There are some interesting books, such as

• Ricci Flow and the Poincaré Conjecture
  John Morgan and Gang Tian
• Mirror Symmetry
  Cumrun Vafa and Eric Zaslow
• Strings and Geometry
  Michael Douglas, Jerome Gauntlett, Mark Gross

They are all wonderful books. The Mirror Symmetry book, which I often use as a reference, is the most complete literature survey I have found on the subject; it is nearly a 1000 pages! Thanks to the PDF version, now I can just grab my laptop and move around the world with all the heavy books right next to me!

I also gave a talk on this subject for the Part III return conference in Cambridge, which was held for theoretical physicists that did Part III last year. It was nice to see all my friends again. Here is a cute teaser by Flip for the fifth Part III return conference. I was initially planning to blog about the conference, but I decided against it. Mostly because I didn’t understand most of the talks, except for Cyril’s talk on spin geometry.

Since all of the dust has settled down, I will (attempt to) start blogging regularly. I can’t promise anything now, but I am hoping to write about generalized complex geometry. There is one problem, though, I have to learn it beforehand! So patience and silence might be needed.

Pauli and Schwinger had formulated CPT theorem before people learned that in weak interactions, actually P is not conserved. (Even though P is not conserved in weak interaction, CPT theorem is not violated in the end. – I don’t know how to show this, but I know it is well-known (to others).) Before Pauli knew the results of the experiment of Wu, he wrote to Weisskopf in a letter dated January 17, 1957[1]:

[...] I do not believe the Lord is a left-hander, and I am ready to bet a very high sum that the experiment will give symmetric angular distribution of the electrons. [...]

Georges M. Temmer was on a laboratory tour in western Europe at that time, and Pauli asked him what was the latest news from the States. Temmer said that parity is no longer assumed to be “conserved”. Pauli replied with great surprise “that’s total nonsense”. Temmer said “I assure you that the experiment says not”. Pauli curtly, “then it must be repeated!”. [2]

After knowing the results of the experiment, Pauli wrote to Weisskopf in January 27, 1957.

[...] Now the first shock is over and I began to pull myself together. [...] It is good that I did not make a bet. I can afford making a fool out of myself, but cannot afford loosing money. [1, and re-translated by my friend]

But stories around weak interaction does not end on Pauli. T. D. Lee and C. N. Yang are also well known to have been in bad relationship for a long time.

It was when C. N. Yang was in Princeton. T. D. Lee came from Chicago and visited him and had a private conversation in his office. After a few minutes, people in the hallway could hear both of them shouting at each other, and T. D. Lee burst out of Yang’s office and slammed the door behind. Unfortunately, nobody around could understand Chinese, and the reason of quarrel has been a mystery since. People guess it was about who, among them, first proposed the idea of parity non-conservation for weak interaction.

There is another episode. It happened when Yang was visiting Malaysia for a conference. The Malaysian press were interested at the Nobel Laureate, and asked “what do you regret most in your life?”. Yang said, “the fact that I selected T. D. Lee”. [3]

Sources
[1] Tung-Mow Yan, Professor C. N. Yang’s Impact on Physics
[2] David R Lide, A Century of Excellence in Measurements, Standards, and Technology
[3] ExtraD, C. N. Yang and T. D. Lee (Korean)

His lecture notes `Riemannian holonmy groups and calibrated geometry‘ is a summary of his books, but it is more geared toward SYZ conjecture. It is similar to the first section of `Calabi-Yau manifolds and Related Symmetry‘ (Springer), but has less material, thus making it ideal for people who do not want to learn all the details in the area.

The lecture notes take the differential geometer’s point of view. Rather than working with indecies intensively, mostly it uses index free notation. It starts from the very basics to establish notation, it defines tensors and forms, connections, holonomy groups, and Berger’s classification of holonomy groups. I really liked this way of approaching manifolds. I must say even though this lecture notes did not have direct impact on my research, when I was reading it, it was one of the happy moments that I couldn’t wait to go back to my office and read more about it.

The lecture notes quickly move onto Kahler manifolds and Calabi-Yau manifolds. Most of the materials are stated without great details, because Dominic wants to move on into calibrated geometry. I guess if you are not familiar with complex manifolds, the lecture notes might feel challenging by this point. For introductory notes on complex manifolds, there are Candelas, Bouchard, Nakahara. For people who are more mathematically minded there is also Huybrechts, but don’t get fooled by its title that claims it is an introduction, it is quite advanced.

After discussing all the preliminary material, Dominic moves on to calibrated geometry. But why calibrated geometry? It is because he wants to discuss about the SYZ conjecture. Calibrated geometry turns out to be the most natural framework to work in. This is how the idea goes roughly: if we define what a calibration is carefully, we can show that a calibrated submanifold is a volume-minimizing manifold in its homology class. This corresponds to BPS states and D-branes in the SYZ conjecture. (To be honest, I do not know how to derive that BPS states and D-branes are calibrated submanifolds. This is originally dicussed in Becker, Becker, Strominger 1995. Later it was noticed that this is equivalent to calibrated geometry, which Harvey and Lawson had already formulated in 1982.)

The lecture notes devote three sections on calibrated geometry. Priori reading these lecture notes, I had no background knowledge in calibrated geometry, but had no problem in following the lecture notes. It gives a very friendly introduction to calibrated geometry and moves on and discuss the deformation of special Lagrangian submanifolds.

Finally, the lecture notes land on the SYZ conjecture, which is one of Dominic’s primary field of research. There is not that much known about the SYZ conjecture, and the SYZ conjecture is a area with high research activity. In the last section Dominic gives an excellent overview of what are the open problems and what has been happening recently.

Riemannian holonomy groups and calibrated geometry lecture notes is a perfect place to learn about calibrated geometry and SYZ conjecture without getting lost in the details. You can see through Dominic’s eyes where mathematical gears fit together and how the engine runs. I highly recommend reading it.

Further readings

1. Gross, Huybrechts, Joyce, Calabi-Yau Manifolds and Related Geometry.
2. Joyce, Riemannian Holonomy Groups and Calibrated Geometry.
3. Joyce, Compact Manifolds with Special Holonomy.
4. Salamon, Riemannian geometry and holonmy groups.
5. Harvey and Lawson, Calibrated geometrie, Acta Mathematica 148 (1982), 47-157.
6. McLean, Deformations of calibrated submanifolds, Comm. in Analysis and Geometry 6 (1998) 705-747.
7. Strominger, Yau, and Zaslow, Mirror Symmetry is T-duality, hep-th/9606040

1. Fulton, Algebraic Topology
2. Gross, Huybrechts, and Joyce, Calabi-Yau manifolds and related geometry
3. Hartshone, Algebraic Geometry
4. Harris, Algebraic Geometry
5. Humphreys, Introduction to Lie Algebras and Representation Theory
6. Lee, Introduction to Topological Manifolds
7. Lee, Introduction to Smooth Manifolds
8. Smith, Kahanpaa, Kekalainen, and Traves, An Invitation to Algebraic Geometry
9. Warner, Foundations of Differentiable Manifolds and Lie Groups

1. Link to the person who tagged you.
2. List 7 random/weird things currently on your mind.
3. Tag more people at the end of your blog and link to theirs.
4. Let the tagged people know by leaving a note on their site.

Seven random/weird things:

1. A canonical line bundle of a complex symplectic manifold is trivial.
   I was asking my friend about this over lunch, but we were not able to come to a conclusion. After talking with him, I am not sure whether that this is even true for all complex symplectic manifold. Any ideas?
2. I like Witten’s style of writing.
   I am currently reading Witten’s “Mirror Manifold and Topological Field Theory” for my “Calabi-Yau Manifolds and Mirror Symmetry” course mini-project. He has a very clear style of writing papers. I like it because he writes comments like “I do not know how to do this”, and he knows which details can be hidden so that the reader could easily get the big picture.
3. Squash is good for the soul.
   I played squash in the morning at 7:30am. It’s a good way to start a new day. Currently, I am working on my backhand and volley. I used to run, do push-ups and sit-ups, but I found out that playing squash is more fun!
4. The end of the term: party!
   At last, all of my lectures for this term is over! Yesterday, I went to Prof. Philip Candelas’ last string theory course for the term. I will miss talking about strings with Philip over coffee after his lectures – he is good at explaining things, but I could use some extra time on reading papers. My friends and I went out for dinner and had some drinks.
5. The ftplugin for php files in vim is overwritten by another file, but can’t figure out which one.
   I am trying to use folding for php files in vim. Folding is quite powerful and convenient when I am trying to understand the overall structure of a document. When I try this for a php file by setting commentstring to \<!–%–\> in ftplugin, it resets to /*%s*/ when I actually open the file. It is strange since all the settings I wrote in the ftplugin for php works, but only the commentstring has a problem. I guess when vim loads it reads the ftplugin and another config file, and overwrites the one I wrote. But I don’t know how to get around this, any ideas?
6. I need a better LCD.
   According to a research from the University of Utah, generally if you get a bigger LCD monitor, the productivity increases. (For full details and the assumption made in the experiment, read the linked document.) My laptop has a 12″ LCD monitor, and I think it is about time to buy a decent one.
7. Firefox sucks.
   Firefox (more precisely, Iceweasel) died while I was writing this post. OK, maybe it wasn’t Firefox’s fault, but I just opened three tabs and it just halted. Maybe it might be the Youtube plugin for Firefox that had a problem. Anyway, I don’t really like Firefox, it is slow and needs to improve plugin management.

Now I am going to tag my friends. But be aware, there are lots of blogs written in Korean. (I am not sure whether my friend will write a blog-meme in Korean.)

• Space, Time & Matter (Korean)
• 11th fear (Korean)
• blog.echo (Korean)
• Cogito, ergo sum! (Korean)
• My Heart Unfolding
• Oxford, Abridged.
• Quantum World (Korean)
• Raven hair (Korean)
• tiroo.net (Korean)
• Youngkuk’s Blog (Korean)
• 프로페셔널 찌질이 (Korean, obviously)
• 피타고라스의 창 (Korean, obviously)
• extraD (Korean)

I personally know all the authors of the listed blogs. I didn’t include other blogs I read, but do not know the blogger personally. I am going to break one of the rule slightly – I am only going to notify two blogs, the ones written in English, that I tagged them.

Path Integral Approach
The book has a rather unconventional approach – it briefly covers canonical quantization, and moves on with path integrals. Considering the fact that older books such as Peskin, introduce path integrals half way through the book, this approach is quite different. I would call it the modern approach to QFT. I found it much easier to work with the path integral picture rather than the canonical picture. Some of my friends who are phenomenologists says that the canonical method is more physical, but I think path integral method is better in giving the big picture.

Organization of the book
The book starts from spin 0, and builds up QFT by introducing particles with more spin. It is amazing to see how this works so well. In a theoretical point of view, I think this approach is natural, since the tools behind spin 0 is significantly simpler than that of spin 1/2.

If one just wants to learn the basics of QFT, I think the spin 0 part of Srednicki is the perfect place to consult. The book presents most of the physical tools/ideas of QFT in spin 0 part. While we don’t have to worry about the representation of the Lorentz group and the spinor indeces, Srednicki introduces loop corrections, scattering amplitude, renormalization, 1PI diagrams, perturbation theory, and more in the first part of the book. If the reader wants to learn more, then s/he can proceed on to spin 1/2 and spin 1 part.

Renormalization is introduced relatively early in the book. This book is the only book I found that explains renormalization in a natural way. Usually, in other books renormalization is introduced after QED, and by then I think the beginner gets used to QFT and it seems strange that it turns out that we should renormalize the theory. When I was learning renormalization in class, my lecturer said that there is a deep reason behind the procedure, but never told us exactly what it is. I had to find it inside this book.

Structure of the book
The mathematics are built as Srednicki moves on to advanced topics. One outstanding feature of the book is that it is split into small modules, and usually the mathematics (and also the physics) are contain in a small self-contained section. In the beginning of each section, prerequisites are clearly stated. This aspect of the book makes it also ideal for a reference book.

Up-to-date
Finally, another advantage of the book is that it is up-to-date. The arguments the author presents are gathered from various sources in a coherent modern approach. In the end of each section, Srednicki gives references for further reading.

Conclusion
I think Srednicki’s QFT is the best QFT book I have read. I have Peskin, Ryder, Ticciati, Weinberg, and Srednicki’s QFT. My favorite book and the book that I always consult first is Srednicki. I think if my friend had encountered this book before, he wouldn’t have ended up being an algebraic topologist! Too good to be true? You can read Srednicki’s QFT draft and check it out yourself!

Zeldovich wrote a textbook called the “Mathematics for beginners and physics and engineering”. In his book a derivative was defined as

The ratio of a functions increment to an argument’s increment, assuming the latter is sufficiently small.

according to his own words,

There is no point to consider increments less than : the structure of space and time is not described by a mathematical continuum for such small intervals.

Not surprisingly this approach made the mathematicians quite upset. In the end, one of the main critic Pontryagin wrote his own book on analysis. According to Arnold, this book was extremely dull, but there was an interesting comment in the preface:

Some physicists assume that one could do analysis correctly without an exhaustive study of its fundamentals.

Then he added: “I agree with this”.

It’s not often that I see software that really changes my world. It’s even rarer that I see software that I know will change the world my sons live in. I can count those times pretty easily. The first time I saw an Apple II in 1977. When Richard Cameron showed me Apple’s Hypercard. Microsoft’s Excel. Aldus’ Pagemaker. And something called Photoshop, all in his West Valley Community College classroom. Later when I saw Marc Andreessen’s Netscape running the WWW. ICQ and Netmeeting which laid the ground for Skype.

from Scobleizer

WorldWide Telescope is a interesting software, but will it impact ourlives as Scoble predicted? Well, I guess the education sector will benefit from it, but I am a bit skeptic whether WorldWide Telescope will change our daily life as much as the other products he listed above.

TED has released the video of the first public presentation of WorldWide Telescope.

Fortunately, the attempts of philosophers to exterminate (after genetics) most parts of theoretical physics (particularly, the theory of relativity and quantum mechanics, and at the same time mathematics with its “non-materialistic” infinities and limits failed in USSR.

When I talk to philosophers, I sometimes get frustrated by their arguments that science is basically wrong, set theory is just a illusion, and so on. It is quite frustrating when they try to argue based on philosophical reasoning, which I find mostly idealistic. I wondered how the scientists in USSR defended against such arguments. Arnold continues in his book, and provides the answer:

[..] it was mainly Kurchatov who saved science in our country. He told Stalin and Beria that to abolish quantum mechanics and the theory of relativity was possible, and technically it was even easier than to abolish genetics (which had already been done), but in this case atomic bombs should be produced by those destroyers themselves and without the help of these sciences (which is impossible).

I guess this would be a good example to point out. Now when I talk to philosophers I got a weapon to defend science!