User mikael vejdemo-johansson - MathOverflow

Answer by Mikael Vejdemo-Johansson for Question about getting Review services

2013-01-10T22:35:01Z

Many if not most reputable journals state up front whether their production is reviewed by Mathematical Reviews and/or Zentralblatt Math. This can work as a heuristic when deciding where to submit a paper you write, to ascertain that it gets the treatment you would expect for it.

MR and ZBM are not venues for peer-review; they are abstraction services to help mathematicians keep up to speed with already published, peer-reviewed mathematics.

Answer by Mikael Vejdemo-Johansson for Why is a ring called a "ring"?

2012-12-27T00:42:46Z

I think the classical argument is that the original rings were Z/nZ, which as you'll remember from “clock arithmetic” has a certain cyclicity to it. This theory is backed up by Wikipedia ( http://en.wikipedia.org/wiki/Ring_(mathematics)#History ) citing Harvey Cohn's number theory book with

According to Harvey Cohn, Hilbert used the term for a specific ring that had the property of "circling directly back" to an element of itself.

(citation from the Wikipedia page as of 26th December 2012)

The Hilbert paper in question is Die Theorie der algebraischen Zahlkörper (Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897), credited with introducing the word Zahlring in the first place.

Answer by Mikael Vejdemo-Johansson for Persistent homology of Gaussian Fields in Euclidean space

2012-12-05T11:50:37Z

The closest I can find spontaneously would be Matthew Kahle's work on random topology; http://arxiv.org/abs/0910.1649 looks like it would be directly related to your question, and http://arxiv.org/abs/1009.4130 seems related too.

Answer by Mikael Vejdemo-Johansson for Noncommutative computational package

2012-09-25T20:49:39Z

Magma can certainly deal with that. Not sure about other packages; Singular has been approaching a non-commutative extension for years, but I'm not sure of its status.

Answer by Mikael Vejdemo-Johansson for Computer aided homology computations

2012-09-12T09:22:43Z

There are several applications and libraries out there that deal with homology computations with various approaches to the computation. One field with a strong focus on efficient computation of homology is persistent homology; which computes classical homology as a side-effect.

For C/C++ use, I would recommend you take a look at Dionysus (http://mrzv.org/software/dionysus/). This library is optimized for computing persistent homology with field coefficients; and is one of the most capable libraries I know of in this field.

As for your wishlist, I would point out that from the persistent homology side, integer coefficients are Just Not Done; small prime field coefficients gets you some of the information contained in integer coefficients, but with a huge gain in computation speed. Computing groups and producing a basis for the homology is done by most systems around; but parallelization is yet basically unsolved.

Of course, you want classical homology, not persistent and you want integer coefficients. I would recommend you spend some time looking around for the field of Smith normal form computation libraries and roll your own from there. There is some interesting research into efficient and parallelizable Smith normal form computation; both Kaltofen and Jäger seem to have papers on the subject, and they could well have implementations you can use.

Answer by Mikael Vejdemo-Johansson for Fiction books about mathematicians?

2012-07-09T02:24:16Z

Rymdväktaren and Nyaga are (admittedly Swedish language) sci-fi books that feature 5-6 mathematicians in the main cast as well as one supercomputer.

Answer by Mikael Vejdemo-Johansson for Fiction books about mathematicians?

2012-07-09T02:19:46Z

The First Circle by Solzhenitsyn features a mathematician as the main character.

Answer by Mikael Vejdemo-Johansson for Real-world applications of mathematics, by arxiv subject area?

2009-10-26T04:57:02Z

math.GR Group Theory

Group theory provides methods for understanding the Rubik's cube, and for generating algorithms for solving the cube remarkably quickly from any state the cube may be in.
Groups find various applications in chemistry, eg. in the study of crystal structures and spectroscopy.
Cryptography - various hard algorithmic problems about groups are used to design crypto-systems.
Groups of symmetries are used to reduce the dimension of parameter spaces in engineering models to make model verification more tractable.
Potentially fast matrix multiplication; see this MO question.
Card tricks that don't work by sleight of hand, but via the arrangements of the cards. e.g. Sim Sala Bim, see this site for description and a graphical explanation. If you think about it, the symmetric group explains the trick and shows you how you to extend it past three piles of seven cards, but to N piles of M cards.

Reference Request: Smith Normal Form for maps between free _graded_ modules

2012-04-11T11:25:20Z

I feel like this should be easy, but I cannot quite find a literature reference for this: We know (i.a. from the Kaplansky reference in http://mathoverflow.net/questions/31275/does-smith-normal-form-imply-pid) that sufficient for Smith normal form as well as Hermite normal form to work is that the underlying ring be a PID.

I am interested in the case where the ring is $k[t]$, for some field $k$, and all modules involved are $\mathbb N$-graded with the "obvious" grading of $k[t]$. For a matrix $M$ representing a map between two graded $k[t]$-modules $S\to T$, it seems obvious to me that Smith normal form is computable, and about as efficient as one might hope over any ring. The presence of a grading seems to imply one should take some minute care — but the care needed seems to be almost non-existent.

Has anyone dealt with this sort of setting in the literature already? I'd rather have a good reference for this than develop everything in analogy with well-known results myself.

Answer by Mikael Vejdemo-Johansson for What programming languages do mathematicians use?

2010-01-08T04:05:19Z

APL as such is probably less in use nowadays - but its ASCII sibling J is quite usable, and has interesting grammatic and mathematical constructions.

Answer by Mikael Vejdemo-Johansson for Are there some original papers or books related to applications of algebraic topology and algebraic geometry in complex dynamic systems

2011-12-21T19:05:40Z

One place where papers on applications of algebraic topology — to dynamical systems as well as to statistics, data analysis, bio-medicine, computational geometry and other areas — gets aggregated is on the webpage of the Computational Topology group at Stanford. This page has a running listing of relevant preprints and papers that may form a good starting point for you.

Answer by Mikael Vejdemo-Johansson for Are there some original papers or books related to applications of algebraic topology and algebraic geometry in complex dynamic systems

2011-12-21T19:03:53Z

Another good place to start is to track the output of Marion Mrozek and Konstantin Mischaikow, and their various co-authors. There is a whole group at the University in Krakow centered on Mrozek doing algebraic topology applications to dynamical systems.

Answer by Mikael Vejdemo-Johansson for The second homotopy group of a simple CW-complex

2011-12-15T13:07:50Z

So, the one 0-cell forces the 1-skeleton to be a figure-8. And we attach three 2-cells to this figure-8. These cells can be attached to:

loop 1, with some winding number n.
loop 2, with some winding number m.
the 0-cell, and it's degenerate 1-cell.

In the last case, we get a generator for $\pi_2$ from the resulting sphere; and without any 3-cells, any generator that shows up will produce non-trivial homotopy.

Suppose, thus, that the last case does not occur. Then we would be distributing three 2-cells on 2 loops. Regardless of how we do this, at least two 2-cells attach to the same loop, possibly with different winding numbers. Unless all three 2-cells attach to the same loop, the fundamental group will be trivial. If $\pi_1$ is indeed trivial, then because $H_2(X)=Ab \pi_2(X)$, it follows that $\pi_2(X)$ is indeed non-trivial. If all three 2-cells attach to the same loop, then the space is a wedge of a circle and the CW-complex on 1 0-cell, 1 1-cell and 3 2-cells. Being a wedge, if the homotopy on a factor is non-trivial, the entire homotopy will also be, and for the factor of the three attached 2-cells, the above argument with the abelianization also works out.

... or at least, that's how I would approach it. Would those here who know homotopy theory now please tell me why this cannot possibly work? ;-)

Answer by Mikael Vejdemo-Johansson for Real-world applications of mathematics, by arxiv subject area?

2009-10-26T04:53:31Z

math.AT Algebraic Topology

Algebraic Topology finds applications in sensor network design, coverage analysis for sensor networks, and in expanding data analysis techniques to give better visualizations for large data sets.
It has also been applied to computer vision and pattern recognition algorithms, for instance here.
Algebraic Topology can be used in robotics. Motion planning and behavioral algorithms for robotics have been studied with topological tools.
Knot theory is used when dealing with protein folding and other analysis of DNA function. There are enzymes called 'topoisomerases' that change the knottedness of loops of DNA. In fact, when bacteria (which have circular 'chromosomes' called plasmids) reproduce, they make use of an enzyme whose specific role to to unlink Hopf links! There are antibiotics that target this enzyme.
Model categories have been used in the study of concurrency. See this paper by Gaucher.
Nash's proof (Ann. of Math, Vol. 54, No.2; 1951) that every finite non-cooperative game has an equilibrium point in mixed strategies is a direct application of Brouwer's fixed point theorem, and spurred a great deal of interest in applications of game theory to economics (cf. this survey article). Game theory itself has applications in computer science and mathematical finance.

Answer by Mikael Vejdemo-Johansson for When are (finite) simplicial complexes (smooth) manifolds?

2011-07-27T14:06:27Z

There is the idea of a simplicial manifold, which works by checking that the complex is pure (all facets of the same dimension) and that each codimension 1 face is included in the correct number of facets.

Beyond this answer to your question a), I believe b) and d) to be potentially really difficult. It would seem to me that almost no simplicial complexes are in themselves smooth (unless you give an explicit embedding, in which case you need to check smoothness for each face separately, or something like that), but that all simplicial manifolds can be deformed into a smooth manifold.

As for c), it seems to reconnect to the criterion for a), but it is not clear to me whether algorithmics exist.

You might want to check out the work by Benjamin Burton.

Answer by Mikael Vejdemo-Johansson for The relationship between low dimensional topology and dynamics

2011-04-19T15:38:02Z

Robert Ghrist started out his (prolific) research career investigating knotted trajectories in dynamical systems. http://www.math.upenn.edu/~ghrist/preprints-dynamics.html

Konstantin Mischaikow has been working on topological classification of dynamical behaviours. http://www.math.rutgers.edu/~mischaik/pub_page/paperlist_dynamics.html

One thing I have heard people show interest in is a decomposition of a low-dimensional dynamic system into cells with homogenous behaviour -- much in the vein of Morse theory: you pick out attractive and repulsive submanifolds, and regions of homogenous flow (starting out and ending up in the same sources and sinks), and this gives you (hopefully) a CW complex describing the dynamics.

There probably is more to it, but these are the things I have heard of.

Answer by Mikael Vejdemo-Johansson for Are the numbers of elements of two distinct prime orders not equal in finite groups?

2010-12-18T06:04:19Z

Consider $C_6$. It has 1 element of order 1, 1 element of order 2, 2 elements of order 3, and 2 elements of order 6.

Notice here that the numbers of elements of order 2 and of order 3 differ.

Answer by Mikael Vejdemo-Johansson for What is correct name of the following construction?

2010-12-01T20:13:49Z

This is the image of $I$ in the localization $\mathbb Q[x_1,x_2,\dots,x_n]_{I}$.

There is an issue here though. If $I$ is not a prime ideal, then its complement is not multiplicatively closed, and therefore not a good set to invert in a localization. If $I$ is not a prime ideal, then there are some $f, g$ in $I^c$ such that $fg\in I$. Thus, e.g., $\frac{1}{f}\cdot\frac{1}{g}$ becomes a non-allowed coefficient, while $\frac{1}{f}$ and $\frac{1}{g}$ both are.

The set you describe can still be defined, obviously, but it will lack interesting structure. If $I$ is a prime ideal, then your set is the ideal in the localization ring that I describe above.

Answer by Mikael Vejdemo-Johansson for Advanced Math Jokes

2010-11-21T18:24:55Z

The first time I ran into the carry operation from grade school addition presented as a non-trivial group cocycle generating part of the group cohomology of $\mathbb Z/10$, it was introduced as a joke embedded completely within mathematics.

Specifically, for those who haven't seen this yet, the carry operation $c(n,m)$ is defined as $c(n,m) = 0$ if $n+m < 10$ and $c(n,m) = 1$ for $n+m ≥ 10$. You can verify the cocycle condition reasonably easily, and then it remains to check there is no endomap $g:\mathbb Z/10\to\mathbb Z/10$ with $c$ as its coboundary.

Answer by Mikael Vejdemo-Johansson for How to show that a space has the homotopy type of wedge of spheres ?

2010-10-26T20:30:57Z

(If I remember correctly) a shellable (simplicial) complex automatically has the homotopy type of a wedge of spheres: if you could find a shellable triangulation you should be done.

Of course, this'll only work if you're lucky enough that the structure you have admits nice combinatorial structures that happen to be shellable — but it's one way you can get to a wedge of spheres.

Answer by Mikael Vejdemo-Johansson for Is a shellable polytopal complex a PL-ball or a PL-shere?

2010-09-18T15:29:59Z

A shellable complex has the homotopy type of a wedge of spheres; but since it may well have the homotopy type of a wedge of more than one sphere -- the number of spheres given by the h-vector of the complex, shellable complexes in general are not homeomorphic to a ball or a sphere. See, e.g., section 4 of Anders Björner: Shellable Nonpure Complexes and Posets

Answer by Mikael Vejdemo-Johansson for Journals for undergraduates

2010-08-27T08:41:53Z

If you read German, you might also be interested in die Wurzel.

Answer by Mikael Vejdemo-Johansson for What is the process of computing homology?

2010-08-04T19:04:25Z

One way of thinking about this is that homology gives a way to almost answer the questions you would try to answer by homotopy, but in a way that allows you to use linear algebra to get the answer. You don't get all the way - which is why homology and homotopy are actually different entities - but anything that would have been identified by homotopy certainly will be identified by homology.

Key in this process is the transferral from topological arguments to algebraic arguments: we would be interested in, say, non-trivial loops under the equivalence relation given by homotopies of functions, but that's topological and potentially hard. Instead, we find correspondences in algebra to what non-trivial and loop means.

Thus, we triangulate the space to get a simplicial approximation of the space, and instead of looking at bona fide loops, we look at cycles. These are like loops in that neither have non-trivial boundaries - the loops don't have non-trivial boundaries because of their geometry, the cycles don't by the way we define cycles: namely as things that are in the kernel of the boundary operator. (drawing pictures of simplicial 1-cycles at this stage helps understand why these definitions give us the geometric results we want to approximate by the algebraic approach)

Similarily, we want to capture homotopic loops as somehow the same. Specifically, this means we should be able to wiggle our loops around as long as they stay in the space. Once we go to the simplicial side, the wiggles have to jump across an entire simplex in one go, changing an edge to the other two edges of a triangle, say. The upshot of all this is that in order for homotopic things to be identified in homology, we need to identify anything that differs by the action of pulling a cycle across a higher-dimensional simplex.

Which is, once the pictures are drawn and the arguments you'd find in any introduction to algebraic topology are made, the same as taking the factor group by the image of the boundary operator.

In the end, to compute $H_1(X)$, what you really compute is a complete enumeration of all possible cycles, under the condition that things you get by wiggling a cycle a little bit should be considered the same.

Now, the next question would be Why do we care about homotopic functions? which is a very different question that I won't start answering here.

Answer by Mikael Vejdemo-Johansson for What are some mathematical sculptures?

2010-07-19T13:39:34Z

The Mathematical Research Institute in Oberwolfach have a sculpture on their grounds depicting Boy's surface:

Answer by Mikael Vejdemo-Johansson for Statistics for mathematicians

2010-07-13T08:45:52Z

I'm currently working my way through Cramér's Mathematical methods of statistics. It starts out with a half-book primer on all measure theory, Lebesgue integration et.c. you might possibly need for anything, and then goes through first probability and then statistics with this backdrop.

Skipping the introductory analysis it might be what you're looking for.

Algorithms for the Lakes of Wada

2010-05-16T17:36:24Z

The Lakes of Wada partitions the unit square in to three regions, all of whom share a common boundary. The Wikipedia entry (http://en.wikipedia.org/wiki/Lakes_of_Wada) gives a construction approach, and a picture of a first few steps of the construction.

Is there a good algorithm available somewhere to explicitly list the partition up to some specific iteration? Or a closed form expression for the boundary in the limit of the process?

I'm hoping for something I can implement myself, or a piece of software that already does it, and in the end get a picture to arbitrary high levels of detail and arbitrary high iterations of the construction.

Answer by Mikael Vejdemo-Johansson for Finding a representative of a branch in hierarchical clustering

2010-05-04T18:51:13Z

If you do some sort of centroid linkage algorithm to construct your hierarchal clustering, you should be able to pick out the centroid values just before merging clusters as representatives at those levels. It'd require tweaking the algorithm to get the points to tag the corresponding tree, but shouldn't be too hard to write up.

Alternatively, reinterpret the hierarchal clustering as persistent homology and pick out representative cycles for the homology classes associated to each bar in the barcode. jPlex ( http://comptop.stanford.edu/programs/jplex/index.html ) or javaPlex ( http://code.google.com/p/javaplex ) should both be able to do just that if I remember correctly.

Answer by Mikael Vejdemo-Johansson for Hopf algebra structure on $\prod_n A^{\otimes n}$ for an algebra $A$

2010-04-25T22:03:37Z

You may want to look at the work by Ron Umble with various coauthors on $A_\infty$-Hopf algebras and bialgebras. I don't recall the details, but I think they are trying to deal with a rather similar situation.

http://arxiv.org/abs/0709.3436

http://arxiv.org/abs/math/0406270

Answer by Mikael Vejdemo-Johansson for `Topos' with alternate subobject lattice?

2010-04-18T23:39:33Z

Probably not the complete picture, but my impression was that one of the reasons that Heyting lattices play such a large role with topoi is that the Heyting lattice definition captures the properties we expect a set theory to have: the joins, meets and arrow capture, cleanly, the and, or and implies predicates.

Of course we could try to mimic the topos constructions basing it all on some different lattice structure - but my guess is that unless we restrict to special kinds of Heyting lattices, the result will no longer correspond to anything sensible.

Answer by Mikael Vejdemo-Johansson for The Interrelationship Problem Of Modern Mathematics- How To Deal With it In First Year Graduate Courses?

2010-04-14T18:01:07Z

I've almost uniformly studied the homological algebraic aspects before I got around to studying the corresponding results from algebraic topology. It did get somewhat artificial at points - specifically triangulated categories make a lot more sense once you've seen Serre fibrations than before you do.

I felt quite well motivated by the approaches I encountered though; with the study of Ext and Tor to divine interesting ring properties taking the forefront in homological algebra, with a side dish of approximating modules by things that are free everywhere that matters, but sacrifice degree concentration to achieve it.

My personal feeling is that it probably depends to a large extent on whether whoever is teaching the material wants to teach homological algebra or algebraic topology: if you're happier thinking about topology, then homological algebra will feel desolate and artificial almost no matter what you do about it; while if you are genuinely interested in homological algebra on its own, it's much easier to sprinkle in the off-ramps as you go, pointing out where certain concepts have roots outside the current area, and how to get more information about the roots.

Comment by Mikael Vejdemo-Johansson

2012-11-25T15:43:36Z

In particular, by stepping stone I mean to look for computational topologists who are interested in more complexity knowhow in their own workgroups, and use your participation in a postdoc in such a group as a way to bootstrap yourself into computational topology. After 2-3 years you'll be prolific in your new field instead. It is a gamble, but it is far from impossible.

Comment by Mikael Vejdemo-Johansson

2012-11-25T15:42:09Z

I shifted after my PhD: from computational homological algebra to computational and applied algebraic topology. It took several years, and I have yet to see if my career eventually benefitted from it, but if anything I'd recommend trying to get contacts now to help you through, and to make your postdoc time a stepping stone for the shift.

Comment by Mikael Vejdemo-Johansson

2012-11-25T09:13:01Z

You might want to take this question to math.stackexchange.com where the homework-level of the question is more tolerated.

Comment by Mikael Vejdemo-Johansson

2012-11-12T14:42:34Z

Does $a=h\text{coker}(k)$ even exist? It seems to me that $\text{coker}(k)$ should be a subobject of $G$, while $h$ takes input from $H$.

Comment by Mikael Vejdemo-Johansson

2012-09-15T06:20:42Z

I know Dionysus can deal with several million, not sure where the upper limit resides. In particular, the upper limit probably depends a lot on what size computer you have access.

Comment by Mikael Vejdemo-Johansson

2012-09-05T17:59:27Z

Hey unknown? If you paid attention to BR's response, you'd notice he is directing you to where these experts go when they want to help people out with random questions. We go to math.stackexchange.com and stats.stackexchange.com when we want to help out with questions about the basics, and we go to mathoverflow.com when we want to discuss current research: it's like the difference between helping out with tutoring vs. going to a research seminar.

Comment by Mikael Vejdemo-Johansson

2012-07-11T04:03:25Z

… as indeed is visible from the quote the link goes to. Yes. It's been a favorite quote at the applied algebraic topology conference I organized last week. ;-)

Comment by Mikael Vejdemo-Johansson

2012-05-30T11:44:38Z

Saying this in a comment now, since the question closed just as I was working out an actual answer: the tension between teaching and popularization on the one hand, and the pure research careerism on the other is present all over the world — and often it leads to a polarization; like the divide between Liberal Arts colleges and research universities in the US. To answer your actual question: in order to be a respected research mathematician, yes, you do have to publish papers. If all you want to do is to teach well, your innovative approaches might be enough to gain respect. Maybe.

Comment by Mikael Vejdemo-Johansson

2012-04-18T21:48:21Z

This might be more appropriate at math.stackexchange.net. That said, in vectorspaces very few morphisms fail to split.

Comment by Mikael Vejdemo-Johansson

2012-04-11T13:52:23Z

Ralph: Sorry, yes. Graded degree 0 $k[t]$-linear.

Comment by Mikael Vejdemo-Johansson

2012-04-11T13:51:04Z

$S$, $T$ are free of finite rank. They both have homogenous bases, but the basis elements need not all have the same degree as each other.

Comment by Mikael Vejdemo-Johansson

2012-04-11T13:50:12Z

The presence of a grading means, for instance, that you cannot use all the tricks available for doing Gaussian reduction on the kinds of matrices I'm interested in — you could not reduce rows in lower degrees using rows in higher degrees. I suspect that similar restrictions are in place for computing Smith normal forms, but also that the restrictions do not actually limit things very much.

Comment by Mikael Vejdemo-Johansson

2012-04-11T12:15:28Z

Graded $k[t]$-linear.

Comment by Mikael Vejdemo-Johansson

2012-04-11T12:15:07Z

I know this, and as I mentioned, I expect this to actually be rather easy — which is why it's a reference request, not a question; but in my setting, the rows of the matrix have inherent degrees, and I worry somewhat about keeping everything compatible with the grading.

Comment by Mikael Vejdemo-Johansson

2012-03-14T15:25:34Z

This seems like a perfect question for community-wiki.