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Arithmetic Chern-Simons

Low Dimensional Topology

The most compelling aspect of Quantum Topology for me is its connection to analytic number theory. Today I’d like to draw your attention to recent work of Minhyong Kim on Arithmetic Chern-Simons Theory (see his paper for more details). I was fortunate to hear him give a talk about this last Wednesday and a colloquium talk on related subjects the day before. People have been talking about such “quantum topological number theory” for a long time- e.g.this 2010 MO question – but we haven’t seen much of an uptake so far. This isn’t an easy direction to pursue because one needs to know both quantum topology and analytic number theory, but I was left with the strong feeling that “There’s gold in them thar hills”, both for topologists and for number theorists.

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Noether’s Theorem: Quantum vs Stochastic

Azimuth

guest post by Ville Bergholm

In 1915 Emmy Noether discovered an important connection between the symmetries of a system and its conserved quantities. Her result has become a staple of modern physics and is known as Noether’s theorem.

Photo of Emmy Noether

The theorem and its generalizations have found particularly wide use in quantum theory. Those of you following the Network Theory series here on Azimuth might recall Part 11 where John Baez and Brendan Fong proved a version of Noether’s theorem for stochastic systems. Their result is now published here:

? John Baez and Brendan Fong, A Noether theorem for stochastic mechanics, J. Math. Phys. 54:013301 (2013).

One goal of the network theory series here on Azimuth has been to merge ideas appearing in quantum theory with other disciplines. John and Brendan proved their stochastic version of Noether’s theorem by exploiting ‘stochastic mechanics’ which was formulated in the network theory series to…

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Random walking trough a lattice

This post is a draft

Problem setup

Consider the following lattice:

 

Possible path in a lattice

Possible path in a lattice

Suppose you want to move from point A to C, but such that you can only go one step to the right or upwards at a time. For an n x m lattice, there are

(n + m)! / (n! m!)

posible paths. This is so due to the fact that if we denote r as going to the right and u as going upwards, then for every permutation of the symbols r and u with n times r and m times u, there exist exactly one path, and conversely, every path from A to C can be represented with exactly one such permutation.

Let say a path corner is a node in the lattice such that the trajectory changes direction there, either from going right to upwards or viceverse.

Given a path from A to C, what is the probability that it has R corners?

First note that for every corner, there is one upward and one right movement, so that there are at most 2 min(n, m) – 1 corners in a valid path.

 

An odd trajectory in a lattice

An odd trajectory in a lattice

Fix a path in the lattice, such that its first movement is to the right and the last one is upwards. No matter the path shape, it will always have an odd number of corners. Suppose an odd number R in the range 1,…,(2 min(n, m) – 1) is given, in order to calculate the probability for a path having R and due to the fact that the paths are uniformly distribuited, we must count how many paths are there with exactly such a number of corners.

Consider the previous figure, with the present hypotesis of the first movement being to the right, and labeling the corners in the order shown in the figure, then the upward movements correspond begin in the odd labeled vertixes, moreover, we must accomodate (R – 1) / 2 odd numbered vertixes in n – 1 horizontal cell boundaries, which is accomplished with the combinatoric formula

C_{(n – 1), (R – 1) / 2}

For each one of those horizontal partitions, we must select one possible vertical partition, such that que horizontal step begins with an even numbered label, in the range 2,…, R – 1. There are (R – 1) / 2 even labels and m – 1 posible vertical boundaries where to accomodate them, therefore, there are

C_{(m – 1), (R – 1) / 2}

ways to do so, and the total different paths we can get considering the horizontal and vertical ordering of the vertixes is

C_{(n – 1), (R – 1) / 2} C_{(m – 1), (R – 1) / 2}

The case where the walk starts upwards and finishes to the right is simmilar, with n and m reserved, and therefore the formula is the same.

Consider the case when the movement start and finishes going to the right, like in the next figure:

 

A path in a lattice with an even number of corners

A path in a lattice with an even number of corners

in contrast with the previous cases, there is an even number of corners, and every leftwise corner in a row, except the first, is even labeled, so that from the m – 1 inner rows in the lattice, there should be chosen (R – 2) / 2 rows to accomodate the corners. Likewise, from the n – 1 inner columns in the lattice, we must choose R / 2 columns in order to acoomodate the corners. In total, there are

C_{(n – 1), R / 2} C_{(m – 1), (R – 2) / 2}

different ways to accomodate the corners.

The final case is analogous, and therefore there should be

C_{(n – 1), (R – 2) / 2} C_{(m – 1), R / 2}

different paths with first and last movement upwards and R corners.

Counting corners

For the details, see here.

a(nother) media player

aplayer is my last weekend programming project. It is an experiment in how to use qt5 to write useful code fast. Besides, however how comfortable I have been with auditive as my media player, there are some things that it lacks and that I find difficult to write in vala.

For instance, after I tried spotify’s media player, I was convinced about how important is to have a media library, and how convenient is to use a networking framework to ease the process of downloading meta information from the web.

Sure, there are very good options out there (I guess), but I wanted to try something of my own.

The result looks like this:

screen

So it was a good weekend, qt5 is a solid framework for fast and easy development. In my next vacation I should try implementing some of these ideas:

  1. A database.
  2. Art visualization with the graphics framework.
  3. It would be pretty cool to have a backend database in the cloud, maybe exporting your entire database to a cloud based music service.

Of course, the goal here is to learn what qt5 has to offer in terms of functionality, so that for end users there are very good options driven by the community (amarok, banshee, etc, to name a few) and some others developed by online companies (for instance spotify‘s player which motivated me to try doing my own), but for a developer, I think the most valuable lesson is how far she can get in a few days with the right tool.

Setting a WPA connection with netctl

Last week I tried to setup my laptop wireless adaptor to work at school. It turns out that on arch linux, the default network manager is netctl, an interesting piece of software, compliant with the arch way, that is, it’s configuration is mostly on the terminal with a text editor, as arch’s wiki explains.

However, what the wiki did not explain, is how to setup the network with a WPA2 configuration, and the examples that  I found were not of much help.

Finally, after a few hours of struggle at school, I could make my precious connection to the wireless network.

As my memory is very limited by so many years of watching the Simpsons show, I decided to save my configuration with a gist, just in case I need it in a later occasion.

The gist is here: https://gist.github.com/ixxra/6920140

It’s Alive: Running ipython’s notebook from within sage

I followed http://www.liafa.univ-paris-diderot.fr/~labbe/blogue/categorie/ipython/ and could install the notebook. The first time I tried was unsuccessful, so I had to upgrade sage. Nevertheless, after upgrading everything ran smoothly =D.