept. 26 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. Tues. Sept. 24 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. Thurs, Sept 12 Qingyun Wang On the Tracial Rokhlin Property (continuing) Tues, Sept 10 Yanli Song, David Barmherzig, and Qingyun Wang Geometric K-Homology and [Q, R]=0 problem 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). Thur Sept 5 Luis Santiago Tues, Sept 3 Qingyun Wang and David Barmherzig Thurs Aug15 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. Thurs Aug 8 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). Tues Aug 6 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 Thurs August 1 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. Tues Jul 30 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. Thurs Jul 11 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. Tues Jul 9 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. Mon Jul 8, 4.10 pm in BA6183 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. Thurs Jul 4 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. Tues July 2 Vitali Vougalter (University of Cape Town) Solvability in the sense of sequences for some non Fredholm operators 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.