TQC, Hopf algebras and TPM

Hello everyone,
It has been a long time since I got time to update. The last few months have been rough as well as exciting. To start off  here are the project updates :
Between the last update and this, I gave my second internals at the university which went pathetic, made it to the GSoC reunion in California where I had worked on the documentation and few failing cases, and have started to read more of TQC, Hopf algebras and material on Topological Phases of Matter.
Coming to the project, the issues with the documentation had been resolved, I could not get time to revamp the code and get it working for the failing cases. Recently Miguel chipped in with the revamped code and most of the cases seem to work. The library is open for testing and it would be great to hear some comments on it.
Coming to the trip to California it was a great experience (not before a lot of travel … changed around 7 flights on the way and back … no more flights without leg space), I had an opportunity to interact with people from various other open source communities (some of them which I contributed and attempted to contribute to) and legends like Linus Torvalds, Peter Norvig. On my return,a lot of work had already been piled up and I had to catch up with various deadlines. Somehow I managed to finish them but not without the support of my friends.
Lately I have started reading through the work of Prof. Wang who has been an inspiration from the time I started to go through the topics of TQC. I have been making attempts to understand his paper titled “Quantum Computing : A Quantum Group Approach”. My understanding of it is the anyonic systems can be modeled through UMC’s (Unitary Modular Categories). The symmetries of crystals can be explained by Group Theory but not all phases of matter can be classified by this and so to classify these Topological phases of matter we use Fusion Categories (More of it can be found from the paper on String Net condensation, I had a look at it but nothing concrete). So to continue I had the understanding of Ribbon Categories (which are rigid braided monoidal categories with an additional isomophim which maps the object to the dual of the dual). This mathematical structure is in total sync with the anyonic system, as the rigidity property relates to the anti particle of the anyon, the braiding isomorphism relates to the braiding of anyons, and the monoidal structure explains how anyons can be put together and the interaction with the trivial anyon, then to explain the twist isomorphism (this can be viewed as the link formed when an anyon is braided around a trivial anyon is not same as not braiding at all) so we use the additional isomorphism (I think it can be called as a pivotal structure). So anyon can be related as an object in this category. To explain the fusion rules we require the semisimplicity and in addition to that we require the condition of modularity and that would make the system complete. This also can be related through Hopf algebras which share similar structure.
How can this be related to TQFT’s and link invariants ??
n+1 TQFT can be defined as a functor from n+1 bordism category to the category of finite dimensional vector spaces. (I just took the definition) from my understanding there can be some results which can be obtained in the 3+1 TQFT using such kind of a formulation.
Link invariants :
The axioms of RFC are used to define topological invariants of braided trivalent framed graphs in the plane. This statement is again from the paper and I am trying to still understand this.

I had the opportunity to realize why Jones polynomial computation is faster in quantum realm than in classical realm but I really cant write it down firmly I have an idea of how it works but I will make an attempt to explain it the next time.
I recently got into relating graphs and knots, and this might be clinging on for some time in the near future. I have read through Kauffman bracket formulation using graphs and its a simple enough idea on how we construct a graph out of a knot. We shade the bounded regions and mark as nodes the regions, which ever regions shares a crossing we draw an edge across it and we extend the rules.

I am again sorry if something is not clear, I am not good at writing mathematics in blogs but I hope I could provide the underlying connections and ideas.

Cleaning, Other projects, permissions

Hello Everyone ….
Its been a long time since I have posted anything on the work I have been doing. There has been a lot of work at hand the academics at the university as well as other constraints arising from the other projects. I would like to post about
1. The project and its current status (Knot Theory for sage)
2. Permissions on the Linux file system (And a bit of my project I have been working on, using raspberry pi)
3. Quantum Computation using Categorical ideas (This might be a little bit harsh as I am not using mathjax as of now for the symbols)

I.
So starting off with the project, the status report is as follows :
1.There has been some reviewing going on the project and corrections have been made. I have been committing the work on sage trac ticket 17030, we have moved from github to the trac server.
2. I am working on the plot method based on the ideas that Miguel has presented me with. I am not getting enough time to get through it all at once. This will be really important and will be focusing in the coming days.
3. The ncomponents method needs edits and the construction of the various representations must be in the form of tuples.
4. The documentation is not building, there are some errors as I try to build it up so I need to get this working.
5. After the above work (which is already pending and a lot) I need to steer my efforts in bringing up the two important knot invariants the HOMFLY polynomial and Khovanov homology.(These are big targets and will be great additions to the project).

II.

We are working on developing a framework for controlling the GPIO pins on Raspberry Pi over small to medium range using wireless techniques. We are more focused on the software and not on the hacking the hardware. So we had this idea of file creation and watcher watching this event on the pi whenever there is a communication between pi and the user. So where does file permissions come in : GPIO pins cant be accessed as a user, one needs root privileges. So what are the alternatives :
1. We found this pigpio daemon which could control GPIO pins without the root access.
2. We thought we could tweak a further and there in came the file permissions, so how do you do it :
set the setuid bit so that the file is run with the permissions of the owner rather than the user, but this does not work on scripts so compile a C file with the file permissions setting the uid and then running the file. This works … But the whole idea is yet to be adjusted to this adjustment.

(ITS 4_ _ _ for setting the uid) 😛

III.
Off late I have started to focus on the study of Topological Quantum Computation through Categorical Framework. I have reached upto the understanding of the Ribbion Category structure. Next would be modular tensor categories which are ribbon categories with additional structure like the semisimplicity and modularity. I have simultaneously started working on Hopf Algebras (Quantum Groups) to get a better understanding of the current design. The next targets are to understand these structures through Categorical Framework. There is some close relation between Hopf Algebras and knot invariants (I might be wrong on this) but yes it would be great to complete the incomplete picture to realize the artistic math in it. I have messed it up here with the jargons and I hope I could have put more concretely and precisely for everyone to understand. That’s it from me … Hope you didnt curse at me while reading this last instance … Anyways thanks for scanning through … 🙂

Further edits

Hello Everyone,
This week I have worked on several minor issues. The main ones are the following:
1. Check on the pd_code
Have added additional conditions, to check whether each number is given exactly twice.
2. Work on the imports.
I tried various ways before sending in a mail to Miguel in what I was missing. Finally we could figure it out and now it is possible to just create a link on the fly no need to import the module.
3. One of the most exciting things I have worked on is I have started using the SMC (sage math cloud) for development. There are some latency issues but that said wow!!! I am amazed working on SMC. I am still tweaking around with the editor but till now it has been amazing !!!!
4. So if you want to test out my work :
(a) Create an account on sage math cloud (cloud.sagemath.com)

(b) create a terminal (you have an instance in hand)

(c) git clone -b week15 <link to my github repo)

(d) move into the sage folder and hit make (while it builds grab a coffee and read my previous posts)

Test out the latest work :). Just do a pull if you are interested in the latest work. Dont forget to build once you have pulled.

I am still working on few other methods (the ncomponents is one which needs a bit of focus as we have ignored the trivial circles) and the pull request below shows the work that needs to be done.

https://github.com/miguelmarco/sage/pull/26

Edits, Evaluations and university …

Hello Everyone,
This week we focused on edits (minor ones, some related to documentation and small others in the code). The code is in the review phase and Miguel is giving it a thorough check. Once we are done with it we would be creating a ticket on trac server and then I hope the works gets merged in the master. A lot of documentation has been added on how the input works. The constructor has been completely revamped, some conditions have been added to check on the input. Last week we could identify what kind of input it was, but this time we have enhanced it to check whether it is in the required format or not. There is an conditional check to be performed on the PD code and that should be done this week. I do not remember mentioning in the previous post about the jones polynomial. We were initially working in the LaurentPolynomialRing but we have changed it to SymbolicRing so that the answer is returned with least damage. So once the support for the rational powers chips in we can revert back to LaurentPolynomialRing for Jones polynomial. Many such edits were made in the documentation.

We had the final evaluations on Friday and I have passed the evaluations. I am happy with the way things have been moving, this has been a great learning curve for me and I will be continuing my efforts in making this module more functional and robust.  The next target would be making the code reach the required standards before adding further functionality which would be the homfly polynomial and then the plot. I am very much excited in implementing the other invariants but that said the most important thing is to prepare the present code for the master.

There are a lot of people to thank who have kept me motivated and inspired during the program. Miguel has been very much supportive and has helped me a lot. He has helped me tackle some questions with ease. I would like to thank the Sage community for having accepted me and making me a part of such a wonderful team. I would like to thank David Horgan for having followed my work and inspiring me with the physics applications of the subject. Thanks to Nisanth, I still remember it all started off with us discussing FOSS and for giving some invaluable tips. Thanks to Manoj and Srikanth for all the chat we had , it would have been difficult to move forward without some of those chats we had. Hoping to continue the work and getting some results. The university has resumed and a great summer has finally come to an end. Lots of targets and lots to learn. That’s it from me. Thanks for reading through.