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Michi’s blog » archive for 'Mathematics'

 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.

 Recursively counting numbers with fixed bit counts

  • May 13th, 2012
  • 1:04 am

I ran across this problem in a reddit side-bar job-ad, and was intrigued by the task (description paraphrased to decrease googleability):

Write a function

uint64_t bitsquares(uint64_t a, uint64_t b);

such that it return the number of integers in [a,b] that have a square number of bits set to 1. Your function should run in less than O(b-a).

I think I see how to do it in something like logarithmic time. Here’s how:

First off, we notice that we can list all the squares between 0 and 64: these are 0, 1, 9, 16, 25, 36, 49, and 64. The function I will propose will run through a binary tree of depth 64, shortcutting through branches whenever it can. In fact; changing implementation language completely, I wonder if I cannot even write it comprehensively in Haskell.

 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

 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

 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?

 [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.

 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

 So that must mean I’ve been a mathematician since 2005?

  • December 8th, 2008
  • 4:03 am

http://www.thesun.co.uk/sol/homepage/features/article2011061.ece

This is a rather atrocious article giving yet another ad hoc “formula” to compute some numeric measurement of something-or-other. In this particular case, it’s about cleavage, and how to avoid showing too much of it, but these “formulae” plague us every time some journalist wants to math up their reporting.

What caught my eye in this particular case was the people they lined up to back up the story.

Mathematician William Hartston, who holds an MA in Maths from Cambridge University, reckons this will save a lot of showbiz blushes on the red carpet.

 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.

 On the chromatic number of Lichtenstein

  • October 28th, 2008
  • 7:41 pm

Following the featuring of the internal political structure of Lichtenstein on the Strange Maps blog, Brian Hayes asks for the chromatic number of Lichtenstein.

Rahul pointed out that I made errors in transferring the map to a graph. Specifically, I missed the borders Schellenberg-Eschen and Vaduz-Triesen. The post below changes accordingly.

Warning: This post DOES contain spoilers to Brian’s question. If you do want to investigate it yourself, you’ll need to stop reading now. Apologies to those on my planet feeds.

As a first step, we need to build a graph out of it. I labeled each region in turn with the exclaves numbered higher than the “main” region of each organizational unit. And then I build a .dot file to capture them all:

 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.

 RIP Henri Cartan

  • August 22nd, 2008
  • 6:29 pm

As John Armstrong said, I didn’t know he was still alive!

On the algebraic topology mailing list, the announcement came today that Henri Cartan, once co-founder of Nicholas Bourbaki, died August 13, 2008:

La Soci

 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 :-).

 On purity and essence of mathematics

  • June 15th, 2008
  • 12:06 pm

I seem, lately, to be so densely planned that all I can do for my blog is to react on blog posts from Ben Webster at the Secret Blogging Seminar.

He has, recently, written a post inspired by the xkcd comic on purity in the sciences. The comic is funny, and rings true, but Ben brings up a severe criticism of the premises of the comic that rings back to my own years as a hotheaded undergraduate.

You should read all of Ben’s post, but if you don’t, you should at least read the following:

And I think one of the key points here is this: mathematics is not science. Mathematics is often lumped in with science, and is often used by scientists. Mathematicians often know more science than normal people, and certainly scientists know more mathematics. But mathematics and science are fundamentally different activities, as different as making a gun and fighting in a battle. I mean, no one would claim there are no links between those occupations, or that gun-makers don

 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.

 Parallel and cluster computing with MPI4Py

  • May 18th, 2008
  • 11:46 am

First off, I’d ask your pardon for the lull in postings – this spring has been insane. It has been very much fun – traveling the world, talking about my research and meeting people I only knew electronically – and also very intense.

To break the lull, I thought I’d try to pick up what I did last summer: parallel computing on clusters. It’s been a bit of blog chatter about SAGE and how SAGE suddenly has transformed from a Really Good Idea to something that starts to corner out most other systems in usability and flexibility.

Matlab? SciPy bundled with SAGE and the Python integration seems to be at least as good, if not better.
Maple? Mathematica? Maxima? Singular? GAP? SAGE interfaces with all those that it doesn’t emulate.

 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.

 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.

 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!

 AMS-MAA JMM 2008 Liveblogging, day 1

  • January 7th, 2008
  • 5:04 am

I’m exhausted.

I’m completely exhausted.

And I just got through the first day.

However, I also managed to meet up with S from the university interested in me. We had a really nice chat, and I feel rather good about it.

Other things done today – listened to an interesting talk generalizing Koszul algebras based on the highest degree ring generator of the Ext algebra. Listened to bits and pieces of a talk on Koszul and Verdier duality. Saw Flatland – The Movie (with Martin Sheen playing the main character, Arthur Square).

I also chatted with Cliff Stoll – whose sales pitch for the Klein Bottles is immensely entertaining; NSA – who don’t want me; Maplesoft – who are interested in me; Mathematica – who pointed me to their website; various e-Learning companies; and many many other exhibitors.

Also, got tired, hungry and WET. It’s bloody raining here.

 Checking email 4000 times a day

  • November 20th, 2007
  • 6:12 pm

In a recent column at The Chronicle of Higher Education, the columnist writes

I’m a latecomer to it, in part because I have a very hit-or-miss interest in new technologies. (I still don’t own a cell phone, for example, though I check my e-mail 4,000 times a day.)

Now. There are 24 hours in a day. 1 440 minutes. 86 400 seconds. Thus, checking e-mail 4 000 times in a day would require you to check your inbox every 21.6 seconds. Day and night.

Either the author is innumerate or hyperactive.

 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.

 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&#8217;_{x_1},h&#8217;_{x_2},\dots,h&#8217;_{x_r};g&#8217;)= \\
(h_{x_1}h&#8217;_{gx_1},h_{x_2}h&#8217;_{gx_2},\dots,h_{x_r}h&#8217;_{gx_r};gg&#8217;)
(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.

 Blogging seminars – a new fad!

  • July 6th, 2007
  • 11:46 pm

They simply do not end. Now, Cornell grads and pre-grads have started the Everything Seminar – which has absolutely brilliant discussions about the forbidden minor theorem in graph theory as well as a fascinating overview over constructing homological algebra as embedded in the theory of modules over \mathbb C[\epsilon]=\mathbb C[x]/(x^2).

Connected to this comes the observation that by constructing calculus using the tricks used in synthetic differential geometry, we end up with – again – modules over \mathbb C[\epsilon], and some very fascinating discussions are sparked as to subtle and interesting connections between these two viewpoints!

How on earth I am going to keep up with the interesting sprouting discussion group blogs I shall never know. Maybe it’s getting to the point where we’ll start an A_\infty-blog?

 Linking in hushed voices

  • June 12th, 2007
  • 6:49 pm

All the cool kids are doing it, so I’ll tag in too.

There’s a secret blogging seminar going on. And the people seem to be writing interesting things – what little they managed to write before the hushed rumour mill started.

 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.

 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.

 Interview with a blogger

  • April 30th, 2007
  • 4:40 pm

The website/forumsite Mathetreff, run by the Bezirksregierung (region government) D

 Modular representation theory – when Maschke breaks down

  • April 21st, 2007
  • 1:43 pm

This post is dedicated to Janine K

 looksay – today’s Haskell snippet

  • April 18th, 2007
  • 9:34 am
nextLookSay = foldr (\xs -> ([length xs, head xs]++)) [] . group
lookSay = iterate nextLookSay [1]
 

Conway’s Look-and-say sequence

 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.

 Sudden responsibilities

  • March 28th, 2007
  • 5:41 pm

I just met up with the workgroup in the Deutsche Mathematikervereinigung (German Association of Mathematicians) with interest spanning “Information and Communication” – which turns out to mean that they care about libraries, about communicative tools for mathematicians, and spend their time thinking about these things, and meeting at conferences.

Met up with is to say participated in their workshop.

Stunned them all with the relevation that I blog. “Wow, a real, live blogger! Here!”

And promptly got elected to their executive committee.

Kinda unexpected result of going to a conference.