User josh roberts - MathOverflow

Real analysis has no applications?

2010-06-03T12:37:44Z

I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications to other fields of science. None. It is pure mathematics." This seems like a false statement. My first thought was of probability theory. And isn't PDE's sometimes considered applied math? I was wondering what others thought about this statement.

Classifying space of a crossed complex

2009-10-27T01:06:28Z

Brown defines the classifying space of a crossed complex in the following way.

Given a filtration X_* of a space X, define the fundamental crossed complex by: C_0 = X_0, C_1=\pi(X_1,X_0) (the fundamental groupoid), C_n = the family of groups \pi(X_n,X_n-1,p) for all p in X_0.

Now let Δ^n be the cell complex of the standard n-simplex, with its skeletal filtration. The crossed complex \pi(π^n) is then written \pi[n]. The nerve NC of a crossed complex C is defined to be the simplicial set given in dimension n by (NC)n = Crs(\pi[n],C), where Crs(-,-) is the internal hom in the category of crossed complexes.

The think I don't understand is that each of the n-simplices are contractible, so why wouldn't the fundamental crossed complex associated to the filtration of Δ^n be trivial?

Cures for mathematician's block (as in writer's block)

2009-11-07T06:27:00Z

What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening and playing music can stimulate mathematical thinking. Any particular style that someone finds helpful?

Other things that help?

(In case you haven't figured it out, I've been stuck on some things lately.)

Geometric interpretation of the fundamental groupoid

2010-06-08T16:59:33Z

Motivation

The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy groups are how higher dimensional spheres behave (up to homotopy in both cases, of course). Even better, for nice enough spaces the (integral) homology groups count $n$-dimensional holes.

A groupoid is a category where all morphisms are invertible. Given a space $X$, the fundamental groupoid of $X$, $\Pi_1(X)$, is the category whose objects are the points of $X$ and the morphisms are homotopy classes of maps rel end points. It's clear that $\Pi_1(X)$ is a groupoid and the group object at $x \in X$ is simply the fundamental group $\pi_1(X,x)$. My question is:

Is there a geometric interpretation $\Pi_1(X)$ analogous to the geometric interpretation of homotopy groups and homology groups explained above?

Crossed module structure on homotopy groups.

2010-08-18T15:56:01Z

A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy

$\partial(g\cdot c)=g(\partial c)g^{-1}$, and
$cc'c=(\partial c)\cdot c'$

Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, $$ \pi_2(X,A) \stackrel{\partial}{\rightarrow} \pi_1(A) $$ is a crossed module. Simply put, my question is what does this give us, other than an extra bit of structure? Does knowing that this is true aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)

Cite articles or book where I first found the result?

2011-05-15T03:57:56Z

I'm writing up a paper and I'm not sure how to cite a few things. It concerns a conjecture made by Quillen. Some work has been done showing it's true in some cases, false in others. These results I found in a book that had a chapter about the conjecture; of course, the book gave all the references. In the background of my paper, I want to briefly sum up these results. My question is, should I cite the articles, the book, or both? I wasn't sure what the etiquette/rules are in this situation.

Answer by Josh Roberts for Textbook recommendations for undergraduate proof-writing class

2011-04-25T03:03:38Z

A "book" that satisfies all of your criteria is a set of notes from the Journal of Inquiry Based Mathematics called "Introduction to Proof" by Ron Taylor. linky

The chapters are

Symbolic Logic
Proof Methods
Mathematical Induction
Set Theory
Functions and Relations

There are two appendices: one on mathematical writing and one on Style (By James Munkres).

It is a set of notes for an IBL class, so the assumption is that the students will be doing virtually all of the proofs themselves. I've never used this set of notes for teaching, but I've used others from the journal. I like them very much.

Their copyright notice allows free use and printing as long as attribution is given and no charge for the students other than printing costs. Similar sets of notes that I've used have cost the students about $6.

Others from the journal's website about intro to proof/foundations are http://www.jiblm.org/downloads/dlitem.aspx?id=17&category=mathnerdscollection http://www.jiblm.org/downloads/dlitem.aspx?id=16&category=mathnerdscollection http://www.jiblm.org/downloads/dlitem.aspx?id=14&category=mathnerdscollection

(These last three haven't been refereed by the journal, but they still gives links to them.)

Classifying space of large category?

2011-03-28T01:20:43Z

Whenever I've seen the definition of the classifying space of a category, the category is always specified to be small. I understand the definition well enough for my purposes (I think), but it occurred to me today, why small?

Is there a reason we only take nerves of small categories? Does the definition fail if the objects are too big?

"Surprising" categorical equivalences

2011-03-06T03:50:12Z

This is inspired by this question about the equivalence between the category of finite sets and non-negative integers. Now this question was (rightly, I guess) closed, but the fact was surprising to the OPer. I didn't know about this, but it was easy to verify. I had a little bit of difficulty understanding the nuances between categorical equivalences and category isomorphism until I thought about the analogy with homotopic spaces and homeomorphic spaces.

I was wondering about other equivalences that might unexpected and/or not so straightforward to prove. I would appreciate answers to a general audience...things that a beginner or an expert might find of interest.

Another group cohomology cup product question

2010-11-01T18:31:10Z

I am wondering if there is a way to see the cup product, in some cases, without using cochain complexes. The situation I am interested in is the following:

Let $G=F/R$ be a finitely presented group and $k$ a finite field. Then $H_1(G;k)$ is easy to find as a finitely generated abelian group. Since I'm taking field coefficients, $H^1$ is isomorphic to this abelian group. If I keep track of normalizing the relations matrix I can even get a set of generators in terms of $G$. Using Hopf's formula for $H_2(G;k)$, I can get generators for $H^2$. Is there a way to see what the cup product of terms from $H^1$ is in $H^2$? Can I get elements of $H^2, H^3, H^4$ this way?

Smooth classifying spaces?

2009-10-18T23:40:21Z

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular construction if G is finitely presented, etc.

What I'm wondering about, is there a notion of a smooth classifying space? That is, when can a classifying space for a group be given a smooth structure?

Answer by Josh Roberts for Where have you used computer programming in your career as an (applied/pure) mathematician?

2010-09-01T11:27:11Z

I wrote a series of algorithms to find bounds on the homology of finitely-presented groups and implemented them in GAP in http://www.intlpress.com/HHA/v12/n1/a3/. Graham Ellis has written the GAP package HAP, http://www.gap-system.org/Packages/hap.html, which does some group (co)homology calculations.

Answer by Josh Roberts for Why aren't there more classifying spaces in number theory?

2010-08-31T17:38:05Z

I suppose we should also mention algebraic k-theory. Quillen defined the k-groups as the homotopy groups of certain classifying spaces. For a unital, associative ring $R$, $$ K_n(R):=\pi_n(BGL(R)^+), $$ where $GL(R)$ is the direct limit of the general linear groups and $^+$ is Quillen's plus-construction on spaces whose fundamental groups have perfect subgroups. Now I'm not sure how useful this has been for computation (these are homotopy groups, after all), but the classifying space is used. And there are number theory applications of algebraic k-theory.

Answer by Josh Roberts for presentation for GL(n,K)

2010-06-06T05:41:09Z

You might want to look at Cohn's paper "On the structure of the ${\rm GL}_{2}$ of a ring", MR0207856.

Answer by Josh Roberts for Database of finite presentations of used groups

2010-06-06T05:37:37Z

In my dissertation I developed an algorithm for finding bounds on $H_2$ of a finitely presented group with finite field coefficients. I was motivated by a conjecture Quillen on the (co)homology of linear groups. As such, I included an appendix with presentations of several linear groups and the homology calculations using my algorithms. I didn't include the list of presentations for publication, but if these types of groups are of interest I could get it to you.

Essential theorems in group (co)homology

2010-01-21T13:46:26Z

I'm trying to fill in the gaps in my understanding of group (co)homology and I'm wondering what are considered the "must know" theorems and concepts. I'm thinking of things along the lines of

Hopf's formula - If $G$ has presentation $F/R$, then $H_2(G)=R \cap [F,F]/[F,R]$
If $G$ has torsion then $H_n(G)$ has no top dimension
$H_n = Tor_n$ so is the left derived functor of $\otimes$
$H^n = Ext ^n$ so is the right derived functor of $Hom$
If $G$ is discrete, then $H_n(G)=H_n(K(G,1))$

Explicit classifying spaces for crossed complexes

2010-01-21T02:22:09Z

I'm trying to understand the topology behind a certain group which fits into a truncated crossed complex, so I've been trying to understand Brown's construction of the classifying space of a crossed complex. I asked a similar question here and the answer cleared up some of my misunderstanding.

What I'm looking for now is an explicit construction of such a space. I've not found any papers that give explicit examples of such a construction, but hopefully there is one somewhere. I'm looking for something along the lines of the easy to follow of the construction of the classifying space of cyclic groups.

Answer by Josh Roberts for Intuition for Group Cohomology

2010-01-06T05:33:51Z

I'm not sure if this is what you're looking for, but I always think of group (co)homology in terms of the homology of the classifying space for your group. Assuming $G$ is discrete, then there is a topological space $BG$ with the property that $\pi_1 BG=G$ and the higher homotopy groups vanish. By construction, $BG$ has a contractible cover $EG$ so that $EG/G=BG$.

$H_n(BG)$ is the same as the algebraically defined $H_n(G)$ since, if we take the cellular chain complex of $EG$ we end up with a resolution of the integers by $G$-modules because of the action of $G$ on $EG$ passes to the chain groups. Then tensoring by the integral group ring of $G$ just divides out the $G$ action and we get the cellular chain complex of $BG$.

Answer by Josh Roberts for Killing the torsion in homotopy

2010-01-05T16:10:46Z

It might go without saying, but there is a procedure for non-simply connected spaces if you're killing a perfect torsion subgroup. It's just Quillen's plus construction used in the construction of algebraic k-theory.

Examples of the varying strengths of topological invariants

2010-01-02T18:21:11Z

In my first algebraic topology class, I remember being told that the simplest reason for homology was to distinguish spaces. For example, if is X=circle and a Y= wedge of a circle and a 2-sphere then X and Y have the same fundamental group, so the fundamental group isn't strong enough to distinguish them. We need to look at the other homotopy groups or homology to tell them apart. I'm looking for a variety of other examples of this nature. The examples I'm wondering about are

Same homology groups
Same cohomology groups, but different cohomology rings
Same cohomology rings (but maybe different Steenrod operations?)

If I put more thought into it, I could come up with others questions like these. Any other examples/thoughts along these lines would be very welcome! (I have examples for the first one, but I'm wondering what others will say.)

Answer by Josh Roberts for Cocktail party math

2009-11-14T14:11:05Z

There are some good ones in topology (or maybe I just know these examples because it's my field).

People always seem to think the hairy ball theorem (already mentioned) is interesting because it explain why people have cowlicks on their head, not to mention the name itself usually gets a few giggles.
The Meteorological Theorem (Borusk-Ulam) implies there are antipodal points on the surface of the Earth where the temperature and barometric pressure are the same.
My favorite one is similar to the plate trick above. If you hold a coffee cup in your hand and rotate your wrist until the cup is oriented the original way--your arm is all tangled up, but if you rotate it again, your arm straightens out. It demonsrates that the fundamental group of the group of rotations in R^3 is Z_2.
Vin de Silva, who does works in applied topology, has the best one though. Take a piece of paper draw a few dots on it and ask what the shape it. Draw more until it's clear that you're "sampling" points from a circle. Then tell them that math (persistent homology in this case) lets you find the shape of a sampling of points. Leads to simple discussions of using math to solve lots of applied problems.

Then you can start making jokes using the work "functor." ("Functor? I hardly know her!" or just randomly say "Clusterfunctors!")

Answer by Josh Roberts for Estimating the number of clusters

2009-10-21T11:40:54Z

Carlsson has developed methods from applied topology to do clustering work. Robert Ghrist called a talk about this "Clusterfunctor" since it involves a functor from metric spaces to "persistent sets". It's talked about in Topology and Data, a survey article about using topology to do data analysis.

Answer by Josh Roberts for What's a groupoid? What's a good example of a groupoid?

2009-10-19T01:05:49Z

A groupoid is a generalization of a group. The easiest definition, IMO, is as a category in which all arrows are isomorphisms. So a group is just a groupoid with one object and arrows the elements of the group.

The best example is the fundamental groupoid of a topological space. Build a groupoid by taking the objects to be the points in the space and an arrow from point x to point y to be equivalence classes of paths from x to y. This genearlizes the idea of the fundamental group.

They are useful and Ronald Brown has a whole project of building higher dimensional group theory using them. The great thing about the fundamental groupoid is that there is a version of Van Kampen that gives the fundamental group of the circle (without using covering space theory as is the standard way to do it using only the fundamental group).

A good link is http://www.bangor.ac.uk/~mas010/nonab-a-t.html

ETA: That link might not be working. Google Ronald Brown's Topology and Groupoids book for a good introduction and motivation.

Answer by Josh Roberts for Two finite groups with the same identical relations?

2009-10-15T21:50:05Z

Are you assuming the two groups are quotients of the same free group?

Differentials in the Lyndon-Hochschild spectral sequence

2009-10-15T12:17:05Z

The Lyndon-Hochschild(-Serre) spectral sequence applies to group extensions in a manner analogous to the Serre-Leray spectral sequence applied to a fibration.

Does anyone know of a good description (or reference) of the transgression maps in the Lyndon-Hochschild spectral sequence? MacLane describes them in terms of an additive relation, but I don't find this helpful in computing them.

More generally, I don't know how to calculate the differentials in this spectral sequence. In the Serre spectral sequence, I can see how an exact couple arises and the differentials are straightforward to see if not easy to calculate. But the LHSS arises from a double complex and I'm not sure how to get an exact couple from this.

Answer by Josh Roberts for Does finite math need the Axiom of Infinity?

2009-10-15T03:42:55Z

Doesn't "there is no largest prime" implicitly assume the existence of the natural numbers?

Answer by Josh Roberts for How do you show that $S^{\infty}$ is contractible?

2009-10-14T21:50:07Z

Are follow up questions allowed? What does "version with all but finitely many components zero" mean? Is this different from taking the limit of n-spheres?

Answer by Josh Roberts for Motivation for algebraic K-theory?

2009-10-14T21:37:15Z

Here's a reference that gives some of the history of algebraic k-theory. It might have something you're looking for. http://www.math.uiuc.edu/K-theory/0343/khistory.pdf. Also Rosenberg's book "Algebraic K-theory and Its Applications is good.

Answer by Josh Roberts for References for homotopy colimit

2009-10-14T21:31:16Z

A good (if kind of old) reference is Vogt's "Homotopy Limits and Colimits". I can't find a free reference for it, but if you can't access it, I could email a pdf (if that's allowed here).

Also, in the IMA's video library there's a video of Gunnar Carlsson giving a talk on homotopy limits & colimits. http://www.ima.umn.edu/videos/?id=870

Comment by Josh Roberts

2013-04-12T14:03:37Z

what definition did you use!?

Comment by Josh Roberts

2012-07-03T03:06:06Z

That doesn't really answer my question, but I misread your OP. I saw "low dimensional homology" and you wrote "low dimensional CW complex". At any rate, you can get H_1 by abelianizing the group. For H_2, you might try the singular package on GAP, gap-system.org/Packages/simpcomp.html, or Javaplex, code.google.com/p/javaplex .

Comment by Josh Roberts

2012-07-03T00:08:42Z

How high of a dimension do you need to go? Can you give one example of a presentation of one of the groups you want to use?

Comment by Josh Roberts

2012-04-22T02:08:16Z

Take a look at en.wikipedia.org/wiki/….

Comment by Josh Roberts

2012-04-19T01:40:17Z

The Lyndon-Hochschild-Serre SS is associated to a group extension $A /to B \to C$. The Serre-Leray (or Serre) SS is associated to any fibration $F \to E \to B$. @Zuriel Try Section 6.2 in McCleary's book for the trangression of the SLSS and chapter 8b for the LHSSS. Also, you might find the answers to mathoverflow.net/questions/590/… useful.

Comment by Josh Roberts

2012-04-19T01:27:54Z

"More Concise..." is a book that they published, not an article. So you should probably check the library rather than trying to find it online.

Comment by Josh Roberts

2012-04-18T19:26:52Z

Have you looked at McCleary's User's Guide?

Comment by Josh Roberts

2011-12-06T19:56:06Z

Cohen's A course in simple-homotopy theory is good, but out of print, I think. Also, look at some of Tom Chapman's (a former professor of mine) work from the 70's.

Comment by Josh Roberts

2011-08-18T15:44:19Z

There are some partial results in intlpress.com/hha/v12/n1/a3. If G is finitely presented, and you're okay with field coefficients, there is an algorithm that can give an upper bound to H_2.

Comment by Josh Roberts

2011-06-28T00:23:08Z

Look up the closed topologist's sine curve. This is probably going to get closed, so maybe ask on math.stackexchange.com if you want more info.

Comment by Josh Roberts

2011-06-19T18:12:12Z

Try asking at math.stackexchange.com.

Comment by Josh Roberts

2011-05-15T11:32:01Z

There seems to be two main opinions: cite, cite, cite (as Ben said below) and don't cite unless you understand what you're citing. To clarify my situation, I'm not going to be using any of the results or techniques of the papers in question. The techniques I've found give a new proof of some of the known cases and prove the conjecture in some open cases. I'm not trying to inflate the list of references, rather I want to give as complete a picture as possible about status of the conjecture.

Comment by Josh Roberts

2011-05-12T00:16:19Z

Are you talking about Ryan Budney's comment on this question? mathoverflow.net/questions/15217/… You're probably need to give some more details.

Comment by Josh Roberts

2011-03-06T19:33:20Z

@Martin: I didn't mean to imply that the thread inspiring this question actually was surprising. When I saw the question it sounded like something that should be true and was easy to show.

Comment by Josh Roberts

2011-03-06T19:22:01Z

*"link" should be "thread" in the above comment.