User jon bannon - MathOverflow

Answer by Jon Bannon for What does a mathematician expect from mathematics education?

2013-04-20T01:33:48Z

I recall working with a reasonably reputable mathematics educator once, teaching a calculus class. At one point it became evident that the guy wasn't comfortable summing a geometric series. One thing I expect from mathematics education is that mathematics educators know some mathematics. Saying this any less bluntly would lose too much of the sentiment I'm trying to convey, so there it is.

As a mathematician, I'd personally like to see work in mathematics education that helps me teach what it is that I actually do as a mathematician. I was introduced by Ken Appel to some of the ideas of Hy Bass on mathematics education, e.g. the "granularity" concept which asserts that at different levels of sophistication mathematicians allow different jump sizes in their arguments. Awareness of granularity made explicit like this really has changed the way I organize my undergraduate course material and for me was revolutionary.

Other ideas of Bass that I'd like to see followed up include the idea of a common structure problem set. Such an idea might help to get a large chunk of a difficult aspect of the mathematical aesthetic into the curriculum.

In general, I'd like to see mathematics education address how the practices of the best mathematicians can be brought to the graduate and undergraduate population in universities, and how we can bring more of mathematics itself as experienced by mathematicians to our students. I'm more interested in this than this than studies of how to improve calculus course assessment, for example. I've always been frustrated that nobody seems to study the learning approaches of successful mathematicians rather than average students. I'd personally like to see more of our best practices being studied and propagated. This last paragraph is a bit ignorant, I know, but it's my honest impression.

Is there a deep reason for the fecundity of involutions?

2013-04-04T19:43:25Z

You might have come across the book of involutions in your travels. A colleague of mine asked whether there is a natural global reason (versus ad-hoc trickery) for considering involutions in mathematics. The above book provides many situations that suggest such a global perspective. Having witnessed the extraordinary power of certain involutions in operator algebras (e.g. in Tomita-Takesaki theory), I'd be interested in hearing about such a global perspective in summary from an expert. I'm aware that this question as I've asked it risks being trite...perhaps warranting the answer "it's the simplest nontrivial symmetry" but the existence of the above book might suggest otherwise:

Question: What are some "global" reasons for considering involutions in mathematics?

Answer by Jon Bannon for Learning through guided discovery

2013-01-26T18:51:12Z

This guided discovery approach goes by other names, as well. One such name is "Inquiry Based Learning" or IBL. A list of guided discovery problems is often referred to as an "IBL script". Many such scripts are available from the Journal of Inquiry Based Learning in Mathematics (JIBLM): http://www.jiblm.org/

The Riemann Hypothesis and the Langlands program

2013-01-19T13:27:46Z

On page 263 of this book review appears the following:

Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic number theory from the standpoint of L-functions and their analytic properties), but in fact the properties of L-functions traditionally of interest to analytic number theorists - for example, the location of zeroes in the critical strip (the Generalized Riemann Hypothesis) - have historically had little to do with the preoccupations of the Langlands program. Thanks largely to the efforts of a few charismatic and determined individuals, this is beginning to change and Langlands himself has in recent years turned to methods in analytic number theory in an attempt to get beyond the visible limits of the techniques developed over the last few decades.

I'd like to ask for a big picture exposition of how such questions about the location of zeroes of L-functions appear and interact with the Langlands program. My interest is mainly cultural and the answer should be tailored for the outsider to number theory (I'm viewing Langlands program algebraically as the pursuit of a nonabelian class field theory.)

A more crude question is:

Does the Langlands program say anything about the Grand Riemann Hypothesis or vice versa?

This is almost certainly too crude a question for MO, but Langlands seems to have such an amazing unifying appeal, that I feel a temptation to see how much it subsumes. I fully expect an answer like "It is impossible to coherently discuss this without years of training". Thank you for any attempt to explain things to someone who is not a number theorist, in advance!

Excellent mathematical explanations

2012-12-19T01:57:35Z

In the Stanford Encyclopedia of Philosophy there is an entry on mathematical explanation.

The basic philosophical question is: What makes a proof explanatory?

Two main "models" of mathematical explanation are mentioned:

1) Steiner's model, which asserts that an explanatory proof can be distinguished from a non-explanatory proof if only the explanatory proof contains a so-called "characterizing property", which is (roughly) a property unique to a particular entity or structure in a family of structures. Here `family' is taken as primitive. (For a more complete description, follow the above link.)

2) Kitcher's model, which asserts that a proof is explanatory if it provides a unification of disparate methods. (Again, follow the link for a more complete discussion.)

Philosophers of mathematics find neither of these models adequate. The above philosophical accounts of mathematical explanation proceed from the top down, but in each case it has been possible to find good examples of mathematical explanations which don't fit. Quoting the linked entry above:

Recent work has shown that it may be more fruitful to proceed bottom-up, by first providing a good sample of case studies before proposing a single encompassing model of mathematical explanation.

I confess, I've been unable lately to figure out how to ask a good `philosophy of mathematical practice' question here on MO and so this is my attempt to do so. I think we can provide the philosophers (and each other) with a storehouse of proofs that are also excellent explanations, and reason why we think so. If this works, I may send the link to Paolo Mancosu as a gesture of good will toward the philosophers who are studying contemporary mathematical practice.

Please volunteer an example of an excellent explanatory proof, and the reason you think the proof also provides a good explanation of the phenomenon it deals with.

Since the philosophers are looking for case studies, I think that a link to the proof in question will be enough (if the proof is not short). I'm sure that you can be contacted later to explain.

(I'm still not sure if philosophical language is appropriate for MO, but the above question has clear value and will admit precise mathematical answers!)

Is there an i.c.c. nonamenable simple group that is inner amenable?

2010-06-06T12:26:12Z

A finitely presented, countable discrete group $G$ is amenable if there is a finitely additive measure $m$ on the subsets of $G \backslash${$e$} with total mass 1 and satisfying $m(gX)=mX$ for all $X\subseteq G \backslash${$e$} and all $g \in G.$

A countable discrete group $G$ is inner amenable if there is a finitely additive measure $m$ on the subsets of $G \backslash${$e$} with total mass 1 and satisfying $m(gXg^{-1})=mX$ for all $X\subseteq G \backslash${$e$} and all $g \in G.$

The growth $b:\mathbb{N} \rightarrow \mathbb{N}$ of $G$ (with respect to a given word length metric on $G$) is defined as the number of elements $b(n)$ in $G$ lying inside the ball of radius $n$ around $e$.

It is possible to detect the amenability of $G$ in terms of the growth of G (c.f. R. I. Grigorchuk, “Symmetric random walks on discrete groups”, UMN, 32:6(198) (1977), 217–218).

Can the growth of G detect inner amenability?

I'd like to know if there is an i.c.c. discrete nonamenable simple group that is inner amenable?

On a related note, what about an answer to Owen's question below?

What is the physical difference between states and unital completely positive maps?

2012-04-07T13:59:06Z

Mathematically, completely positive maps on C*-algebras generalize positive linear functionals in that every positive linear functional on a C*-algebra $A$ is a completely positive map of $A$ into $\mathbb{C}$. Furthermore, we have the Stinespring construction as a powerful generalization of the GNS construction.

Certainly, the relationship between completely positive maps and positive linear functionals can only go so far. I am curious about what physics has to say about this analogy/generalization. It seems that completely positive maps should serve as generalized states of a quantum system, but I've mostly seen cp maps arise in the discussion of quantum channels and quantum operations. I'd like to know precisely in what sense a completely positive map can be viewed as a generalized physical state.

Question: What is a completely positive map, physically? Particularly, in what precise sense can a completely positive map be regarded as a generalized (physical) state?

If there are nice survey papers discussing the above relationship, such a reference may serve as an answer to my question.

Is the von Neumann algebra associated to a unitary representation of an amenable group always injective?

2012-11-30T16:37:37Z

I should be tarred and feathered for not knowing at least the status of the following question.

Question: Let $\Gamma$ be a discrete amenable group. If $\pi:\Gamma \rightarrow B(\mathcal{H})$ is a unitary representation of $\Gamma$ on a separable Hilbert space $\mathcal{H}$, is the von Neumann algebra $\pi(\Gamma)''$ necessarily injective?

Flippantly one imagines that the answer to this question is yes, by Theorem 2.2 of Bekka's paper on amenable representations. But this result only says that the images of group elements are in the centralizer of a non-normal state...it isn't immediately clear why the entire von Neumann algebra should lie in the centralizer of such a state. If one tries to sidestep this by looking at a proof using almost invariant vectors, one is busted by the fact that a representation that is "$H$-amenable" isn't necessarily amenable in Bekka's sense.

EDIT: Makoto's nice answer below provided me with some closure. I'm still worried that I can't see a more or less direct way to this result from Connes's '76 paper on the classification of injective factors. If this paper can, in a more or less direct and self-contained way, be used to resolve the question, please feel free to include another answer.

Proof synopsis collection

2010-10-27T22:15:08Z

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself!

Definition (Fraleigh): A proof synopsis is a one or two sentence synopsis of a proof, explaining the idea of the proof without all the details and computations.

My question is this:

What is your favorite proof synopsis of a theorem we all should know?

(I'm sorry, I'll do my time in big-list hell...)

Clarifying the link between deformation/rigidity and dual cocycles

2012-11-28T23:37:07Z

Suppose that a type $II_{1}$ factor $M$ decomposes in two ways as a group von Neumann algebra, e.g. as $L\Gamma$ and as $L\Lambda$. The decomposition $L\Gamma$ gives rise to a comultiplication $$\Delta_{\Gamma}:M\rightarrow M\overline{\otimes}M$$ that sends a canonical unitary $u_g$ to $u_{g}\otimes u_{g}$. There is an analogous comultiplication $\Delta_{\Lambda}$. On page 8 of Stefaan Vaes's notes, it is shown that if $\Omega$ is a unitary element in $M\overline{\otimes}M$ such that $$\Delta_{\Gamma}(x)=\Omega \Delta_{\Lambda}(x)\Omega^{*}$$ for all $x\in M$, then a certain set of equations hold which, if the group $\Gamma$ were instead abelian, would yield a symmetric 2-cocycle on the dual compact abelian group. Such cocycles cobound, and this fact leads Ioana, Popa and Vaes to prove an analogous fact in the nonabelian setting.

To obtain the set of equations in question, one writes down an element $$Z=(\Delta_{\Gamma}\otimes id)(\Omega)(\Omega \otimes 1)(1 \otimes \Omega^{*})(id \otimes \Delta_{\Gamma})(\Omega^{*})$$ which is unitary because of coassociativity properties of the two comultiplications. The set of equations referred to in the previous paragraph follow because $M$ is a factor.

Question: Is the consideration of this unitary $Z$ completely ad-hoc, or is there some deeper reason for considering an element of this form that is perhaps related to the fundamental unitary that encodes duality in the theory of locally compact quantum groups?

I think the idea of developing the link between the quantum group picture and $II_{1}$ factors is intriguing, and so posted this question instead of sending S. Vaes an e-mail, in hope that an expert would provide an answer for the OA community.

Is there an upper bound on the dimension for irreducible representations of a continuous trace $C^{*} $-algebra?

2012-10-23T18:37:04Z

The following are questions of Don Hadwin:

If $A$ is a unital continuous trace C*-algebra, is there an upper bound on the dimension of all the irreducible representations?

It is known that all irreducible representations are finite-dimensional?

Spectrum of the sum of generators for irrational rotation algebra

2010-08-17T14:28:16Z

Let $\theta \in \mathbb{R}\backslash\mathbb{Q}$. The irrational rotation C*-algebra $\mathcal{A}_{\theta}$ is the universal C*-algebra generated by unitary elements $u$ and $v$ with $vu=e^{2\pi i \theta}uv$.

What is the spectrum of u+v?

(Note: in the case where the unitary elements u and v are the standard generators of a free group factor, the spectrum of u+v is computable by work of Haagerup and Larsen.)

Answer by Jon Bannon for Spectrum of the sum of generators for irrational rotation algebra

2012-10-26T12:02:18Z

The question is nicely resolved here: http://arxiv.org/abs/1210.4771

Collection of equivalent forms of Riemann Hypothesis

2010-09-25T12:29:06Z

This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include statements that imply RH or its negation.

Here is what I am suggesting we do:

Construct a more or less complete list of sufficiently diverse known reformulations of the Riemann Hypothesis and of statements that would resolve the Riemann Hypothesis.

Since it is in bad taste to directly attack RH, let me provide some rationale for suggesting this:

1) The resolution of RH is most likely to require a new point of view or a powerful new approach. It would serve us to collect existing attempts/perspectives in a single place in order to reveal new perspectives.

2) Perhaps the resolution of RH will need ideas from many areas of mathematics. One hopes that the solution of this problem will exemplify the unity of mathematics, and so it is of interest to see very diverse statements of RH in one place. Even in the event where no solution is near after this effort, the resulting compilation would itself help illustrate the depth of RH.

3) It would take very little effort for an expert in a given area to post a favorite known reformulation of RH whose statement is in the language of his area. Therefore, with very little effort, we could have access to many different points of view. This would be a case of many hands making light work. (OK, I guess not such light work!)

Anyhow, in case this indeed turns out to be an appropriate forum for such a collection, you should try to include proper references for any reformulation you include.

Overview of the interplay of Harmonic Analysis and Number Theory

2011-03-31T21:32:37Z

I'm kind of disappointed that the question here was never sharpened.

The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ is the fundamental domain of, say, a congruence subgroup $\Gamma$ of $Sl_{2}(\mathbb{Z})$. Eigenfunctions of the discrete spectrum of $\Delta$ are real analytic solutions to $\Delta (\Psi)=\lambda \Psi$ that are $\Gamma$-equivariant functions in $L^{2}(D, dz)$, where $dz$ is the Poincare measure on the upper half-plane. These eigenfunctions evidently carry quite a bit of number theoretic information. Frankly, this point of view on number theory sounds incredibly interesting...

Question: Would someone please suggest a readable introductory account that tells this story?

(I imagine that answers will include the words Harish-Chandra, Langlands, etc...)

Also, if experts are inclined to write a short overview as an answer, that would also be much appreciated.

Answer by Jon Bannon for Assumptions on a commutative C*-algebra to get a nice C(X) - space

2012-08-29T22:59:38Z

The space $X$ is second countable if and only if $C(X)$ is separable for the norm. This is proved, for example, as Theorem 2.4 of the little article "Notes on the Separability of C* algebras" by Chun-Yen Chou. Actually, the short proof given there works also for locally compact Hausdorff spaces and therefore non-unital $C^{*}$-algebras.

Is there a Dedekind-Frobenius group determinant for infinite groups?

2012-08-27T13:00:53Z

If $G$ is a finite group and $\lbrace x_{g} \rbrace_{g\in G}$ are commuting formal variables, then one can form a matrix whose $(g,h)$ entry is $x_{gh^{-1}}$. The determinant of this matrix is a polynomial with integer coefficients and is called the group determinant. Considering the factorization of this determinant led Frobenius to discover seminal results in the representation theory of finite groups. Later on, it was shown that a group can be recovered from its group determinant.

Many people have asked me the following question over the years, and I haven't a good answer for it. One might think about looking at something like a generalized determinant over a polynomial ring in infinitely many variables indexed by the group.

Question: In the literature, does there exist a more or less direct attempt to generalize the Dedekind-Frobenius group determinant to the setting of infinite groups?

Moreover, if such a thing exists, can it determine the group from which it is constructed?

Answer by Jon Bannon for Universities not listed on mathjobs

2012-08-17T13:02:19Z

A big reason not to advertise on MathJobs is that one can apply for jobs simply by clicking on a job. This produces an enormous pool of applications to swim through. It would be nice if there were some measures in place on MathJobs that would prevent such "capricious applying". (That said, for a conscientious search committee chair the large list provides many opportunities for hiring that otherwise may never have happened. I'm not suggesting making applying too much harder, only that some speedbump be put in place.)

This said, it is absolutely a good idea to send applications to all the schools you can find (respectfully inquiring about the availability of a position, and asking to be kept on file for consideration should a job become available). Often small schools are looking for certain stability criteria in an applicant, like a geographic constraint, that help promise that a "good catch" would stay even when presented with future opportunities to leave.

The prudent thing for a department to do in this market is to hold a national search for any permanent position, but if you have independently contacted the department in advance you will have a better chance of not being missed in the electronic pile 1000 MathJobs CVs.

When is a $\ast$-algebra a $C^{\ast}$-algebra?

2012-08-16T14:31:44Z

The purpose of this question is to collect sufficient conditions on a unital $\ast$-subalgebra $\mathcal{A}$ of the algebra of bounded linear operators $B(\mathcal{H})$ on a separable Hilbert space $\mathcal{H}$ that guarantee that $\mathcal{A}$ is actually a $C^{*}$ algebra (is closed in the operator norm). Please provide links and references. At least, I'd like a reference or proof for the following:

"Thm:" If $\mathcal{A}$ is a unital $\ast$-subalgebra of $B(\mathcal{H})$ and whenever $A\in\mathcal{A}$ is self-adjoint it follows that $A_{+}$ and $A_{-}$ both lie in $\mathcal{A}$, then $\mathcal{A}$ is norm-closed.

(Here, $A_{+}$ and $A_{-}$ live naturally in the $C^{*}$-algebra generated by $A$ and $I$, isomorphic to $C(\sigma(A)))$, where $A$ corresponds to the function $f(x)=x$, $A_{+}$ corresponds to $max[f,0]$ and $A_{-}$ to $min[f,0]$.)

(Edit: Nik has pointed out that the "Thm" is false. The broader question stands: Is there any other interesting abstract characterization of a C*-algebra that doesn't obviously say the algebra is norm-closed?)

Strong monotone limits and dense subalgebras of von Neumann algebras, again

2012-07-27T13:19:20Z

Edit: I just realized that this question is related to Andreas Thom's very interesting question here. I think the question below is more crude...

Michael's question here reminded me of the first lemma of this paper of Kadison, establishing that those $C^{*}$-algebras $\mathcal{A}\subset B(\mathcal{H})$ for which the set of self-adjoint elements $\mathcal{A_{s.a.}}$ is strong-operator closed under taking monotone increasing limits are, in fact, von Neumann algebras. I've often wondered about the following

Question: What are necessary and sufficient conditions on a $*$-subalgebra $\mathcal{A}\subset B(\mathcal{H})$ such that strong operator closure of $\mathcal{A_{s.a.}}$ under monotone increasing nets guarantees that $\mathcal{A}$ is closed in the strong operator topology (i.e. is actually a von Neumann algebra)?

I'd like to know if there are weaker conditions than $\mathcal{A}$ being a $C^{\ast}$-algebra for which Kadison's conclusion holds. For example, conditions including things like "$\mathcal{A}$ is closed under continuous functional calculus" and such, which are essentially equivalent to $\mathcal{A}$ being a $C^{*}$-algebra in the (singly-generated) abelian case, but not in general.

(My impression is that this question is really tough, but I hope I'm wrong. It would be nice even to have an expert's digression on precisely why this question should be tough...as I'd learn some new things from that insight!)

It is also possible to ask about analogues of Pedersen's "up-down" theorem, which says that any self-adjoint element in the strong closure of a $C^{\ast}$-algebra $\mathcal{A}$ on a separable Hilbert space $\mathcal{H}$ is the strong limit of a monotone decreasing sequence of self-adjoint elements each of which is a strong limit of a monotone increasing sequence of self-adjoint elements. Can one weaken the $C^{\ast}$-condition on $\mathcal{A}$ and still get this result? (I've tried to modify the argument in Pedersen's $C^{*}$-algebras and their automorphism groups but if I remember correctly this seems to use the definition of a $C^{*}$-algebra in an essential way. Is there a way around this?!)

Answer by Jon Bannon for congruences for Fourier coefficients of modular forms

2012-07-27T19:49:36Z

Also, have a look at Ken Ono's The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and Q-Series. This book is full of what you are looking for.

Answer by Jon Bannon for Good notions of "perfect number" for rings of integers more general than ${\Bbb Z}$?

2012-07-21T10:10:21Z

Definition 2.0.5 in the following dissertation gives one possible definition of perfect number:

http://eprints.maths.ox.ac.uk/741/1/smallbone2.pdf

Answer by Jon Bannon for Stochastic processes with random matrices

2012-07-18T10:22:13Z

Yes. A quick search turns up papers like:

http://arxiv.org/pdf/math-ph/0402061.pdf

and

http://arxiv.org/pdf/1004.0301v2.pdf

Answer by Jon Bannon for Classroom platonism

2012-06-18T23:55:55Z

The platonization you are looking at seems related to the idea of reification that appears in mathematics education literature (roughly the compression of a mathematical process to a mathematical object). Note: This is clearly distinct from what you are asking about, but shares the feature of bestowing objecthood on something in a way that requires the shift to (what is at first) a radically different viewpoint.

A few things on this that may be helpful can be found here.

Particularly, here are a couple of papers on reification.

Regarding Cayley Graphs of Property (T) Groups

2012-06-01T17:44:39Z

A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of finite quotients of property (T) groups can give us families of expanders (see Exercise 14 of Tao's blog post). The construction seems to critically use the finite quotients to obtain the unitary representations required to employ the definition of property (T). It would be very nice to have an answer to the following:

Question: Is any of the behavior of expander graphs reflected in the (infinite) Cayley graph of a property (T) group with respect to a finite, symmetric generating set?

Please provide a reference to anything in the literature that sheds some light on this.

A somewhat broader (related) question that may be helpful:

What are some qualitative properties of the Cayley graph of a property (T) group?

For example, does the Cayley graph of a property (T) group exhibit any sort of (local) concentration of measure phenomena using the word metric w.r.t. a finite generating set?

What are some useful intuitions for the Cayley graph of a property (T) group? (Here I'm wondering if there is anything akin to the image of "thin triangles" for hyperbolic groups.)

Positive definite functions on G from Hilbert space vectors?

2012-05-25T13:28:19Z

Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a positive definite function on $G$ using $\xi$?

This question is rather vague and open-ended so I've made this CW, but I'd be very appreciative of any references that try to do this.

Positive definite function zoo

2011-08-22T16:36:09Z

I've asked the following question on math.stackexchange but there has been no response so I'll ask it again here:

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a "diagonal" coefficient of a unitary representation of $G$.

For a definition and discussion of positive definite function see here.

I've often wished I had a collection of diverse examples of positive definite functions on groups, for the purpose of testing various conjectures. I hope the diverse experience of the participants of this forum can help me collect a list of such examples.

To clarify what I'd like to see:

What is an example of a positive definite function on a group $G$ that is not easily seen to be a coefficient of a unitary representation of $G$? What are some positive definite functions that arise in contexts sufficiently removed from studying the coefficients of unitary representations?

Also, the weirder the group $G$ the better. I'd like a collection of quirky beasts...

Cake-cutting and amenable groups

2012-05-23T15:57:14Z

I recently heard Alan Taylor speak about envy-free fair division and started wondering if questions like these make sense if we replace finitely additive measures with invariant means on amenable groups. Valerio has suggested a few tweaks for this question (thanks!) so I'll post a broad version and the earlier narrow one.

Along these lines, I'll ask the following broad question:

What are some fruitful modifications of cake-cutting fair division problems which replace the cake by an amenable group and the partygoers' preferences by invariant means?

Here's an attempt at such a modification in the discrete case, which was the body of the original question:

Let $G$ be an infinite discrete amenable group with $n$ given distinct left-invariant means $\mu_{1},...,\mu_{n}$. Is it possible to partition $G$ into $n$ parts $\lbrace K_{i} \rbrace_{i=1}^{n}$ so that $\mu_{j}(K_{l})\leq\mu_{j}(K_{j})$ for all $l,j\in \lbrace 1,2,...,n \rbrace$ and $l \neq j$?

Answer by Jon Bannon for What are the advantages and disadvantages of the Moore method?

2012-05-14T23:00:05Z

The Moore Method has other variants grouped together under the "IBL" (Inquiry Based Learning) umbrella. I just finished teaching an intro (undergrad) real analysis course using an IBL script instead of a textbook and am amazed at how much more solidly all of my students absorbed the information in the course.

Certainly, this method doesn't move as quickly as traditional methods, but students assimilate the information so much more fully this really doesn't matter.

BTW: If you are interested in resources for running such a course, there are refereed IBL scripts in the journal of inquiry based learning in mathematics.

Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$?

2012-05-10T16:55:12Z

A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset ${ x_{1}, x_{2},..., x_{n} } \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $M$ with $\tau (u)=0$ and $||ux_{j}-x_{j}u||_{2}<\epsilon$ for all $1 \leq j \leq n$. (Here $||T||_2=(\tau(T^{*}T))^{1/2}$ for $T\in M$.)

I should mention that if the left group von Neumann algebra of an i.c.c. group has property $\Gamma$ then the group is inner amenable, however there exist i.c.c. inner amenable groups whose group von Neumann algebras don't have $\Gamma$, as recently shown by Stefaan Vaes.

Given a non-residually finite Baumslag-Solitar group $$BS(m,n) = \langle b,s\mid s^{-1}b^ms = b^n\rangle$$ does its group von Neumann algebra have property $\Gamma$?

It is known that all such groups are inner amenable, and it recently has been shown that the associated group factors have no Cartan subalgebra, are prime and yet are not solid.

Comment by Jon Bannon

2013-05-07T14:22:25Z

This question needs to be sharpened.

Comment by Jon Bannon

2013-04-27T10:49:23Z

Thanks, Ronnie Brown!

Comment by Jon Bannon

2013-04-20T10:39:28Z

@S.Carnahan: I knew someone would bring this up! Implicit in my answer is an earlier comment I deleted: Personally, I'm not as interested in educating the general populace as I am in educating undergraduate and graduate students. (This isn't something I'm proud of!) So I'm not interested in a renewal of the "new math".

Comment by Jon Bannon

2013-04-20T10:37:56Z

@Henry: To my chagrin, I only find mention of it in the paper to which I linked above as "grain size". Maybe that's how it is referred to in the literature. All I know is what I wrote above, and that was enough to make a difference for me. If anyone has a source, I'd love to have it, too.

Comment by Jon Bannon

2013-04-12T17:23:22Z

This set of notes is excellent!

Comment by Jon Bannon

2013-04-05T17:21:32Z

Thanks for asking this, Jiang! I should mention that in the above link to our paper, certain bounds (4/49 etc.) were not right. In the actual paper: sciencedirect.com/science/article/pii/… The correct bounds appear.

Comment by Jon Bannon

2013-04-04T21:34:23Z

!@Mariano: From where I stand, that is very reasonable!

Comment by Jon Bannon

2013-03-24T20:32:50Z

Find out what kind of problems you want to work on...then pick where to go. OA and NCG is not specific enough, in my opinion.

Comment by Jon Bannon

2013-03-24T12:18:22Z

@Harald Hanche-Olsen: One needs to use a "strong sum" (take the closure of the sum of the two closed operators) and "strong product". Indeed, one has to be careful: plms.oxfordjournals.org/content/s3-23/1/…

Comment by Jon Bannon

2013-03-05T20:56:38Z

Welcome, Dmitri!

Comment by Jon Bannon

2013-01-27T13:39:44Z

@Théophile: I'm happy to pass this along.

Comment by Jon Bannon

2013-01-26T00:35:05Z

@Ronnie: Trite as saying this is, I find the question "What is and should be a theorem?" very interesting. I wish there were a way to ask this philosophical question here on MO without the danger of it encouraging too much discussion.

Comment by Jon Bannon

2013-01-22T11:53:37Z

This is a nice flyover! Thank you for the response.

Comment by Jon Bannon

2013-01-21T17:47:22Z

It does answer the question as posed... Perhaps there will be more to say, but this is pretty good.

Comment by Jon Bannon

2013-01-21T17:46:02Z

This is pretty strong. If I don't get any other answers to this thing, I think I'll accept this one.