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 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.

 Species, derivatives of types and Gröbner bases for operads

  • December 18th, 2010
  • 2:05 am

This is a typed up copy of my lecture notes from the combinatorics seminar at KTH, 2010-09-01. This is not a perfect copy of what was said at the seminar, rather a starting point from which the talk grew.

In some points, I’ve tried to fill in the most sketchy and un-articulated points with some simile of what I ended up actually saying.

Combinatorial species started out as a theory to deal with enumerative combinatorics, by providing a toolset & calculus for formal power series. (see Bergeron-Labelle-Leroux and Joyal)

As it turns out, not only is species useful for manipulating generating functions, btu it provides this with a categorical approach that may be transplanted into other areas.

For the benefit of the entire audience, I shall introduce some definitions.

Definition: A category C is a collection of objects and arrows with each arrow assigned a source and target object, such that

 [Stanford] MATH 198: Category Theory and Functional Programming

  • August 29th, 2009
  • 7:19 am

Category theory, with an origin in algebra and topology, has found use in recent decades for computer science and logic applications. Possibly most clearly, this is seen in the design of the programming language Haskell – where the categorical paradigm suffuses the language design, and gives rise to several of the language constructs, most prominently the Monad.

In this course, we will teach category theory from first principles with an eye towards its applications to and correspondences with Haskell and the theory of functional programming. We expect students to previously or currently be taking CS242 and to have some level of mathematical maturity. We also expect students to have had contact with linear algebra and discrete mathematics in order to follow the motivating examples behind the theory expounded.

Wednesdays at 4.15.

Online notes will appear successively on the Haskell wiki on http://haskell.org/haskellwiki/User:Michiexile/MATH198

 Soliciting advice

  • July 22nd, 2009
  • 3:28 pm

Dear blogosphere,

come this fall, I shall be teaching. My first lecture course, ever.

The subject shall be on introducing Category Theory from the bottom up, in a manner digestible for Computer Science Undergraduates who have seen Haskell and been left wanting more from that contact.

And thus comes my question to you all: what would you like to see in such a course? Is there any advice you want to give me on how to make the course awesome?

The obvious bits are obvious. I shall have to discuss categories, functors, (co)products, (co)limits, monads, monoids, adjoints, natural transformations, the Curry-Howard isomorphism, the Hom-Tensor adjunction, categorical interpretation of data types. And all of it with explicit reference to how all these things influence Haskell, as well as plenty of mathematical examples.

But what ideas can you give me to make this greater than I’d make it on my own?

 Gröbner bases for Operads — or what I did in my vacation

  • May 8th, 2009
  • 6:28 pm

This post is to give you all a very swift and breakneck introduction to Gröbner bases; not trying to be a nice and soft introduction, but much more leading up to announcing the latest research output from me.

Recall how you would run the Gaussian algorithm on a matrix. You’d take the leftmost upmost non-zero entry, divide its row by its value, and then use that row to eliminate anything in the corresponding column.

Once we have the matrix on echelon form, we can then do lots of things with it. Importantly, we can use substitution using the leading terms to do equation system solving.

The starting point for the theory of Gröbner bases was that the same method could be used – with some modification – to produce something from a bunch of polynomials that ends up being as useful as a row-reduced echelon form.

 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

 Dr rer nat, Magna cum laude

  • July 17th, 2008
  • 2:43 pm

After about 5 semesters, one paper, one erratum (submitted to JHRS) and one thesis, and after taking two oral exams and delivering one 30 minute talk on my research, I am now (modulo the week or two it takes to produce my certificate) entitled to the title of doctor rerum naturalium.

Next stop is the topology in computer science workgroup at Stanford, where I have accepted an offer for a postdoc research position up to 3 years (conditional on my good behaviour :-).

 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).

 Introduction to Algebraic Geometry (3 in a series)

  • March 4th, 2008
  • 3:37 pm

I’m going to move on with the identification of geometric objects with functions from these objects down to a field soon enough, but I’d like to spend a little time nailing down the categorical language of this association. Basically, we have two functors I and V going back and forth between two categories. And the essential statement of the last post is that these two functors form an equivalence of categories.

Now, first off in this categorical language, I want to nail down exactly what the objects are. In the category \mathcal{AV}ar_k the objects are solution sets of systems of polynomial equations. And in the category \mathcal{RA}lg_k, the objects are finitely presented Noetherian reduced k-algebras.

The functor V:\mathcal{RA}lg_k\to \mathcal{AV}ar_k acts on objects by sending an algebra R to the solution set of the polynomial equations generating the ideal in a presentation of the algebra.

 Introduction to Algebraic Geometry (2 in a series)

  • February 21st, 2008
  • 10:43 pm

I want to lead this sequence to the point where I am having trouble understanding algebraic geometry. Hence, I won’t take the usual course such an introduction would take, but rather set the stage reasonably quickly to make the transit to the more abstract themes clear.

But that’s all a few posts away. For now, recall that we recognized already that any variety is defined by an ideal, and that intersections and unions of varieties are given by sums and intersections or products of ideals.

This is the first page of what is known as the Algebra-Geometry dictionary. The dictionary is made complete by a pair of reasonably famous theorems. I won’t bother proving them – the proofs are a good chunk of any decent commutative algebra course – but I’ll quote the theorems and discuss why they matter.

 Introduction to Algebraic Geometry (1 in a series)

  • February 21st, 2008
  • 12:33 pm

I’m growing embarrassed by my lack of understanding for the sheaf-theoretic approaches to algebraic (and differential) geometry. I’ve tried to deal with it several times before, and I’m currently reading up on Algebraic Geometry again to fill the void that the finished thesis, soon arriving travels and non-existent job application responses produce.

So, why not learn by teaching? It’s an approach that has been pretty darn good in the past. So I thought I’d write a sequence of posts on algebraic geometry, introducing what it’s supposed to be about and how the main viewpoints develop more or less naturally from the approaches taken.

Varieties

The basic objective of algebraic geometry is to study solution sets to systems of polynomial equations. That is, we take some set f_1,\dots,f_r of polynomials in some polynomial ring k[x_1,\dots,x_n] over some field k. And we write V(f_1,\dots,f_r) for the set of all simultaneous roots to all these polynomials:
V(f_1,\dots,f_r)=\{p\in k^n:f_1(p)=0, \dots, f_r(p)=0\}

 PROPs and patches

  • February 15th, 2008
  • 7:59 pm

Brent Yorgey wrote a post on using category theory to formalize patch theory. In the middle of it, he talks about the need to commute a patch to the end of a patch series, in order to apply a patch undoing it. He suggests a necessary condition to do this is that, given patches P and Q, we need to be able to find patches Q’ and P’ such that PQ=Q’P', and preferably such that Q’ and P’ capture some of the info in P and Q.

However, as such, this is not enough to solve the issue. For one thing, we can set Q’=P and P’=Q, and things are the way he asks for.

Now, I wonder whether we can solve this by using PROPs (or possibly di-operads or something like that). Let’s represent a document as a list of some sort of tokens. We’ll set D_n the set of all lists of length n, and we’ll set P_n^m to denote operations that take a list of length n and returns a list of length m.

 Algebraic surface toys!

  • January 25th, 2008
  • 5:58 pm

At the start of the German Year of Mathematics, the Oberwolfach research institute has released an exhibition and the software they used to produce it. The software, surfer, is a really nice GUI that sits on top of surf and lets you rotate and zoom your algebraic surfaces as well as pick colours very comfortably.

They have a whole bunch of Really Pretty Images at the exhibition website, and I warmly recommend a visit. If you can get hold of the exhibition, they also have produced real models – with a 3d-printer – of some of the snazzier surfaces, so that one could have a REALLY close encounter with them.

But also, I’d really like to show you some of my own minor experiments with the program.

Tuba Mirum - the innards of a Klein Bottle
This is the interior of a Klein Bottle, using the “standard” realization as an algebraic surface given by Mathworld. In other words, I’m using
(x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2)+16*x*z*(x^2+y^2+z^2-2*y-1)=0
for the defining equation. It kinda looks a bit like a Sousaphone in my opinion.

 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!

 Wreath products

  • October 29th, 2007
  • 10:11 pm

In a conversation on IRC, I started prodding at low-order wreath products. It turned out to be quite a lot of fun doing it, so I thought I’d try to expand it into a blog post.

First off, we’ll start with a definition:

The wreath product H \wr_X G is defined for groups G,H and a G-set X by the following data. The elements of H \wr_X G are tuples (h_{x_1},h_{x_2},\dots,h_{x_r};g)\in H^{|X|}\times G. The trick is in the group product. We define
(h_{x_1},h_{x_2},\dots,h_{x_r};g)\cdot
(h’_{x_1},h’_{x_2},\dots,h’_{x_r};g’)= \\
(h_{x_1}h’_{gx_1},h_{x_2}h’_{gx_2},\dots,h_{x_r}h’_{gx_r};gg’)
(or possibly with a lot of inverses sprinkled into those indices)

Consider, first, the case of G=H=X=C_2 with the nontrivial G-action defined by gx=1, g1=x. We get 8 elements in the wreath product H \wr_X G. Thus, the group is one of the groups with 8 elements – C_8, C_4\times C_2, C_2^3, Q, D_4. We shall try to identify the group in question using orders of elements as the primary way of recognizing things. Consider an element ((x,y),z).

 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.

 Coq and simple group theory

  • August 5th, 2007
  • 9:15 pm

Trying to make the time until my flight leaves tomorrow go by, I played around a bit with the proof assistant Coq. And after wrestling a LOT with the assistant, I ended up being able to prove some pretty basic group theory results.

And this is how it goes:

Section Groups.

Variable U : Set.
Variable m : U -> U -> U.
Variable i : U -> U.
Variable e : U.

Hypothesis ass : forall x y z : U, m x (m y z) = m (m x y) z.
Hypothesis runit : forall x : U, m x e = x.
Hypothesis rinv : forall x : U, m x (i x) = e.
 

This sets the stage. It defines a group as a group object in Set, but without the diagonal map. It produces a minimal definition – the left identity and inverse follow from the right ones, which we shall prove immediately.

 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.

 Enumerating the Saneblidze-Umble diagonal in Haskell

  • June 11th, 2007
  • 3:47 pm

IMPORTANT: Note that the implementation herein is severely flawed. Do not use this.

One subject I spent a lot of time thinking about this spring was taking tensor products of A-algebras. This turns out to actually already being solved – having a very combinatorial and pretty neat solution.

Recall that we can describe ways to associate operations and homotopy of associators by a sequence of polyhedra Kn, n=2,3,.., called the associahedra. An A-algebra can be defined as being a map from the cellular chains on the Associahedra to n-ary endomorphisms of a graded vector space.

If this was incomprehensible to you, no matter for this post. The essence is that by figuring out how to deal with these polyhedra, we can figure out how to deal with A-algebras.

 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.

 Modular representation theory – when Maschke breaks down

  • April 21st, 2007
  • 1:43 pm

This post is dedicated to Janine K

 Modular representation theory: Simple and semisimple objects

  • April 2nd, 2007
  • 3:56 pm

Representations of categories

The basic tenet of representation theory is that we have some entity – the group representation theory takes a group, the algebra representation theory most often a quiver, and we look at ways to view the elements of the structure as endomorphisms of some vectorspace. The attentive reader remembers my last post on the subject, where \mathbb R^2 was given a group action by the rotations and reflections of a polygon surrounding the origin.

There is a way, suggested to me by Miles Gould in my last post, to unify all these various ways of looking at things in one: groups – and quivers – and most other things we care to look at are in fact very small category. For groups, it’s a category with one object, and one morphism/arrow for each group element, and such that the arrow you get by composing two arrows is the one corresponding to a suitable product in the group. A quiver is just a category corresponding to the way the quiver looks – with objects for all vertices, and arrows for all edges.

 Representation theory – basics

  • March 21st, 2007
  • 11:42 am

Many interesting groups have a very geometrical definition: transformations that fix certain symmetries is one of the historical origins of group theory.

Thus, one of the most interesting classes of finite groups are the rotation and reflection symmetries of a regular polygon. These are called D_n, for a n-sided polygon. Thus, for a triangle, we can label the corners a,b,c, reading clockwise, and enumerate the possible transformations by the positions the corners end up in. Thus we get the elements:
identity: abc -> abc
τ: abc -> bca (rotation by 120 degrees)
τ2: abc -> cab
σ: abc -> acb (reflection fixing a)
στ: abc -> bac
στ2: abc -> cba

Now, if we fix one equilateral triangle – say the one spanned by the points a=(1,0), b=(-\frac12,\frac{\sqrt3}2) and c=(-\frac12,-\frac{\sqrt3}2), then these transformations of the triangle can be extended to rotations and reflections of the entire 2-dimensional plane \mathbb R^2. As such, we can write down matrices for the group elements, starting with
\sigma=\begin{pmatrix}0&1\\1&0\end{pmatrix} and
\tau=\begin{pmatrix}\frac{\sqrt3}2 & -\frac12 \\ -\frac12 & -\frac{\sqrt3}2\end{pmatrix}
The rest of the group elements we can realize as matrices by just multiplying these with each other.

 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.

 Computational Group Theory in Haskell (1 in a series)

  • October 18th, 2006
  • 3:37 pm

This term, I’m listening to a lecture course on Computational Group Theory. As a good exercise, I plan to implement everything we talk about as Haskell functions as well.

The first lecture was mainly an introduction to the area, ending with a very na

 Localisation and ring depth

  • September 5th, 2006
  • 1:24 pm

So, we’re back at the point where I’m hesitating whether what I tried to work out even made sense or not. So I’ll do a write up of all the things I feel certain about asserting, and ask my loyal readership to hunt my errors for me. :)

Don’t laugh. This is less embarrassing for me than asking my advisor point blank.

We say that a (graded) commutative ring R has depth k if we can find a sequence of elements x_1,\dots,x_k with x_1 not a zero-divisor, each x_i not a zero-divisor in the quotient R/\langle x_1,\dots,x_{i-1}\rangle and R/\langle x_1,\dots,x_k\rangle a ring without non-zero divisors. This definition, of course, being the first obvious point where I may have screwed up.

Now, we know (from looking it up in Atiyah-MacDonald), that for SR the localisation of R in a multiplicatively closed subset S, S(R/I)=SR/SI, that injections carry over to injections, and that the annihilator over SR of an element is the localisation of the annihilator of the element.

 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 (3 in a series)

  • March 7th, 2006
  • 4:29 pm

This post will concern tuesday morning. Tuesday evening will be in a later post.

With the morning thus came, again, the pain in the legs. However, I’m told it’ll be better if I keep on skiing.

The mathematics in this report will come sooner than in the last; mainly because the lectures start at 8.30 and not at 17.00. :)

First out is Bênoit Fresse with

Little cubes operad actions on the bar construction of algebras

The reduced bar construcion of augmented associative algebras is given by fixing a field F, and for an augmented associative algebra A giving a chain complex B(A) such that B_n(A)=\hat A^{\otimes n} where \hat A denotes the augmentation ideal of A with the differential \partial(a_1\otimes\dots\otimes_n)=\sum_{i=1}^{n-1}a_\otimes\dots\otimes a_ia_{i+1}\otimes\dots\otimes a_n

If the product of A is commutative, then the shuffle product of tensors provides B(A) with the structure of a differential commutative algebra. In the talk, Fresse starts looking at the algebraic structure of B(A) for algebras with a homotopy commutative algebra:

 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.

 Report from Villars (1 in a series)

  • March 5th, 2006
  • 8:37 pm

So, I’ve arrived in Villars sur Ollon for the Alpine Operad Workshop. The travel was long and at times annoying, mainly because the heavy snowfall over München and Zürich and some other places in the region triggered extreme delays. As we were supposed to board, the poor attendant at Nürnberg airport told us that the plane had not yet departed from Zürich.

Except for that, though, the travel went fine, and after being treated to some immensely beautiful views (glittering lake of Genéve with rows and rows of snowcovered grapevines in front, anyone?) and reminded of just how much I miss the deep-snow winters, I got up on this mountain in southwestern (very much frenchspeaking) Switzerland to the Hotel du Golf. The receptionist told me, straight off, that a number of my colleagues had already arrived, and then I went to eat (Crêpes – expensive and not even correctly delivered…) and started wrestling the connector dance. Y’see, half of the connectors used in civilized Europe work here. The other half don’t. And those who do work, only do work if they’re impeccably straight. So I had to work for quite a while to actually, y’know, get my laptop, my mp3player, my loudspeakers (I sleep with music, mmmkay? Headphones are NOT very nice to sleep in, mmmkay?) and my cell phone all connected. I think I disconnected 75% of the room lights in the process.

 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

 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.