User cdouglas - MathOverflow

Is Mac Lane still the best place to learn category theory?

2011-07-01T11:26:34Z

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ...

Is Mac Lane still the best place to start?

Or has the day arrived when it is possible to directly learn ($\infty$,n)-categories, without first learning ordinary category theory? (So the next generation will be, so to speak, natively derived.) If so, via what route? If not, what's the most efficient path through the classical core material to a modern perspective?

When does the converse to Schur's Lemma hold?

2009-10-24T18:28:58Z

Let $R$ be a commutative ring, let $A$ be an $R$-algebra, and let $M$ be an $A$-module. If $M$ is simple, then End $_{A-mod}(M)$ is a division ring.

A common use is when $R$ is the complex numbers $\mathbb{C}$, and $M$ is such that End $_{A-mod}(M)$ is finite dimensional. Then End $_{A-mod}(M) = \mathbb{C}$ .

Under what circumstances (regarding $R$ and/or $A$) is the converse true, that the endomorphism ring being a division ring, or being just $R$ itself, implies that $M$ is simple?

Answer by cdouglas for Geometric Interpretations of Homotopy Theoretical Constructions

2012-06-30T23:05:55Z

Though it is often satisfying and useful to have an appropriate way to visualize a construction, in fact the real innovation in homotopy theory that lead to the constructions you mention, among many others, was the realization that you need not limit your attention to geometric, visualizable operations and spaces but can be content knowing only certain relevant properties.

This realization led to what Mike Hopkins (and perhaps others before him) called designer homotopy theory. The simplest example of a designer homotopy type is the Eilenberg-MacLane spaces---once you convince yourself they exist, you don't concern yourself with what they look like (a horrendous infinite CW complex, perhaps), but use only their characteristic relationship to homotopy groups. Needless to say this particular example led to spectacular advances in the hands of Serre and others.

What is the first interesting matric Toda bracket in the stable homotopy of the sphere?

2012-06-20T13:02:54Z

Feel free to gloss `interesting' as you see fit. One way:

1. What is the lowest degree matric Toda bracket in $\pi_*(S)$ that doesn't contain zero?

By `degree', I mean total homotopical degree, i.e. the $*$ in $\pi_*$. By `matric' I mean to exclude ordinary Toda brackets, that is matric Toda brackets all of whose entries are one-by-one matrices, and also to exclude brackets that are trivially determined by ordinary Toda brackets.

I'm also interested in the title question with `first' replaced by `simplest'. For instance:

2. What is the lowest order matric Toda bracket in $\pi_*(S)$, all of whose matrix entries are sums of products of Hopf elements, that doesn't contain zero?

By `order' I mean the number of entries in the bracket, i.e. whether it is 3-fold or 4-fold or 5-fold or ... .

Now I'd like to know the same, but with the proviso that the bracket be detectable in the classical Adams spectral sequence:

3. What is the first or simplest, interesting matric Toda bracket in $\pi_*(S)$ that is detectable in the $\mathrm{E}_2$ term of the classical Adams spectral sequence?

Some remarks:

I believe the matric Massey product $\langle h_2^2, h_0, \left(\begin{array}{cc} h_1 h_3 & h_2^2 \end{array} \right), \left(\begin{array}{c} h_1 \\ h_2 \end{array}\right)\rangle $ in the $\mathrm{E}_2$ term of the Adams spectral sequence is the class $e_0$, but that is not a permanent cycle.
Kochman lists the matric Toda bracket $\langle \sigma, \left(\begin{array}{cc} A[31] & \nu\end{array}\right),\left(\begin{array}{cc} \eta & 0 \\ 0 & \eta \end{array}\right),\left(\begin{array}{c} \nu \\ \eta A[30] \end{array}\right)\rangle$ in degree 44 as containing an element of order 8, where $A[30]$ and $A[31]$ refer to certain elements of order 2 in those degrees.

Is the 4-line of the E_2 term of the classical Adams spectral sequence known?

2012-05-28T23:14:18Z

In other words:

What is $\mathrm{Ext}_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?

If the 4-line is not known, how much is known about it?

Here, $\mathcal{A}$ is the 2-primary Steenrod algebra, $4$ is the homological degree corresponding to the Adams filtration, and $t$ is the internal grading degree. Those $\mathrm{Ext}$ groups make up the fourth row of the classical Adams spectral sequence $E_2 = \mathrm{Ext}_{\mathcal{A}}^{s,t}(\mathbb{Z}/2,\mathbb{Z}/2)$ converging to the 2-adic completion of the $(t-s)^{\mathrm{th}}$ stable homotopy group of the sphere.

For context,

the 1-line is generated by the classes $h_i$, $i \geq 0$, ($\mathrm{deg}\: h_i = (1,2^i)$),
the 2-line is generated by the product classes $h_i h_j$, subject to the relations $h_i h_{i+1} = 0$ and $h_i h_j = h_j h_i$,
the 3-line is generated by two sets of classes,
1. the product classes $h_i h_j h_k$, subject to the relations implied by $h_i h_{i+2}^2 = 0$, $h_{i+1}^3 = h_i^2 h_{i+2}$, $h_i h_{i+1} = 0$, and $h_i h_j = h_j h_i$,
2. the Massey products $\langle h_{i+1},h_i,h_{i+2}^2 \rangle$.

Answer by cdouglas for Simplest examples of rings that are not isomorphic to their opposites

2011-05-09T13:34:54Z

A particularly simple example of an algebra not isomorphic to its (graded) opposite is the $\mathbb{R}$-algebra $\mathbb{C}$, where $1$ is even and $i$ is odd. This is the ($\mathbb{Z}/2$-graded) real Clifford algebra $Cl(-1) = \langle f \mid f^2 = -1 \rangle$. Its opposite is the Clifford algebra $Cl(1) = \langle e \mid e^2 = 1 \rangle$, whose underlying ungraded algebra is isomorphic to $\mathbb{R} \oplus \mathbb{R}$.

Per the discussion in the other answers, these two algebras represent $1$ and $-1 = 7$ in the graded Brauer group $\mathbb{Z}/8$ of $\mathbb{R}$.

Answer by cdouglas for Ideas on how to prevent a department from being shut down.

2011-04-29T18:45:01Z

Petitions can carry more weight when they are part of a personalized effort including other forms of appeal, particularly:

Letters
Phone calls

You could provide the community (here on MO, and on the AT list, and at SBS) the address and phone number of a key decision maker or administrative body in this process (or more than one). If that person begins receiving letters and calls from faculty at numerous international universities, I suspect it would make a strong impression. In particular it might make the number of signatures on the online petition more real and harder to ignore. Also, getting even a few people in the Netherlands (pure mathematicians, applied mathematicians, physicists, and others) to go and visit the relevant administrators in person could be extremely effective.

Another approach:

Press

You can find a reporter at De Telegraaf to write and publish a story about the VU's threat to close down the pure mathematics section and on the international uproar over this threat. If the decision makers see a public article exposing the ongoing damage to the international reputation of VU as a result of their proposal, it will become harder for them to argue that this action is in the best interest of the university and could prompt a rapid reconsideration. Similarly you could have an article in the Chronicle of Higher Education, either as a pure news story about the threat and the online response, or as a discussion piece about the state of the tenure system in the Netherlands.

Another source of support:

Alumni

You could contact alumni of the VU mathematics department and have them write letters to the administrators communicating both how important the department was in their lives and how disappointed and disgruntled they would be to see the department decimated. If they are donors to the university, they could also indicate that the outcome of this decision will affect, in addition, their future contributions.

Another avenue to consider:

Alternative funding options

You could bring forward alternative proposals to directly cover some of the budget gaps that are being `addressed' by the proposed measure. These could involve temporary distributed pay cuts to existing faculty and staff, along with fundraising efforts, or perhaps corporate donations or partnerships and other less traditional means. Even if some of the proposals are not appropriate in the end, an effort to contribute to the solution of the administration's funding problem might make them more receptive to revising their overall plan.

Drawing of the eight Thurston geometries?

2010-05-14T03:09:01Z

Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries?

I am imagining something akin to the standard picture (of a sphere, plane, and saddle) used to illustrate the three constant curvature geometries in dimension two. Of course, it takes more doing to illustrate representative three-manifolds, and there are more choices for natural examples, but I was surprised when I couldn't find such a picture. Another option would be to depict or indicate some of the geometries in less direct ways, for instance via the structure of stabilizers.

Is there a volume conjecture for closed 3-manifolds?

2010-02-25T03:00:52Z

A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is

Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of the knot complement by $$ 2 \pi \lim_{N \rightarrow \infty} \frac{1}{N} \log | J_N(K; \exp(2\pi i / N)) | = Vol( S^3 \backslash K).$$

A refinement of the conjecture is $ 2 \pi \lim_{N \rightarrow \infty} \frac{1}{N} \log J_N(K; \exp(2\pi i / N)) = Vol( S^3 \backslash K) + i \; CS( S^3 \backslash K) \; (\mod \pi^2 i)$ where CS is the Chern-Simons invariant. Both sides of the conjecture can be formulated for 3-manifolds more general than knots in S^3 and their complements. In particular, we might ask about closed 3-manifolds without a knot at all.

Question: Is there an analogous volume conjecture for (some) closed 3-manifolds, or for closed 3-manifolds with embedded knots, and if so where in the literature are these formulations discussed?

How do you compute the space of lifts of an E-infinity map?

2010-02-15T21:37:03Z

Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g: B \rightarrow X$ such that $pg = f$.

Q1: What spectral sequences or other technology exists for computing the (homotopy groups of the) space of such E-infinity lifts?

Any $E_\infty$ lift $g: B \rightarrow X$ provides a lift of the map of commutative monoids $\pi_0 B \rightarrow \pi_0 Y$ across the map $\pi_0 X \rightarrow \pi_0 Y$.

Q2: Do all the techniques for computing E-infinity lifts in effect require that you first solve this $\pi_0$ lifting problem in commutative monoids, and begin an obstruction calculation from there, or are there techniques that solve both the $\pi_0$ problem and the E-infinity problem 'simultaneously' and perhaps in a way that eases both computations?

I am particularly interested in merely knowing if there exists an $E_\infty$ lift, thus might have asked the seemingly more basic question:

Q1': When is there an E-infinity lift of an E-infinity map?

I think the obstruction groups answering Q1' are liable to come packaged in the answers to Q1/Q2, but if there are separate techniques for the existence question, that would also be helpful.

Remark: I could imagine that one answer to Q1 involving relative Andre-Quillen cohomology might be extracted from Goerss-Hopkins, Moduli Problems for Structured Ring Spectra, but perhaps there are more elementary means. I'd be very interested for answers to Q1 along those or especially other lines of thought, and any ideas about Q2. Thanks!

Answer by cdouglas for Proofs of Bott periodicity

2009-12-18T01:43:12Z

By studying natural filtrations of loop groups, viewed as affine Grassmannians, S. A. Mitchell, in "The Bott Filtration of a Loop Group", describes an elegant and in its way elementary proof of Bott periodicity. Homotopy theorists will appreciate that it comes down to the inclusion of a cellular skeleton being highly connected, while combinatorist and Lie theorists delight when Mitchell reads off Bott periodicity from the Dynkin diagrams.

How many embeddings are there of super-Virasoro into n Fermions?

2009-10-19T23:42:38Z

What is the space of N=1 super-Virasoro vertex superalgebras inside the c=n/2 free fermion vertex superalgebra? [Said differently, how many Neveu-Schwartz vectors are there in n fermions?] Answers in terms of VOAs, CFTs, or nets all welcome, and any interpretation of "N=1 super" that makes you happy is okay.

Reference for iterated homotopy fixed points?

2009-11-11T19:01:32Z

What are (good) references for results about iterated homotopy fixed points? That is, suppose G is a topological group acting on a space (or spectrum) X, and H is a normal subgroup of G. Then one would like to first compute the homotopy fixed points of X with respect to H, and use that as a stepping stone to compute the homotopy fixed points of X with respect to G.

(I am independently interested in both the space and spectrum versions, so am happy with pointers, comments regarding either.)

Answer by cdouglas for squares in stable homotopy

2009-11-10T19:11:51Z

Appendix 3 of Ravenel's Green Book, http://www.math.rochester.edu/u/faculty/doug/mu.html#repub, has a chart of stable homotopy groups including much of the multiplicative structure. Figure A3.1 depicts some of this structure visually, while Table A3.3 lists the elements out by name and degree.

The next example of a square after \eta^2 is the element in the sixth stable stem, which is the square of the Hopf map \nu in the third stable stem.

Though stable homotopy is not multiplicatively finitely generated, you can consider Toda brackets, which are a form of higher multiplication in homotopy analogous to Massey products in cohomology, and it is known that the entire stable homotopy groups of spheres are generated by Toda brackets on the Hopf elements 2, \eta, \nu, and \sigma.

Answer by cdouglas for Most helpful math resources on the web

2009-11-05T03:19:50Z

The nLab is an excellent resource, often containing more detail, explanation, and discussion than wikipedia, along with much more specialized and contemporary topics.

(nLab was mentioned in the answer by Justin Hilburn, but it was listed there after other resources, and I think people scanning under the one-resource-per-answer dictum will miss it.)

Answer by cdouglas for Tools for Organizing Papers?

2009-10-28T03:50:23Z

As for physical papers:

I have two coexisting systems. The first is a file cabinet organized by author. Organizing large numbers of papers by subject or date or whatnot ravels out of control. The second is a series of magazine racks labelled by project, which contain papers directly relevant to the corresponding projects.

As for electronic papers:

I used to put them in folders by author, with helpful filenames like

Fukaya-FloerHomologyFor3ManifoldsWithBoundaryI.pdf
HatcherLawson-StabilityTheoremsForConcordanceImpliesIsotopyAndHCobordismImpliesDiffeomorphism.pdf
KustermansRognesTuset-TheModularSquareForQuantumGroups.pdf

but maintaining this got old. I tried Papers, and thought it was going to be fabulous, but like Scott, wasn't won over in the end.

But now search has gotten good enough there is much less need for explicit organization. You can just put the pdfs all together in a ginormous folder, and whenever you need something just search for it. It's the google way. (And if you use Google Desktop, then literally so.)

Here is the one thing I would like to be able to add to this system: it would be great to be able to add tags to papers, which would even further facilitate targeted retrieval and browsing. Does anyone know an easy way to do this?

Answer by cdouglas for How do I describe a fusion category given a subfactor?

2009-10-24T17:41:05Z

An answer to question (5):

Arthur, Andre, and I proved that

{conformal nets [of vN algebras], conformal defects, sectors, intertwiners}

form a 3-category CN. Given a net N \in CN, the endomorphisms of the identity on N, that is End(1_N), is a braided tensor category, automatically. (Indeed, the most compact definition of a braided tensor category is a 3-category with one object and one 1-morphism.) We might call End(1_N) the representation category of the net, and denote it Rep(N). The category Rep(N) is related to Drinfeld centers, but that's probably a separate discussion.

(By the way, you don't need to know what a conformal defect is to say what Rep(N) is: a representation of a net N is just a Hilbert space with compatible actions of all the vN algebras occuring in the net on the circle.)

(Evan Jenkins took very faithful notes from a talk I gave about this stuff, which you can find here:

http://www.math.uchicago.edu/~ejenkins/notes/nwtft/25may-cd.pdf

Discussion of the functor End(1_...) from nets to tensor categories occurs on pages 2 and 6.)

Comment by cdouglas

2012-08-16T05:57:44Z

Note that DMNO only discuss the possibility that the unitary part of the Witt group, rather than the whole Witt group, might be generated by the quantum group categories.

Comment by cdouglas

2012-06-20T15:51:19Z

Ah, so probably the bracket $\langle \begin{smallmatrix} \nu^2 & \eta \end{smallmatrix}, \begin{smallmatrix} \nu & \epsilon \\\ \epsilon & \nu \end{smallmatrix}, \begin{smallmatrix} \nu^2 & \eta \\\ \eta & \nu^2 \end{smallmatrix}, \begin{smallmatrix} \nu \\\ \epsilon \end{smallmatrix} \rangle$ is interesting, and detectable in Adams $\mathrm{E}_2$.

Comment by cdouglas

2012-05-31T10:38:20Z

A rare happy moment when there is a preexisting paper devoted precisely to answering the question. Thanks!

Comment by cdouglas

2011-10-27T11:53:06Z

The link to the reask of the question is broken (... well, it redirects to a question about numerical methods for calculating digits of pi ...).

Comment by cdouglas

2011-06-25T01:14:31Z

I'd quibble with saying 'this subject has a massive dearth of examples' --- spectral sequences are almost exclusively developed by and for examples, of which there are a spectacular plethora. The (well, a) real problem is that until recently typesetting spectral sequences was sufficiently difficult that people did it quite infrequently, even in research papers whose main content was spectral sequence computations. Tilman Bauer's sseq package was a big improvement, but many kinds of SS displays are still painful to tex.

Comment by cdouglas

2011-05-11T17:38:46Z

I don't know at what stage the finder begins to have trouble. Certainly a thousand files in one folder is no problem. My guess is that ten thousand is okay too, but I haven't tried.

Comment by cdouglas

2010-05-23T02:01:33Z

Thanks! I especially like the hyperbolic glass image.

Comment by cdouglas

2010-02-24T00:16:29Z

Thanks, glad to have the perspective on how relative TAQ enters, even if you confirmed my fears about the difficulty of the $\pi_0$ problem.

Comment by cdouglas

2010-02-24T00:12:41Z

Thanks, it's good to keep in mind that to be in spectra-land, it's sufficient for the group completion to be injective. Unfortunately, in my cases I expect the group completion to be a brutal operation on $\pi_0$. I'll contemplate whether some carefully chosen submonoid might have better properties.

Comment by cdouglas

2009-12-18T08:39:52Z

Thanks very much for this detailed answer! Much appreciated. (And sorry the thanks took so long in coming, my notification seems to be on the blink.)

Comment by cdouglas

2009-12-18T01:12:58Z

A small point: Atiyah-Bott-Shapiro did not prove Bott periodicity in their paper on Clifford Modules, but rather used Bott periodicity to show that the Grothendieck group of Clifford modules is the K-theory group. In fact, they specifically mention that it would be nice to have a proof of Bott periodicity that followed from the algebraic periodicity of Clifford modules.

Comment by cdouglas

2009-11-11T23:21:22Z

Primarily the former, that (X^hH)^hG/H = X^hG, but I'd also be happy for any references to computational approaches or examples.

Comment by cdouglas

2009-10-28T04:12:07Z

I'd like the Virasoro inside the super-Virasoro to agree with the Virasoro of the free fermions (that is the n-th tensor power of the Virasoro of the free fermion). The 'space' is then the space of extensions of the given Vir to an sVir.