Skip to Content »

Michi’s blog » archive for 'Topology'

 Classifying spaces and Haskell

  • May 15th, 2014
  • 3:27 am

I’ve been talking with some topologists lately, and seen interesting constructions. I think these may potentially have some use in understanding Haskell or monads.

Simplicial objects

A simplicial object in a category is a collection of objects $C_n$ together with n maps $d_j:C_n\to C_{n-1}$ and n maps $s_j:C_n\to C_{n+1}$ subject to a particular collection of axioms.

The axioms are modeled on the prototypical simplicial structure: if $C_n$ is taken to be a (weakly) increasing list of integers, $d_j$ skips the $j$th entry, and $s_j$ repeats the $j$th entry. For the exact relations, I refer you to Wikipedia.

Classifying spaces

Take a category C. We can build a simplicial object $C_*$ from C by the following construction:

$C_n$ is all sequences $c_0\xrightarrow{f_0}c_1\xrightarrow{f_1}\dots\xrightarrow{f_{n-1}}c_n$ of composable morphisms in $C$.
$d_j$ composes the $j$th and $j+1$st morphisms. $d_0$ drops the first morphism and $d_n$ drops the last morphism.
$s_j$ inserts the identity map at the $j$th spot.

 Topology of Politics — Popular topology lecture

  • September 21st, 2012
  • 1:39 pm

I will be speaking on my research into topological data analysis for political data sets at the University of Edinburgh, at 4:10pm on 15th October, in Room 6206, James Clerk Maxwell Building.

Data Analysis on politics data

Data analysis has played a growing role in politics for many years now; analyzing polling data to predict outcomes of elections is perhaps the most well-known application.

A different approach that has gotten more and more traction lately is to analyze the voting behaviour of elected representatives as a way to understand the inner workings of parliaments, and to monitor the elected representatives to make sure they behave as they once promised. Sites like GovTrack and VoteView bring machine learning and data analysis tools to the citizens, and illustrate and visualize the groupings and behaviour in political administration.

 ATMCS 5 in Edinburgh

  • March 22nd, 2012
  • 7:20 am

ATMCS 5 – Algebra and Topology, Methods, Computing, and Science

Second Announcement

This meeting will take place in the period July 2-6, at the ICMS in Edinburgh, Scotland. The theme will be applications of topological methods
in various domains. Invited speakers are

J.D. Boissonnat (INRIA Sophia Antipolis) (Confirmed)
R. Van de Weijgaert (Groningen) (Confirmed)
N. Linial (Hebrew University, Jerusalem) (Confirmed)
S. Weinberger (University of Chicago) (Confirmed)
S. Smale (City University of Hong Kong) (Confirmed)
H. Edelsbrunner (IST, Austria) (Confirmed)
E. Goubault (Commissariat à l’énergie atomique, Paris) (Confirmed)
S. Krishnan (University of Pennsylvania) (Confirmed)
M. Kahle (The Ohio State University) (Confirmed)
L. Guibas (Stanford University) (Confirmed)
R. Macpherson (IAS Princeton) (Tentative)
A. Szymczak (Colorado School of Mines) (Confirmed)
P. Skraba/ M. Vejdemo-Johansson (Ljubljana/St. Andrews) (Confirmed)
Y. Mileyko (Duke University) (Confirmed)
D. Cohen (Louisiana State)
V. de Silva (Confirmed)
There will be opportunities for contributed talks. Titles and abstracts should be send to Gunnar Carlsson at gunnar@math.stanford.edu.

 Geometric realization of simplicial sets

  • March 1st, 2011
  • 9:27 pm

This post is an expansion of all the details I did not have a good feeling for when I started with for page 7 of Goerss-Jardine, where the geometric realization of simplicial sets is introduced.

The construction works by constructing a few helpful categories, and using their properties along the way. Especially after unpacking the categorical results G-J rely on, there are quite a few categories floating around. I shall try to be very explicit about which category is which, and how they work.

As we recall, simplicial sets are contravariant functors from the category \mathbf{\Delta} of ordinal numbers to the category of sets. We introduce the simplex category \mathbf{\Delta}\downarrow X of a simplicial set X with objects (simplices) given by maps \sigma:\Delta^n\to X and a map from \sigma to \tau being given by a map f in \mathbf{\Delta} such that \sigma = \tau f.

 The Topology of Politics

  • January 4th, 2011
  • 8:09 pm

This is a typed up copy of my lecture notes from the seminar at Linköping, 2010-08-25. This is not a perfect copy of what was said at the seminar, rather a starting point from which the talk grew.

In my workgroup at Stanford, we focus on topological data analysis — trying to use topological tools to understand, classify and predict data.

Topology gets appropriate for qualitative rather than quantitative properties; since it deals with closeness and not distance; also makes such approaches appropriate where distances exist, but are ill-motivated.

These approaches have already been used successfully, for analyzing

  • physiological properties in Diabetes patients
  • neural firing patterns in the visual cortex of Macaques
  • dense regions in \mathbb{R}^9 of 3×3 pixel patches from natural (b/w) images
  • screening for CO2 adsorbative materials

 Coordinatization with hom complexes

  • December 7th, 2009
  • 10:55 pm

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 X\to\mathbb R or possibly some X\to\mathbb C, 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 H_*(X), can we use some map H_*(X)\to H_*(Y) to generate a map X\to Y inducing that map?

 1-manifolds and curves

  • May 2nd, 2009
  • 12:10 pm

I have been painfully remiss in keeping this blog up and running lately. I wholeheartedly blame the pretty intense travel schedule I’ve been on for the last month and a half.

To get back into the game, I start things off with a letter from a reader. Rodolfo Medina write:

Hallo, Michi:

surfing around in internet, looking for an answer to my question, I fell into
your web site.

I’m looking for an answer to the following question:

my intuitive idea is that a one-dimensional connected topological submanifold
of a topological space S should necessarily be the codomain of a curve (if we
define a curve to be a continuous map from an R interval into a topological
space).

Conversely, the codomain of an injective curve, defined in an open R interval,
should necessarily be a one-dimensional topological submanifold of S.

 Applied knot theory

  • March 16th, 2009
  • 2:41 am

Tech note: All figures herewithin are produced in SVG. If you cannot see them, I recommend you figure out how to view SVGs in your browser.

A few weeks ago, my friend radii was puzzling in his server hall. He asked if it was possible to prove that what he wanted to do was impossible, or if he had to remain with his gut feeling. I asked him, and got the following explanation:

He had two strands of something ropelike, both fixed at large furnishings at one end, and fixed in a fixed sized loop at the other. He wanted to take these, and link them fast to each other in this fashion:

I started thinking about the problem, and am now convinced I can prove the impossibility he asked for by basic techniques of knot theory. The argument is what I’ll fill this blog post about.

 Homological Inclusion-Exclusion and the Mayer-Vietoris sequence

  • January 9th, 2009
  • 10:44 pm

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
\chi(A\cup B) = \chi(A)+\chi(B)-\chi(A\cap B)
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
\cdots\to H_{n}(A\cap B)\to H_{n}(A)\oplus H_{n}(B)\to H_{n}(A\cup b)\to\cdots

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, A and B, then the following relationship of cardinalities holds:
|A\cup B| = |A| + |B| – |A\cap B

 More on Lichtenstein

  • October 28th, 2008
  • 11:03 pm

It turns out that there is even more to say on the communes of Lichtenstein.

First of all, there is a 5-clique in the communal graph, as Brian Hayes pointed out. But there are two different excluded subgraphs for planarity – so if we aren’t looking specifically for the chromatic number, but rather how this graph fails to be a “normal” land map, we might want to see whether it realizes BOTH.

It turns out that it does.

The following are two highlighted versions of the Liechtenstein communal graph.


The embedded K5 with edges in blue.


The embedded K33 with blue and red vertices.

 Cup products in simplicial cohomology

  • September 12th, 2008
  • 1:12 am

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
0 \to \mathbb{Z}^8 \to \mathbb{Z}^{12} \to \mathbb{Z}^4 \to 0
with the codifferential given as

\begin{pmatrix}
1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & -1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & -1 & 0 & 0\\
0 & 0 & 1 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & -1 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & -1\\
0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\
\end{pmatrix}
and

\begin{pmatrix}
1 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
\end{pmatrix}

 R and topological data analysis

  • August 23rd, 2008
  • 4:38 pm

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.

idx <- 1:2000
theta <- idx*2*pi/2000
a <- cos(3*theta)
x <- a*cos(theta)
y <- a*sin(theta)
xper <- rnorm(2000)
yper <- rnorm
xd <- x + xper/100
yd <- y + yper/100
cd <- cbind(xd,yd)
 

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

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.

 Restarting high school topology

  • May 21st, 2008
  • 5:41 pm

My two high-school kids came by today. We’ve been trying to get a new teaching session together since early February, but they had a hell of a time all through February, and all our appointments ended up canceled with little or no notice; and then I spent March and April on tour.

We pressed on with knot theory. Today, we discussed knot sums, prime knots, knot tabulation, behavior of the one invariant (n-colorability) we know so far under knot sums, Dowker codes, and we got started on Conway codes for knots. Next week, I plan for us to finish up talking about the Conway knot notation, get the connection between rational knots and continued fractions down pat, and start looking into new invariants.

 Tour dates

  • March 6th, 2008
  • 10:57 pm

Edited to add Galway

I’ll be doing a “US tour” in March / April. For the people who might be interested – here are my whereabouts, and my speaking engagements.

I’m booked at several different seminars to do the following:

Title: On the computation of A-infinity algebras and Ext-algebras
Abstract:

For a ring R, the Ext algebra Ext_R^*(k,k) carries rich information about the ring and its module category. The algebra Ext_R^*(k,k) is a finitely presented k-algebra for most nice enough rings. Computation of this ring is done by constructing a projective resolution P of k and either constructing the complex Hom(P_n,k) or equivalently constructing the complex Hom(P,P). By diligent choice of computational route, the computation can be framed as essentially computing the homology of the differential graded algebra Hom(P,P).

Being the homology of a dg-algebra, Ext_R^*(k,k) has an induced A-infinity structure. This structure, has been shown by Keller and by Lu-Palmieri-Wu-Zhang, can be used to reconstruct R from
Ext_R^{\leq 2}(k,k).

 My topology students move into knot theory

  • February 1st, 2008
  • 2:27 pm

So, here’s the plan for my 10th grade topology students.

Today, we’ll abandon algebraic topology completely, and instead go into knot theory. I’ll want to discuss what we mean by a knot (embedding of S^1 in S^3), what we mean by a knot deformation (thus introducing isotopies while we’re at it) and the Reidemeister moves. Also we’ll discuss knot invariants – and their use analogous to topological invariants.

Later on, we’ll continue with other invariants; definitely including the Jones polynomial, and possibly even covering Khovanov homology. One possible end report would be to explain a bunch of knot invariants and show using examples how these have different coarseness.

Edited to add: I got myself some damn smart students. They figured out the Reidemeister moves on their own – as well as minimal crossing number in a projection being highly relevant – with basically no prompting from me. I’m impressed.

 Building my academic persona

  • January 18th, 2008
  • 4:26 pm

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
  • 4:34 pm

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
  • 4:25 am
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
  • 8:20 pm

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.

So, my MSc advisor, J

 The why and the what of homological algebra

  • July 12th, 2007
  • 7:35 pm

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
  • 1:37 pm

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 A_\infty-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
  • 3:26 pm

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 f\simeq g, if there is a continuous map H\colon X\times[0,1]\to Y 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 f\colon X\to Y,f^{-1}\colon Y\to X such that ff^{-1}=\operator{Id}_Y and f^{-1}f=\operator{Id}_X.

Two spaces X,Y are homotopy equivalent if there are maps f\colon X\to Y,f^{-1}\colon Y\to X such that ff^{-1}\simeq\operator{Id}_Y and f^{-1}f\simeq\operator{Id}_X.

 Modular representation theory – when Maschke breaks down

  • April 21st, 2007
  • 1:43 pm

This post is dedicated to Janine K

 Bright students and topology

  • March 2nd, 2007
  • 3:39 pm

Today, I started an experiment together with the local specialised secondary school. I’ll be taking care of two of their brightest students, meeting them roughly once a week, and taking them on a charge through algebraic topology. At the far end shimmers knot theory and other funky applications; and on the way there, I hope for many interesting spinoffs and avenues.

They got, today, Armstrong’s Basic Topology, and an extract from the German topology book by J

 D8 revisited

  • February 7th, 2007
  • 6:36 pm

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
  • 2:28 pm

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 H^*(D_8,\mathbb F_2).

To do this, I fix the group algebra
\Lambda=\mathbb F_2[a,b]/(a^2,b^2,abab+baba)
and the cohomology ring
\Gamma=\mathbb F_2[x,y,z]/(xy)
with x,y\in\Gamma_1, z\in\Gamma_2

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
\begin{pmatrix}
a&0&bab&0&0&0\\
0&b&0&aba&0&0\\
0&0&a&0&bab&0\\
0&0&0&b&0&aba\\
0&0&0&0&a&b
\end{pmatrix}
for differentials starting in odd degree, and
\begin{pmatrix}
a&0&bab&0&0\\
0&b&0&aba&0\\
0&0&a&0&bab\\
0&0&0&b&aba
\end{pmatrix}
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
\partial_0=\begin{pmatrix}a&b\end{pmatrix}
\partial_1=\begin{pmatrix}a&0&bab\\0&b&aba\end{pmatrix}
\partial_2=\begin{pmatrix}a&0&bab&0\\0&b&0&aba\\0&0&a&b\end{pmatrix}

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 \Lambda^{2k}, keeps the first, third, et.c. elements and throws out the even ordered elements; so x\cdot(a_1,a_2,a_3,a_4,a_5,a_6)=(a_1,0,a_3,0,a_5)
For an element in \Lambda^{2k+1}, the last element gets extra treatment, so
x\cdot(a_1,a_2,a_3,a_4,a_5,a_6,a_7)=(a_1,0,a_3,0,a_5,ab\cdot a_7)
For the lowest degrees, we also have
x_0 = \begin{pmatrix}1&0\end{pmatrix}
x_1 = \begin{pmatrix}1&0&0\\0&0&ab\end{pmatrix}
x_2 = \begin{pmatrix}1&0&0&0\\0&0&1&0\end{pmatrix}

 A-algebras and group cohomology

  • November 23rd, 2006
  • 3:48 pm

In which the author, after a long session sweating blood with his advisor, manages to calculate the A-structures on the cohomology algebras H^*(C_4,\mathbb F_2) and H^*(C_2\times C_2,\mathbb F_2).

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 mi for operations on the group cohomology, and μi 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
  • 4:18 pm

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
  • 3:15 pm

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
  • 4:05 pm

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 \mathbb Z/10 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
f(a,b) = \begin{cases}1&a+b\geq10\\0&\text{else}\end{cases}
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
0 \to \mathbb Z/10 \to \mathbb Z/100 \to \mathbb Z/10 \to 0
with the first map taking a+10\mathbb Z\mapsto 10a+100\mathbb Z and the second taking b+100\mathbb Z\mapsto b+10\mathbb Z

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
  • 3:47 pm

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
  • 9:54 am

In a previous installment, we calculated H^*(D_8) with some amount of success. For that post, I said that I was going to calculate the cohomologies of D_8 and of U(\mathbb H) 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 \mathbb H such that we can work well with it. We all know that \mathbb H=\langle i,j,k|ij=-k, jk=-i, ki=-j, ijk=i^2=j^2=k^2=-1\rangle. So due to ij=-k and i^2=-1, we can just pick any two of the i,j,k and call them x and y. Then x^4=e, y^2=x^2 and iji=ik=j so xyx=y. This gives us the presentation \mathbb H=\langle x,y|x^4,y^2=x^2,xyx=y\rangle

 More group cohomology

  • June 8th, 2006
  • 10:10 pm

My advisor told me to go hit D_8 and U(\mathbb H) 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 \mathbb F_2 with \mathbb F_2D_8, he walked me through his Shiny! Gr

 First academic publication!

  • June 6th, 2006
  • 3:03 pm

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
  • 1:42 pm

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 H^*(C_8,\mathbb F_2) and give explicitly in a series of ways the product structure – as Yoneda splices, as chain map compositions and as cup products.

Now, \mathbb F_2 has a very nice resolution as a \mathbb F_2C_8-module – all cyclic (finite) groups have canonically a really cute minimal resolution – given by

\to \mathbb F_2C_8\to\mathbb F_2C_8\to\mathbb F_2C_8\to\mathbb F_2\to 0

with the last map taking \epsilon\colon a_0+a_1g+a_2g^2+\dots+a_7g^7\mapsto a_0+a_1+\dots+a_7 and the other maps alternatingly being multiplication with \partial_1\colon 1+g+g^2+\dots+g^7 and with \partial_0\colon g-1.

So this gives as a nice projective (in fact: free) resolution to work with. We now can observe that \Hom_{\mathbb F_2C_8}(\mathbb F_2C_8,\mathbb F_2)=\Hom_{\mathbb F_2C_8}(\mathbb F_2,\mathbb F_2)=\mathbb F_2 since any map has to respect the group action, which is trivial on \mathbb F_2, and so any map is determined by its value on 1. Thus we get the sequence of dual modules

0\to\mathbb F_2\to\mathbb F_2\to\mathbb F_2\to\dots

 Reading Merkulov: Differential geometry for an algebraist (4 in a series)

  • April 24th, 2006
  • 2:52 pm

Suppose we have a presheaf \mathcal F of abelian groups over M and pick a point x. On the collection of all abelian groups defined over some neighbourhood of x (disjoint union) we put an equivalence relation which identifies f\in\mathcal F(U) and g\in\mathcal F(V) precisely if there is some open W in the intersection where f and g coincide. (or more precisely, their restrictions coincide). The set of equivalence classes turns out to be an Abelian group \mathcal F_x called the stalk of the presheaf \mathcal F at x.

So, with more fluff introduced, the stalk is all the elements in the presheaf that are defined above any neighbourhood of the point, and counted as the same if they seem to be.

For an open set U and a point x\in U there is a canonical group morphism \rho_x:\mathcal F(U)\to\mathcal F_x which sends an element f\in\mathcal F(U) to its equivalence class. This image is the germ of f at x.

 Question for the mathematical audience

  • April 20th, 2006
  • 3:44 pm

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: \phi\colon G^\prime\to G is a group homomorphism, A\in kG-\operator{mod}, A^\prime\in kG^\prime-\operator{mod}. A can be given the structure of a kG^\prime-module by pulling back through \phi, i.e. we define g^\prime a:=\phi(g^\prime)a for g^\prime\in G^\prime and a\in A.

So far it’s all crystal clear for me. However, it then turns out that we’re highly interested in using a morphism f\in\operator{Hom}_G(A,A^\prime) and I cannot for the life of me find out how such beasts are guaranteed to exist. If it where f\in\operator{Hom}_{G^\prime}(A,A^\prime), 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
  • 12:12 pm

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 \mathfrac g first it’s universal enveloping algebra U\mathfrac g and quantum groups U_q\mathfrac g. From representations of U_q\mathfrac g, one path leads on over braided tensor products to braided tensor categories. Such categories are described by E_2 operads, which occur in the study of Gerstenhaber algebras and their homology.

 Report from Villars (4 in a series)

  • March 8th, 2006
  • 2:09 pm

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 \mathbb R into \mathbb R^d such that outside some finite region in the middle, the embedding agrees with the trivial embedding t\mapsto(t,0,0,\dots,0). The space of all such is denote \mathcal L, and the item of study is more precisely the rational homology and rational homotopy of the fiber of the inclusion of \mathcal L into the space of all immersions of \mathbb R into \mathbb R^d.

 Report from Villars (2 in a series)

  • March 6th, 2006
  • 11:07 pm

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
  • 11:06 am

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 (3 in a series)

  • February 13th, 2006
  • 1:55 pm

Since I cannot concentrate anyway, here comes the third installment of my reading Merkulov. We now get to the funky stuff – introducing sheaves and getting to what should be at the very start of basically any modern geometry course. At least if I am to believe the geometers I know.

So, let’s launch straight to it. A presheaf \mathcal F on the topological space \mathcal M is just a contravariant functor from Top(\mathcal M) to Ab, where Top(\mathcal M) is the category of open subsets of \mathcal M with morphisms being inclusion maps.

So that’s the one-line definition. But what does it mean?
Well, a functor is a map between categories that takes objects to objects and morphisms to morphisms. So we have that \mathcal F(U) is an abelian group for any open set U\subset\mathcal M. For such a map to really be a functor, it has to be sane in a rather precisely defined sense: namely morphism composition should still be associative and the identity endomorphism on a group shouldn’t actually, ya’know, change the morphisms before or after it.
For the functor to be contravariant means precisely that for f:U\to V we get \mathcal F(f):\mathcal F(V)\to\mathcal F(U) – all arrows reverse by application of the functor.

 Reading John M. Lee – Introduction to Smooth Manifolds (1 of 1)

  • February 12th, 2006
  • 11:14 pm

If I’m going to take this renewed interest in trying to understand differential geometry more seriously, I might as well read more than one source on it. So, I’ll start a sequence of posts on this book as well.

Just as Merkulov, Lee starts with a short definition of what a topological manifold is, including definitions for the terms needed for the treatment.

Definition

An n-dimensional topological manifold is a second countable Haussdorff space of local Euclidean dimension n.

Next, Lee goes on to define coordinate charts. I won’t repeat the treatment, since he doesn’t really bring anything Merkulov hasn’t talked about in some manner or other. Atlases, smooth structures and equivalence of atlases also merits some treatment.

 Reading Merkulov: Differential geometry for an algebraist (2 in a series)

  • February 11th, 2006
  • 11:03 pm

So, in the last installment, we got to know smooth manifolds and charts, atlases and some nice topological tricks and tweaks. For this round, we follow Merkulov onward, and pretty soon stumble across category theory and sheaves. The notes I’m following here are from the link on Merkulov’s website. It starts, however, with a nice discussion of temperatures in archipelagos. Go read it – I imagine I’m almost comprehensible at that part of the text.

A map from a subset of a smooth manifold to \mathbb R is called a smooth function on the subset if for every x in the subset and a coordinate chart at x, the n-to-1 variable function f\circ\phi^{-1} is smooth at the point \phi(x).

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

  • February 11th, 2006
  • 2:23 pm

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 M

 Borsuk-Ulam and West Wing

  • February 7th, 2006
  • 11:56 pm

In West Wing 4×20, CJ states that there are two antipodal points with identical temperature on the earth, as an argument why it should be possible to imagine that an egg could stand on its end at the spring equinox. This particular plotline also has her most emphatically claiming that this should not be possible at the autumn equinox. Why this particular physics is complete idiocy will be left as an exercise to the interested reader, and instead I will focus on the other claim.

This is, in fact, true. It’s a corollary to one of the prettiest theorem conglomerates I have ever seen: the Borsuk-Ulam theorem(s). Alas, I haven’t got my sources on it here at the moment, so I won’t give you the deep indepth survey I want to give; but I do want to give a bit of overview as to why the claim CJ supports her insane theory with is actually true.

 Introduction to Algebraic Topology and related topics (I)

  • January 1st, 2006
  • 12:20 am

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.