Coordinatization with hom complexes

• December 7th, 2009

These are notes from a talk given at the Stanford applied topology seminar by Gunnar Carlsson from 9 Oct 2009. The main function of this blog post is to get me an easily accessible point of access for the ideas in that talk.

Coordinatization

First off, a few words on what we mean by coordinatization: as in algebraic geometry, we say that a coordinate function is some or possibly some , with all the niceness properties we’d expect to see in the context we’re working.

A particularly good example is Principal Component Analysis which yields a split linear automorphism on the ambient space that maximizes spread of the data points in the initial coordinates.

Topological coordinatization

The core question we’re working with right now is this:
Given a space (point cloud) X, and a (persistent) view of , can we use some map to generate a map inducing that map?

Homological Inclusion-Exclusion and the Mayer-Vietoris sequence

• January 9th, 2009

This blogpost is inspired to a large part by comments made by Rob Ghrist, in connection to his talks on using the Euler characteristic integration theory to count targets detected by sensor networks.

He pointed out that the underlying principle inducing the rule

goes under many names, among those \emph{Inclusion-Exclusion}, favoured among computer scientists (and combinatoricists). He also pointed out that the origin of this principle is the Mayer-Vietoris long exact sequence

In this blog post, I’d like to give more meat to this assertion as well as point out how the general principle of Inclusion-Exclusion for finite sets follows immediately from Mayer-Vietoris.

Inclusion-Exclusion, and the passage from two sets to many

The basic principle of Inclusion-Exclusion says that if we have two sets, and , then the following relationship of cardinalities holds:

Cup products in simplicial cohomology

• September 12th, 2008

This post is a walkthrough through a computation I just did – and one of the main reasons I post it is for you to find and tell me what I’ve done wrong. I have a nagging feeling that the cup product just plain doesn’t work the way I tried to make it work, and since I’m trying to understand cup products, I’d appreciate any help anyone has.

I’ve picked out the examples I have in order to have two spaces with the same Betti numbers, but with different cohomological ring structure.

Sphere with two handles

I choose a triangulation of the sphere with two handles given the boundary of a tetrahedron spanned by the nodes a,b,c,d and the edges be, ef, bf and cg, ch, gh spanning two triangles.

We get a cochain complex on the form

with the codifferential given as

and

R and topological data analysis

• August 23rd, 2008

This is extremely early playing around. It touches on things I’m going to be working with in Stanford, but at this point, I’m not even up on toy level.

We’ll start by generating a dataset. Essentially, I’ll take the trefolium, sample points on the curve, and then perturb each point ever so slightly.

As a result, we get a dataset that looks like this:

So, let’s pick a sample from the dataset. What I’d really want to do now would be to do the witness complex construction, but I haven’t figured enough out about how R ticks to do quite that. So we’ll pick a sample and then build the 1-skeleton of the Rips-Vietoris complex using Euclidean distance between points. This means, we’ll draw a graph on the dataset with an edge between two sample points whenever they are within ε from each other.

Building my academic persona

• January 18th, 2008

http://arxiv.org/abs/0707.1637
Just got accepted for publication in the Journal of Homotopy and Related Structures.

Damn, this feels good!

High school topology restarting

• November 16th, 2007

Today, I told my two bright students about abstract and geometric simplicial complexes, about the boundary map and the chain complex over a ring R associated with a simplicial complex Δ, and assigned them reading out of Hatcher’s Algebraic Topology.

The next couple of weeks will be spent doing homology of simplicial complexes, singular homology, equivalence of the two, neat things you can do with them; and then we’ll start moving towards a Borsuk-Ulam-y topological combinatorics direction.

I might end up pulling combinatorics papers from my old “gang” in Stockholm on graph complexes, and graph property complexes, and poke around those with them.

Progress

• September 26th, 2007

dynkin:~/magma> magma
Magma V2.14-D250907   Wed Sep 26 2007 13:19:51 on dynkin   [Seed = 1]
Type ? for help.  Type -D to quit.

Loading startup file "/home/mik/.magmarc"

> Attach("homotopy.m");
> Attach("assoc.m");
> Aoo := ConstructAooRecord(DihedralGroup(4),10);
> S := CohomologyRingQuotient(Aoo`R);
> CalculateHighProduct(Aoo,[x,y,x,y]);
z
> exit;
Total time: 203.039 seconds, Total memory usage: 146.18MB

And this is one major reason for the lack of updates recently.

Today seems to be a day for posting…

• July 12th, 2007

ComplexZeta asked me about the origins of my intuitions for homological algebra in my recent post. The answer got a bit lengthy, so I’ll put it in a post of its own.

I find Weibel very readable – once the interest is there. It’s a good reference, and not as opaque as, for instance, the MacLane + Hilton-Stammbach couplet can be at points.

The interest, however, is something I blame my alma mater for. Once upon a time, Jan-Erik Roos went to Paris and studied with Grothendieck. When he got back, he got a professorship at Stockholm University without having finished his PhD. He promptly made sure that nowadays (when he’s an Emeritus stalking the halls) there is not a single algebraist at Stockholm University without some sort of intuition for homological algebra.

The why and the what of homological algebra

• July 12th, 2007

I seem to have become the Goto-guy in this corner of the blogosphere for homological algebra.

Our beloved Dr. Mathochist just gave me the task of taking care of any readers prematurely interested in it while telling us all just a tad too little for satisfaction about Khovanov homology.

And I received a letter from the Haskellite crowd – more specifically from alpheccar, who keeps on reading me writing about homological algebra, but doesn’t know where to begin with it, or why.

I have already a few times written about homological algebra, algebraic topology and what it is I do, on various levels of difficulty, but I guess – especially with the carnival dry-out I’ve been having – that it never hurts writing more about it, and even trying to get it so that the non-converts understand what’s so great about it.

So here goes.

Going to T’bilisi

• May 25th, 2007

In about 23 hours, I’ll step on to the train in Jena, heading for T’bilisi, Georgia.

On Monday, I’ll give a talk on my research into -structures in group cohomology. If you’re curious, I already put the slides up on the web.

I’ll try to blog from T’bilisi, but I don’t know what connectivity I’ll have at all.

Young Topology: The fundamental groupoid

• May 4th, 2007

Today, with my bright topology 9th-graders, we discussed homotopy equivalence of spaces and the fundamental groupoid. In order to get the arguments sorted out, and also in order to give my esteemed readership a chance to see what I’m doing with them, I’ll write out some of the arguments here.

I will straight off assume that continuity is something everyone’s comfortable with, and build on top of that.

Homotopies and homotopy equivalences

We say that two continuous maps, f,g:X→Y between topological spaces are homotopical, and write , if there is a continuous map such that H(x,0)=f(x) and H(x,1)=g(x). This captures the intuitive idea of step by step nudging one map into the other in formal terms.

Two spaces X,Y are homeomorphic if there are maps , such that and .

Two spaces X,Y are homotopy equivalent if there are maps , such that and .

Modular representation theory – when Maschke breaks down

• April 21st, 2007

This post is dedicated to Janine Kühn and her Proseminar-lecture.

We had, in my first representation theory post, a mention of Maschke’s theorem. This states that if the characteristic of our field doesn’t divide the group order, then simple and irreducible mean the same thing.

Now, obviously, the actual proof you normally see first deals with a construction that works for when the characteristic doesn’t divide the group order – which uses 1/|G| at one point. So, what happens when this is impossible to work with? When the conditions of Maschke simply do not hold?

The very simplest answer is that then we can get modules that are glued together by simple modules with some meshing. Such that they aren’t direct sums any more. The ways we can glue together modules are through extensions – i.e. we can glue together A,C by forming a short exact sequence
0 → C → B → A → 0
and we’ll have that B is a module such that B/C=A. Now, the typical such module is the direct sum of A and C – and if Maschke holds, this is indeed all there is.

D₈ revisited

• February 7th, 2007

I have previously calculated the A_∞-structure for the cohomology ring of D₈. Now, while trying to figure out how to make my work continue from here, I tried working out what algebra this would have come from, assuming that I can adapt Keller’s higher multiplication theorem to group algebras.

A success here would be very good news indeed, since for one it would indicate that such an adaptation should be possible, and for another it would possibly give me a way to lend strength both to the previous calculation and to a conjecture I have in the calculation of group cohomology with A_∞ means.

So, we start. We recover, from the previous post, the structure of the cohomology ring as k[x,y,z]/(xy), with x,y in degree 1, and z in degree 2. Furthermore, we have a higher operation, m₄, with m₄(x,y,x,y)=m₄(y,x,y,x)=z.

An A_∞-structure on the cohomology of D₈

• November 30th, 2006

As a first unknown (kinda, sorta, it still falls under the category of path algebra quotients treated by Keller) A_∞-calculation, I shall find the A_∞-structure of .

To do this, I fix the group algebra

and the cohomology ring

with ,

Furthermore, we pick a canonical nice resolution P, continuing the one I calculated previously. This has the i:th component Λⁱ⁺¹, and the differentials looking like

for differentials starting in odd degree, and

for differentials starting in even degree. The first few you can see on the previous calculation, or if you don’t want to bother, they are

Now, armed with this, we can get cracking. By lifting, we get canonical representating chain maps for x,y,z described, loosly, by the following:

x takes an element in , keeps the first, third, et.c. elements and throws out the even ordered elements; so
For an element in , the last element gets extra treatment, so

For the lowest degrees, we also have

A_∞-algebras and group cohomology

• November 23rd, 2006

In which the author, after a long session sweating blood with his advisor, manages to calculate the A_∞-structures on the cohomology algebras and .

We will find the A_∞-structures on the group cohomology ring by establishing an A_∞-quasi-isomorphism to the endomorphism dg-algebra of a resolution of the base field. We’ll write m_i for operations on the group cohomology, and μ_i for operations on the endomorphism dg-algebra. The endomorphism dg-algebra has μ₁=d and μ₂=composition of maps, and all higher operations vanishing, in all our cases.

Elementary abelian 2-group

Let’s start with the easy case. Following to a certain the notation used in Dag Madsen’s PhD thesis appendix (the Canonical Source of the A_∞-structures of cyclic group cohomology algebras), and the recipe given in A-infinity algebras in representation theory, we may start by stating what we know as we start:

A_∞ for the layman

• November 7th, 2006

I recently had reason to describe my PhD work in complete laymans terms while writing letters to my grandparents. This being a good thing to do in order to digest your ideas properly, I thought I might try and write it up here as well.

It will, however, push through some 100-odd years of mathematical development rather swiftly. Try to keep up – I will keep it as light as I can while not losing what I want to say.

Algebra

In the 19th century, a number of different mathematical efforts ended up using more or less precisely the same structures, though not really recognizing that they were the same. This recognition came with Cayley, who first brought the first abstract definition of a group.

A group is a set G of elements, with a binary operation *, such that the following relations hold:

A-infinity and Hochschild cocycles

• October 20th, 2006

This blogpost is a running log of my thoughts while reading a couple of papers by Bernhard Keller. I recommend anyone reading this and feeling interest to hit the arXiv and search for his introductions to A_∞-algebras. Especially math.RA/9910179 serves as a basis for this post.

If you do enough of a particular brand of homotopy theory, you’ll sooner or later encounter algebras that occur somewhat naturally, but which aren’t necessarily associative as such, but rather only associative up to homotopy. The first obvious example is that of a loop space, viewed as a group: associativity only holds after you impose equivalence between homotopic loops.

Carry bits and group cohomology

• July 27th, 2006

Got treated today to a really nice workout in group cohomology; most of which is well worth sharing, since seeing it done once gave me a lot of insight.

So, if we pick and view it as the set 0,1,2,3,4,5,6,7,8,9 and with the group operation given by a*b = a+b % 10, then one standard 2-cocycle is the function

That this actually does form a cocycle would be the same as requiring
f(b,c)-f(a*b,c)+f(a,b*c)-f(a,b)=0
or regrouped
f(a*b,c)+f(a,b)=f(a,b*c)+f(b,c)
which is to say that the number of carry bits generated when adding three digits does not depend on associativity.

This cocycle classifies the group extension

with the first map taking and the second taking

Now, this is a nontrivial extension – which is equivalent to it not being a coboundary – by the following calculation:
Suppose f=dg. Then f(a,b)=g(a)+g(b)-g(a*b). So, since f(0,0)=0, we get g(0)-g(0)+g(0)=0, so g(0)=0. For any b≤8, we also get 0=f(1,b)=g(b)-g(b+1)+g(1), so g(b+1)=g(b)+g(1) and thus by induction, g(b)=bg(1) for all 0≤b≤9.
But, now, 1=f(1,9)=g(9)-g(0)+g(1)=10g(1)=0, which is a contradiction.

Triangulated categories

• July 17th, 2006

One predominant tendency in the algebra/category theory camp is to seek out the minimal set of conditions needed to be able to perform a certain technique, and then codifying this into a specific axiomatic system. Thus, you only need to verify the axioms later on in order to get everything else for free.

One such system is the theory of triangulated categories. This pops up in homological algebra; where you like to work with Tor and Ext – both of which turn out to be derived functors, generalizing the tensor product and the homomorphism set respectively. With the construction of the derived category, we can find a category, in which the tensor product in that category is our Tor, and the hom sets is our Ext.

Group cohomology revisited: Quaternionic unit group

• June 20th, 2006

In a previous installment, we calculated with some amount of success. For that post, I said that I was going to calculate the cohomologies of and of by hand – and I’ve been at it for the latter group since then. With some help from my advisor – mainly with executing the obvious algorithms far enough that I get decent material to work with – I know have it.

The group

So, for starters, we need a presentation of such that we can work well with it. We all know that . So due to ij=-k and , we can just pick any two of the i,j,k and call them x and y. Then , and iji=ik=j so xyx=y. This gives us the presentation

More group cohomology

• June 8th, 2006

My advisor told me to go hit and as my next two cohomology calculation projects; try to do them with resolutions by hand so that I get a feeling for what’s going on. After failing spectacularily both at getting a resolution of with , he walked me through his Shiny! Gröbner base method to get resolutions with free modules over finite p-group algebras. Armed with the minimal resolution, I sat down and started hunting products; and finally found the cohomology ring.

Or … to be exact, I found and then peeked into Carlson, et.al. for the Big List of 2-group cohomologies to see that all interesting stuff happens in anyway.

So for the benefit of any and all readers who want to see what it looks like, I’m going to walk through it again here. Nonono, you don’t need to flee all of you – just skip this entry if it’s that scary!

First academic publication!

• June 6th, 2006

I just received in the mail a bunch of prints. Of my article “Computation of Poincaré-Betti series for monomial rings”, produced from my Master’s thesis for the “School and workshop on computational algebra for algebraic geometry and statistics” in Torino 2004. It is now being published in the Rendiconti di Istituto Matematico di Universita di Trieste, on pages 85-94 of Vol. XXXVII (2005).

Damn, it feels good. Reviewed and everything. If you’re curious, my manuscript can be found at http://math.su.se/~mik/torino.pdf or at the arXiv as math.AC/0502348.

My first group cohomology – did I screw up?

• May 21st, 2006

I thought the seminar on tuesday would possibly benefit from something not very often seen – explicit examples. So I started working through one. I wanted to calculate and give explicitly in a series of ways the product structure – as Yoneda splices, as chain map compositions and as cup products.

Now, has a very nice resolution as a -module – all cyclic (finite) groups have canonically a really cute minimal resolution – given by

with the last map taking and the other maps alternatingly being multiplication with and with .

So this gives as a nice projective (in fact: free) resolution to work with. We now can observe that since any map has to respect the group action, which is trivial on , and so any map is determined by its value on 1. Thus we get the sequence of dual modules

Question for the mathematical audience

• April 20th, 2006

I have now been staring at this particular sentence for way too long, and thus will start using any and all communication lines I can find to get assistance. Either I’m being way too stupid, or the author neglects to mention some salient detail.

Setup: is a group homomorphism, , . can be given the structure of a -module by pulling back through , i.e. we define for and .

So far it’s all crystal clear for me. However, it then turns out that we’re highly interested in using a morphism and I cannot for the life of me find out how such beasts are guaranteed to exist. If it where , I wouldn’t have any problems with it; but then the stuff I need/want to do with it don’t work out.

Report from Villars (5 in a series)

• March 12th, 2006

For the last two half-days of the conference, I managed to take a break in skiing precisely when the conditions were at their very worst; with sight down to a few meters and angry winds. Miles Gould and Arne Weiner, however, managed to sit in a chair lift that kept stopping every 5 meters – AND they managed to break a T-bar lift. Suddenly the rope broke, they told me, and they had to ski down to the warden with the T-bar in the hand.

First out in this mathematical expose, though, is André Henriques, talking about

An operad coming from representation theory

There is a way to connect to a finite Lie algebra first it’s universal enveloping algebra and quantum groups . From representations of , one path leads on over braided tensor products to braided tensor categories. Such categories are described by operads, which occur in the study of Gerstenhaber algebras and their homology.

Report from Villars (4 in a series)

• March 8th, 2006

I haven’t been able to get around to skiing since the last update – I may, or may not, go out in the slopes after this updates. The weather is growing warmer and wetter; and doesn’t really invite to skiing as it previously did.

However, we have had more talks. First out, yesterday evening, was Pascal Lambrechts

Coformality of the little ball operad and rational homotopy types of spaces of long knots

The theme of interest for this talk was long knots; i.e. embeddings of into such that outside some finite region in the middle, the embedding agrees with the trivial embedding . The space of all such is denote , and the item of study is more precisely the rational homology and rational homotopy of the fiber of the inclusion of into the space of all immersions of into .

Report from Villars (2 in a series)

• March 6th, 2006

So we hit the pistes during monday morning, those of us who actually already are here. Me, Bruno Vallette (Hi Stockholm!), Arne Weiner, Miles Gould, Paul Eugene Parents and Jonathan Scott, Dev Sinha and Muriel Livernet. Skiing was MARVELOUS. Me, Arne and Miles shot off on our own, and damn did we have a good time.

As I’m writing this, they’re still out there – I went back when the pain in my legs caused tears in my eyes for just turning on the skis. The techniques were solid as concrete. The muscles not so much. It took half an hour in the sauna to get to the point where I actually was able to walk again.

Monads, algebraic topology in computation, and John Baez

• February 21st, 2006

Todays webbrowsing led me to John Baez finds in mathematical physics for week 226, which led me to snoop around John Baez homepage, which in turn led me to stumble across the Geometry of Computation school and conference in Marseilles right now.

This, in turn, leads to several different themes for me to discuss.

Cryptographic hashes

In the weeks finds, John Baez comes up to speed with the cryptographic community on the broken state of SHA-1 and MD-5. Now, this is a drama that has been developing with quite some speed during the last 1-1½ years. It all began heating up seriously early 2005 when Wang, Yin and Yu presented a paper detailing a serious attack against SHA-1. Since then, more and more tangible evidence for the inadvisability of MD-5 and upcoming problems with SHA-1 have arrived – such as several example objects with different contents and identical MD-5 hashes: postscript documents (Letter of Recommendation and Access right granting), X.509 certificates et.c.

Reading Merkulov: Differential geometry for an algebraist (1 in a series)

• February 11th, 2006

I’ll do this in posts and not pages on further thought…

Sergei Merkulov at Stockholm University gives during the spring 2006 a course in differential geometry, geared towards the algebra graduate students at the department. The course was planned while I was still there, and so I follow it from afar, reading the lecture notes he produces.

At this page, which will be updated as I progress, I will establish my own set of notes, sketching at the definitions and examples Merkulov brings, and working out the steps he omits.

Familiar parts in unfamiliar language

Merkulov begins the paper by introducing in swift terms the familiar definitions from topology of topology, continuity, homeomorphisms, homotopy, and then goes on to discuss homotopy groups, and thereby introducing new names for things I already knew. Thus, I give you, for a pointed topological space

Introduction to Algebraic Topology and related topics (I)

• January 1st, 2006

In the spirit of writing some sort of introductory posts to the things related to what I’m about to spend several years thinking and writing about, I thought I’d try to make a (more or less) layman friendly introduction to Homology and Homotopy.

It’s all residing in the realm of Topology. Topology is the field of mathematics, where those aspects of a shape not dependent on distances are studied. Thus rigidity is not interesting, whereas connectivity is. Narrow/thick is not interesting, but what kind of holes the surface has is. The ultimate thing to be said in topology about two objects is that they are homeomorphic, which technically means that there is an isomorphism between the objects in the category of topological spaces; and more comprehensibly means that there are continuous functions between the shapes such that they are each others inverses.

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