Symmetry and Physics

It’s getting late, but I can’t help myself. Reading too many wrong things about symmetry and physics on Twitter has forced me to do this. And, John Baez says I don’t explain things. So, here’s what the relationship between symmetry and physics really is.

In the language of mathematicians, talking about “symmetries” means you are talking about groups (often Lie groups, or their infinitesimal versions, Lie algebras) and representations. The relation to physics is:

Classical mechanics (Hamiltonian form)

In classical mechanics the state of a system with $n$ degrees of freedom is given by a point in phase space $P=\mathbf R^{2n}$ with $n$ position coordinates $q_j$ and $n$ momentum coordinates $p_j$. Functions on this space are a Lie algebra, with Lie bracket the Poisson bracket
$\{f,g\}$. Dynamics is given by choosing a distinguished function, the Hamiltonian $h$. Then the value of any function on $P$ evolves in time according to
$$\frac {df}{dt}=\{f,h\}$$
The Hamiltonian $h$ generates the action of time translations. Applying the same formula, other functions generate the action of other groups (spatial translations, rotations, etc.). If your function satisfies $\{f,h\}=0$, it generates a “symmetry”, and doesn’t change with time (is a conserved quantity).

Quantum mechanics

Quantization of a classical system is something mathematically obvious: go from the above Lie algebra to a unitary representation of the Lie algebra. This takes elements of the Lie algebra (functions on $P$) to skew-adjoint operators on a Hilbert space, the space of quantum states. There’s a theorem (Stone-von Neumann) that says that (modulo technicalities) there’s only one way to do this, and it gives an irreducible unitary representation that works for polynomials up to degree two. For higher degree polynomials there will always be “operator ordering ambiguities”. The representation is given by
$$1\rightarrow -i\mathbf 1,\ \ q_j\rightarrow -iQ_j,\ \ p_j\rightarrow -iP_j$$
This is a representation because
$$\{q_j,p_k\}=\delta_{jk}\rightarrow [-iQ_j,-iP_k]=-i\delta_{jk}\mathbf 1$$
The right-hand side is the Heisenberg commutation relations for $\hbar=1$.

Posted in Quantum Mechanics | 32 Comments

Glashow Interview

David Zierler, the oral historian at the American Institute of Physics, has done many in-depth interviews with theoretical physicists in recent years. Today I came across a 2020 interview with Shelly Glashow, which was very interesting in general, and also answered a question I had always wondered about. Glashow was my undergraduate advisor at Harvard, where I was a student from 1975-79. From what I remember, his office was more or less next door to Steven Weinberg’s. It was well-known that they had been close friends, in the same class first at Bronx High School of Science, and then at Cornell. Towards the end of my time at Harvard I heard that their friendship was over and they were barely on speaking terms, but I never knew what had happened. In the fall of 1979, they were (together with Abdus Salam), awarded the Nobel Prize for their work on the unified electroweak theory.

In the interview, Glashow explains the story from his point of view:

Glashow:
by the late 1970s I began to think of myself as a Nobel contender. But I was under the impression that my old friend Steven Weinberg was doing everything in his power to keep the prize for himself and Salam. In particular—at a conference that he attended in Tokyo—he went out of his way to avoid mentioning my name at all while presenting the history of weak interaction theory. I got very upset by that omission. It was the issue which terminated our friendship. In the summer of 1979, I was invited to a meeting in Stockholm, to discuss the current state of physics ideas and others. Prior to the meeting, I sent a transcript of my talk to Steve. He was violently against my giving the talk. Because it examined various alternatives to what was then known as Weinberg/Salam theory. In fact, it was an open-minded talk in which I was discussing whether their—or more properly—our theory was a correct one or not. But it was such a heated discussion that I eventually had to simply hang up on him, because I had no intention of revising my talk. And I did not.

Zierler:

Glashow:
I did talk about alternatives to the Weinberg-Salam theory. Yes. I was not yet convinced that it had to be true.

Zierler:
And what was your sense of why this was so unacceptable to him?

Glashow:
He thought it would endanger the Nobel Prize that he had campaigned for and anticipated for Salam and himself.

A copy of Weinberg’s Tokyo paper is here.

In the interview Glashow is scornful about Salam’s work and the campaign to get him a part of the Nobel Prize:

Glashow:
… Recall that Salam made a great deal of noise about why the prize should be given to he, Salam. I’ve been told that there were dozens and dozens of nominations of Salam. In fact, there’s a whole paper written about his shenanigans, which I can refer to you; written by Norman Dombey. Everything he says is true, to my knowledge….

My Nobel Prize depended on that one paper written in 1960. Steve’s Nobel Prize depended exclusively on that one paper he wrote in 1967, a wonderful paper which applied the notion of spontaneous symmetry breaking to the—my electroweak model. So, the question arises, what did Salam do? He introduced the electroweak—the SU(2)XU(1) model in 1964. That was over three years after I did. He copied my work but did not cite me…

Zierler:
Do you want to comment on why then he would have been a co-recipient of the Nobel Prize with you for this copy of your work?

Glashow:
I’ll explain it in a moment. But let me come back to—he also claims to the first to introduce spontaneous symmetry breaking in the paper that he wrote in 1968, one year after Steve wrote his paper. But that paper even cites Steve’s paper, so it is hardly the first time. He did what each of us had previously done, but much later. So why did he get a Nobel Prize? Very simply, he was nominated many times. Because he was Director of the International Center for Theoretical Physics in Trieste, Italy and he was very close with the directors of physics institutes in many countries; almost 100 of different institutions. And many of them wrote letters, by his instruction, using his words in some cases, encouraging the Nobel Committee to give the prize to him and also Steven. All of this documented, in fact, by the paper by Norman Dombey, who had access to Salam’s files in Italy, and has copies of the letters that he sent to other people encouraging them to nominate him. So, I think he shared the prize because he made a point of doing just that.

I wrote something on the blog about Donbey’s claims here.

Zierler also asks Glashow some questions about string theory, a topic on which Glashow’s views have been consistent from the beginning:

Zierler:
In retrospect, Shelly—how well do you think—has both string theory and your criticism of it aged over the past 30 years?

Glashow:
Well, it’s hard to answer that. String theory has become an established part of physics departments throughout the world, more so in Europe than in America. We still have some universities which are proudly string-free, like Boston University. We also have an awful lot of string theorists around who are twiddling their thumbs. It is not clear that string theory is going anywhere. I expect that string theorists would disagree with that assessment. But they are actually considering many other circumstances such as black holes in other spaces than ours, and there are all kinds of interesting things being done in mathematics, in physics, elsewhere by string theorists but with no relationship to the questions that interest me. They cannot answer the questions they set out to answer. That much is clear.

Zierler:
That’s as clear to you—

Glashow:
That was clear from the beginning, I think…

Glashow:
… I no longer feel so strongly about string theory. Why beat a dead horse? String Theory does not answer the questions that I’m interested in. I’m sad about that. I hope that they’re wrong. I have no reason to think that their horse is, in fact, dead, but it’s dead from the point of view of being useful to my way of thinking about physics. And I think that many experimenters feel exactly the same way, because string theorists say nothing about experiments that have or could be done. They only speak of experiments that cannot be done, which is somehow not interesting.

Update: Robert Delbourgo wrote in to point to his description of what happened in 1967. Here’s the relevant part:

I have been asked by the organizers to comment upon the the birth of the standard model during 1967 and Salam’s prominent role in it. This is an excellent occasion to set the record straight and recount my view of its history; if nothing else to refute innuendos which have occasionally surfaced during the 1970s that Salam was not deserving of the Nobel Prize. That autumn of 1967 I had been in charge of organizing the seminars at IC. Because Salam was constantly on the move and hardly spent more than one month at a stretch in London, I arranged with him to give a couple of lectures on his recent research (in October, to the best of my recollection) during his spell at IC to kick off the seminar season, as it was early in the academic year. He agreed to do so even though the audience attending those talks was somewhat thin. Paul Matthews was certainly present, but Tom Kibble was away in sabbatical in the USA. My memory of his lectures is a bit indistinct nowadays, but I do remember that he kept on invoking these k-meson tadpoles which disappeared into the vacuum which induced the spontaneous breaking of the gauge symmetry: what we now know as the expectation value of the Higgs boson. The resulting model looked rather ugly – and it still is – and I admit that I paid little attention to it; nor do I think that Salam himself was especially enraptured by the model’s beauty. A week or so later, I wandered into the Physics Library and came across Steven Weinberg’s Physical Review Letter, which I noticed looked suspiciously like Salam’s attempt. I showed the article to Salam, who was rather troubled that it was almost the same as his own research, but which was of course entirely independent. Matthews and I urged him to publish his work at the earliest opportunity and this happened to be the upcoming Nobel Symposium. As they say, “the rest is history”. I hope that this account of the events at the time scotches all aspersions that Salam should not have been a prize recipient.

Posted in Uncategorized | 28 Comments

Before the Big Bang: The Origin of the Universe from the Multiverse

There’s a new book out this month, Before the Big Bang: The Origin of the Universe from the Multiverse, about which we’re told:

One of the world’s most celebrated cosmologists presents her breakthrough explanation of our origins in the multiverse.

In recent years, Laura Mersini-Houghton’s ground-breaking theory, spectacularly vindicated with observational evidence, has turned the multiverse from philosophical speculation to one of the most compelling and credible explanations of our universe’s origins.

I spent a few minutes today looking through the book in the bookstore, trying to figure out where to find the details of the “spectacularly vindicated with observational evidence.” I didn’t see any references in the book, just a claim that in 2018 the author collaborated with Eleonora Di Valentino on showing vindication by observation. Presumably this is a reference to these three papers, but who knows. I don’t see anything like that in a quick look at the papers.

For many years I’ve spent a significant amount of time reading books and papers purporting to offer scientific evidence for a multiverse, trying to carefully understand the author’s arguments and write about them here (one example involved earlier claims by this author, see here). Few physicists though seem to care that bogus claims and pseudo-science about the multiverse have overrun their field and become its public face. I’ve come to the conclusion that best to not waste more time on this.

Update: Will Kinney reminds me that he wrote a paper about this, see here, as well as here and here for more about the story of that paper. Also see another old posting, here.

Posted in Book Reviews, Multiverse Mania | 17 Comments

Strings Black Holes 2022

Each summer for nearly a quarter-century there has been a big yearly conference bringing together the string theory community. I’ve often written about these conferences on the blog, see here. This year’s version will be held next week in Vienna, for more information see here.

Taking a look at the program, one thing that stands out is that the string theory community has almost completely stopped doing string theory. Looking at the program, only two out of 44 talks seem to be significantly about string theory. One of three parallel discussion sessions is entitled “Strings and the Real World” and will be chaired by Cumrun Vafa. I’m guessing this will mostly be about the swampland, not string theory.

A tradition at these conferences is one or more public talks designed to publicize string theory. This year’s versions will be given by Netta Engelhardt and Andy Strominger. They have nothing to do with string theory, but they do make very clear what the string theory community has found to replace string theory: black holes. Engelhardt’s title is “The Black Hole Information Paradox: A resolution on the horizon?” and Strominger’s is “Black Holes: the Most Paradoxical Objects in the Universe”.

Looking at the talk titles, the most common words in the titles are “holography” and “black holes”, with the center of gravity of the subject now for a couple decades the effort to use holography to say something about black holes. Maldacena’s title is “What happens when you look at supersymmetric black holes for a long time?” which seems also an interesting question about the field itself.

Update: Paolo Bertozzini points out to me that the LQG community has scheduled its big yearly conference LOOPS2022 at exactly the same time as the string theory community one (this week). It’s quite interesting to compare and contrast the two sets of talks. There are some very broad similarities between what both communities are doing, with overlaps in interest around black holes, entanglement, holography (in the form of large symmetry groups at infinity). Another commonality is that both communities are focused on the gravitational field, with nothing to say about particle physics and matter in general. This has been true of LQG since the beginning. In the case of string theory the big selling point originally was that it gave a theory of matter, but the string community has for a long time given up on that. There is a difference in how the communities think about “what are the fundamental degrees of freedom for gravity?” On the string theory side they’ve given up on that, the answer now is that gauge-gravity duality and emergence are supposed to allow you not to care about fundamental degrees of freedom. On the LQG side, people are still hard at work on specific sorts of degrees of freedom and how to quantize them.

Posted in Strings 2XXX | 31 Comments

ICM 2022

The 2022 ICM is starting soon, in a virtual version organized after the cancellation of the original version supposed to be hosted in St. Petersburg (for how that happened, see here). The IMU General Assembly is now going on, moved from St. Petersburg to Helsinki. One decision already made there was that the 2026 ICM will be hosted by the US in Philadelphia. With the 2022 experience in mind, hopefully the IMU will for next time have prepared a plan for what to do in case they again end up having a host country with a collapsed democracy being run by a dangerous autocrat.

Registration for following the talks in real time has now been closed, but the talks are being recorded and will appear on the IMU Youtube channel. The program is here.

There will be quite a few other virtual events affiliated in some way with the main ICM, for a list see here. Some of these are traditional satellite conference which have been moved from their originally scheduled version in Russia. An example is this one organized by Igor Krichever, which was supposed to be held at Skoltech in Moscow, but was moved online and hosted by Columbia.

The Fields Medals will be announced at 10am local time in Helsinki on July 5, there will be a livestream here. This will be 3am here in New York, so I’ll likely be sleeping and find out what happened later in the morning. Since I just got back from vacation and it’s now a holiday weekend, I’ve been out of touch with my usual sources of math gossip and haven’t heard any informed rumors about who the medalists will be. One person who has been mentioned as a possibility is the Ukrainian mathematician Maryna Viazovska.

The last couple times (2014 and 2018) the IMU has put out the news about the Fields Medals to some of the press under unusual embargo terms that made reporting difficult for everyone except Quanta magazine which was given special access (for more about this see here). I haven’t heard anything about whether the same thing is happening this year.

Update: just noticed this, indicating that again press access may be Quanta-only.

Update: Antoine Chambert-Loir claims “serious information” that Viazovska will get the Fields Medal (at least that’s who he seems to be referring to). It looks like press access is going to more organizations than Quanta this time, see this from Nature. Terry Tao has a blog post with some more ICM information.

Update: The medalists are Duminil-Copin, Huh, Maynard and Viazovska, much the list of names that people have been speculating about. There’s much about the winners and their work at the IMU site, and several other press organizations have extensive coverage, including Quanta, Plus Magazine! and the New York Times. Stories about each of the Laureates from Plus Magazine! are featured on the IMU site.

The medalists were chosen quite a few months ago, before the Ukraine war. The interview with Viazovska contains part conducted before the war, as well as a more recent part about Russians and the war (the interviewers were Okounkov and Konyaev).

Update: Barry Mazur was awarded this year’s Chern Medal. During the ICM a new documentary about Mazur will be available for watching, Barry Mazur and The Infinite Cheese of Knowledge.

Update: I enthusiastically recommend that you take a look at Andrei Okounkov’s remarkable set of popular articles about the work of the four Fields medalists, see here, here, here and here.

Posted in Uncategorized | 18 Comments

In a Parallel Universe, Another You

From today’s New York Times, Michio Kaku explains:

In physics, the concept of a multiverse is a key element of a leading area of study based on the theory of everything. It’s called string theory, which is the focus of my research. In this picture, subatomic particles are just different notes on a tiny, vibrating string, which explains why we have so many of them. Each string vibration, or resonance, corresponds to a distinct particle. The harmonies of the string correspond to the laws of physics. The melodies of the string explain chemistry.

By this thinking, the universe is a symphony of strings. String theory, in turn, posits an infinite number of parallel universes, of which our universe is just one.

In this universe I’m on vacation and in no mood to waste time commenting on this crap.

Posted in Multiverse Mania, Uncategorized | 28 Comments

All Langlands All the Time

I’m about to head to Paris on vacation, quite possibly there will be less blogging for the next couple of weeks. Here are a few Langlands-related items:

Posted in Langlands, Uncategorized | 3 Comments

Physicists Discover Never-Before Seen Particle Sitting On a Tabletop

I woke up this morning to find out that a new Higgs particle which could explain dark matter has been discovered, in a table-top experiment at Boston College. For some of the news stories about this, see here and here. Wikipedia now has an entry for this that explains:

The Axial Higgs boson is a fundamental particle whose discovery was announced by American researchers in Nature on June 8, 2022.[1]

Of course this is complete nonsense. The paper Nature just published (the preprint is here) is about a condensed matter experiment that has nothing at all to do with the Higgs (effective fields in a description of a condensed matter system have nothing to do with fundamental fields).

Who is responsible for misleading the public and discrediting science with this kind of behavior?

• The authors, who begin their abstract with

The observation of the Higgs boson solidified the standard model of particle physics. However, explanations of anomalies (for example, dark matter) rely on further symmetry breaking, calling for an undiscovered axial Higgs mode.

which has nothing to do with the result in their paper.

• The editors and referees at Nature, who should never have allowed such an abstract.
• Boston College, which put out this press release, which starts out:

Chestnut Hill, Mass. (6/8/2022) – An interdisciplinary team led by Boston College physicists has discovered a new particle

In this case another institution, Oak Ridge, put out a much more responsible press release for the same paper, showing how to do this properly.

Universities desperately want to see this kind of story in the press, and there’s rarely any downside for the scientists and PR people who produce bogus such stories. Boston College needs to take action to retract the press release and make sure this doesn’t happen again. Nature should also take action to issue a correction stating this paper has nothing at all to do with the Higgs field and address the bad editing and refereeing that led to this.

Update: At least the Wikipedia article has been fixed.

Update: The Higgs hype has been extended to add quantum computing hype, see Newly-Observed Higgs Mode Holds Promise in Quantum Computing, and the Wikipedia article now includes this new, extra hype.

Posted in Uncategorized | 19 Comments

Memories of a Theoretical Physicist

Joe Polchinski’s autobiographical Memories of a Theoretical Physicist has just been published, in an open-access version that is freely available. Much of the volume is what appeared here on the arXiv back in 2017, but this has been supplemented with other material, including an introduction by Andy Strominger and detailed bibliographical notes by Polchinski’s student Ahmed Almieri. If you’re interested in the details of Polchinski’s work, I’d recommend also reading Witten’s biographical memoir here, which covers the same scientific material, but from Witten’s perspective.

Polchinski and Witten agree that one of his three major accomplishments was the anthropic string theory landscape, but I’d argue that this was the opposite of an accomplishment. Instead it should be seen as a disastrously bad scientific argument, one that became a wrecking ball that brought to an end most work towards a better, more unified theory of fundamental physics. Polchinski and Susskind were the two most influential figures in pushing for this argument in the theoretical physics community (Susskind wrote a popular book).

Everyone I’ve talked to who knew Polchinski has nothing but positive things to say about him as a scientist and as a person. I never got to meet him in person, but wish that I had, this might have improved our bad relations. Back in 2004, I wrote an early blog entry that seems to have greatly upset him, by describing (accurately I still think) a popular article on the landscape he wrote with Raphael Bousso for a Scientific American issue about Einstein and his legacy as pseudo-science that would have made Einstein gag. I was not the only one with this reaction to his work, his KITP colleague David Gross also had strong things to say on the topic. In the memoir Polchinski refers to the years of these arguments (and of the appearance of my book and Lee Smolin’s) as ones he found emotionally very difficult. At the time he somehow managed to get the arXiv to ban trackbacks to my blog, for that sorry story see here.

Towards the end of his life the landscape pseudo-science he had so vigorously promoted became a dominant point of view among influential theorists, with even Witten coming to accept it. Polchinski remained upset by my continuing complaints about the subject. One of his last papers, (this one, also see here), extensively attacked me personally, claimed that string theory was true with Bayesian probability greater than 97.5 percent and appears to have been partly a reaction to this blog post. In the blog post I made fun of his claims to have calculated the Bayesian probability of a multiverse as at least 94%. I was unaware at the time that he was already sick with the disease that would later take his life.

Much of this autobiographical memoir is rather technical and will be mostly of interest to experts and specialized historians. The complicated story of Polchinski’s career is very much the story of what happened during this time to the field of fundamental theory in physics. This is a very different story than the usual one of a scientific field’s progress towards greater enlightenment.

Update: A correspondent pointed me to something I hadn’t noticed. In section 3.3 Polchinski writes:

This was typical of Mandelstam, how far ahead he was in much of his thinking. Another example, the first paper that one studies in the
Langlands program today, is the first paper that Mandelstam gave me to read forty years ago.

I wouldn’t describe it the way Polchinski does, but I’m guessing the paper he’s referring to is Montonen-Olive.

Posted in Book Reviews | 20 Comments

This and That

Various things that may be of interest:

• MSRI in Berkeley has announced a \$70 million dollar gift from Jim and Marilyn Simons, and Henry and Marsha Laufer. This gift will make up the bulk of a planned endowment increase of \$100 million and is the largest endowment gift ever made to a US-based math institute. The success of the Renaissance Technologies hedge fund is what has made gifts on this scale possible. This summer MSRI will be renamed the “Simons Laufer Mathematical Sciences Institute”, and the directorship will pass from David Eisenbud to Tatiana Toro.
• The journal Inference has just published an article by Daniel Jassby, which gives a highly discouraging view of the prospects for magnetic confinement fusion devices. Jassby, who worked for many years at the Princeton Plasma Physics Lab, argues that performance of magnetic confinement fusion systems has not much advanced in a quarter century, making for very bleak prospects that such designs will lead to a workable power plant in the forseeable future. He sees inertial confinement fusion systems like the National Ignition Facility at Livermore as making some progress, but ends with:

The technological hurdles for implementing an ICF-based power system are so numerous and formidable that many decades will be required to resolve them—if they can indeed be overcome.

• I’ve been spending some time reading Grothendieck’s Récoltes et Semailles, which is a simultaneously fascinating and frustrating experience. I’ve made it almost to the end of the first part, except that there will be another forty pages or so of notes to go. To get to the first part involved starting by reading through about two hundred pages of four layers of introduction. It seems that basically Grothendieck did no editing. Once he was done writing the first part, as he thought of more to say he’d add notes. He distributed copies to various other mathematicians, and then kept adding new introductions, with various references to how this fit in with more technical mathematical documents he was working on (La “Longue Marche” à Travers la Théorie de Galois, À la poursuite des champs).

After the first part, looking ahead there’s the daunting prospect of 1500 pages with the theme of examining his deepest mathematical ideas and what he felt was the “burial” that he and his ideas had been subjected to after his leaving active involvement with the math research community in 1970. Quite a few years ago I did spend some time looking through this part to try and learn more about Grothendieck’s mathematical ideas. I’ll see if I can try again, with the advantage of now knowing somewhat more about the mathematical background.

Besides the frustrating aspects, what has struck me most about this is that there are many beautifully written sections, capturing Grothendieck’s feeling for the beauty of the deepest ideas in mathematics. One gets to see what it looked like from the inside to a genius as he worked, often together with others, on a project that revolutionized how we think about mathematics. This material is really remarkable, although embedded in far too much that is extraneous and repetitive. The text desperately needs an editor.

There are various places online one can find parts of the book and other related material, sometimes translated. Two places to look are the Grothendieck Circle, and Mateo Carmona’s site.

• For an up-to-date project on reworking foundations of mathematics (with an eye to eliminating analysis…), Dustin Clausen and Peter Scholze are now teaching a course on Condensed Mathematics and Complex Geometry, lecture notes here.
• I noticed that the Harvard math department website now has an article on Demystifying Math 55. The past couple years this course has been taught by Denis Auroux, and one can find detailed course materials including lecture notes at his website.

The current version of the course tries to cover pretty much a standard undergraduate pure math curriculum in two semesters, with the first semester linear algebra, group theory and finite group representations, the second real and complex analysis. The course has gone through various incarnations over a long history, and has its own Wikipedia page. For various articles written about the course over the years, see here, here (about a Pavel Etingof version) and here (about a Dennis Gaitsgory version).

I took the course in 1975-76, when the fall semester was taught by mathematical physicist Konrad Osterwalder, who covered some linear algebra and analysis rigorously, following the course textbook Advanced Calculus by Loomis and Sternberg. The spring semester was rather different, with John Hubbard sometimes following Hirsch and Smale, sometimes giving us research-level papers about dynamical systems to read, and then telling us to read and work through Spivak’s Calculus on Manifolds over reading period.

My experience with the course was somewhat different than that described in the articles above, partly due to the particular instructors and their choices, partly due to the fact that I was more focused on learning as much advanced physics as possible. I don’t remember spending excessive amounts of time on the course, nor do I remember anyone I knew or ran into being especially interested in or impressed by my taking this particular course. What was a new experience was that it was clear the first semester that I was a rather average student in the class, not like in my high school classes. The second semester about half the students had dropped and I guess I was probably distinctly less than average. The current iteration of the course looks quite good for the kind of ambitious math student it is aimed at, and it would be interesting if a new textbook ever gets written.

Update: One more related item. This week Chapman University is hosting a conference about Grothendieck. Kevin Buzzard has posted his slides here.

Posted in Uncategorized | 22 Comments