Tycho Brahe Heat-1X: successful launch of danish spacecraft!

June 3, 2011

– Our mission is very simple. We are working towards launching a human being into space.

Today at 16:31 (GMT+1) , the two danish amateur rocket scientists Kristian von Bengtson and Peter Madsen (founders of the non-profit organization Copenhagen Suborbitals) successfully launched their first standing-room-only spacecraft Tycho Brahe Heat-1X, named after the danish astronomer. The rocket reached a height of about 2.8 km. The rocket was launched after the first attempt on sept. 5 2010, which failed.

Congratulations from here!

Interview: New Scientist

From Polyakov to Perelman: History of the Ricci Flow (and a Millennium Prize)

March 19, 2010

It has just been announced by the Clay Mathematics Institute, that G. Perelman has won the Millennium prize for “resolution of the Poincare conjecture”.

Previously I’ve written about the Poincare conjecture and Perelman’s celebrated proof of it, which was based on the so-called Ricci flow equation. Now let me talk more about how this story is related to physics, and where this equation first appeared in history.

The Ricci flow is an analog of the heat equation for geometry – it is a diffusive process acting on the metric of a Riemannian manifold. It is a special case of so-called geometric evolution equations, that describe the deformation of the metric of Riemannian manifolds driven by their curvature.
Actually, what is usually called the Ricci flow equation, constitutes a very simple equation. Let M be a n-dimensional compact manifold with positive definite metric g_{ij}. The Ricci flow equation is

dg_{ij}/dt = -2 R_{ij}(g)

where t is an external parameter defined on some interval and i,j =1, 2, …, n; the Ricci tensor with components R_{ij} is calculated from the Riemann tensor as the trace R_{ij}=R^k_{ikj}.

In physics it was introduced by Friedan in the 1980’s in the renormalization-group (RG) flow of two-dimensional sigma models. Such sigma models describe the propagation of strings in curved backgrounds. A variant of the Ricci flow appeared already in the 1970’s in the renormalization group studies of non-linear sigma models; the original computation was performed by Polyakov who considered the class of O(n) sigma models with target space metric that of the round sphere on S^{n-1} for n>2. Polyakov (1975) originally considered the O(n) nonlinear sigma models. Here the radius of the sphere is the only parameter of the theory, and as such, it serves as the inverse of its coupling constant, i.e. R ~1/g. These two-dimensional models are not conformally invariant in the quantum regime, as it was found that the coupling constant runs with respect to the world-sheet renormalization scale parameter \Lambda to lowest order in perturbation theory.

The computation of the perturbative beta function was later extended by Friedan (1980) to encompass two-dimensional non-linear sigma models with generalized coupling given by the target space metric g_{ij} of arbitrary Riemannian manifolds. Here

\Lambda^{-1}\frac{\partial g_{ij}}{\partial \Lambda^{-1}} = -\beta(g_{ij}) =-\alpha' R_{ij}-\frac{\alpha'^2}{2}R_i^{\, klm}R_{jklm}+...

where the dots stands for higher order curvature corrections, and the RHS is a perturbative expansion in \alpha'. 
The first-order term in \alpha' is

\Lambda^{-1}\frac{\partial g_{ij}}{\partial \Lambda^{-1}} = -\beta(g_{ij}) =-\alpha' R_{ij}.

Now, to see that the Ricci flow is essentially the same as RG flow, we identify the “time” t with the energy scale \Lambda through t=\log(\Lambda^{-1}), then \Lambda \sim e^{-t} and so large positive t corresponds to small energy \Lambda: flow in t is therefore a flow from the ultra-violet (UV) regime to the infra-red (IR) regime, or from small to large distance scales. The calculation above of Friedan led to the development of the sigma model approach to string theory, since the requirement of conformal invariance for the world-sheet quantum field theory gives rise to the vacuum Einstein equations in target space, R_{ij}=0 plus higher order corrections. Further generalizations can be considered by including other massless modes of the string, such as the dilaton \phi and the anti-symmetric torsion field B_{ij} to the action of the world-sheet sigma model.

Perelman’s proof (2003) of the Poincare  conjecture is related in an interesting way to this physical model. Perelman’s proof uses an “entropy” functional {\cal F}. Let {\cal M} be the space of smooth Riemann metrics on M. Define {\cal F}:{\cal M}\times C^\infty (M)\rightarrow R by

{\cal F}(g,f) =\int_M\left(R+|\nabla f|^2\right)e^{-f}dV

this is basically the Einstein-Hilbert action in general relativity, where f is a “dilaton” field; roughly speaking we go from Perelman to Polyakov by setting f=\phi. To my knowledge, the full extent of the connection between string theory and Perelman’s proof has not really been understood yet.

The Abel Prize won by John Griggs Thompson and Jacques Tits

March 27, 2008

Today, the name of the winner of the 2008 Abel Prize was announced by the president of the Norwegian Academy of Science and Letters. The winner is John Griggs Thompson (U. of Florida) and Jacques Tits (College de France). Their work is in algebra and group theory, and in particular in the classification of finite simple groups.

The Abel Prize is the mathematicians analog of the Nobel Prize. From the homepage of the Abel Fund:

“The Niels Henrik Abel Memorial Fund was established on 1 January 2002, to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003.”

The prize was first proposed to be part of the 1902 celebration of 100th anniversary of norwegian mathematician Henrik Abel’s birth; however, for various historical reasons, the prize was first to be awarded beginning in 2002, on the 200th anniversary of Abel’s birth.

The former laureates are: 2007:  S. R. Srinivasa Varadhan, (Courant Institute of Mathematical Sciences, New York University). 2006:  Lennart Carleson (Royal Institute of Technology, Sweden).
2005:  Peter D. Lax (Courant Institute of Mathematical Sciences, New York University). 2004:  Michael F. Atiyah (University of Edinburgh) and Isadore M. Singer (MIT). 2003:  Jean-Pierre Serre (Collège de France).

Dr. Thompson and Dr. Tits will hopefully have a nice trip to Oslo in late May, when they will receive their prizes from King Harald of Norway. Rumor has it, that Tits will buy a gift for his wife, and Thompson will pay for his familys trip to Oslo.

A good review (by Ron Solomon) of the problem of classifying simple finite groups can be found here [pdf].

Unparticles, Unpolitics, and Their Possible Signatures

December 11, 2007

“Unparticles” and “unpolitics” are two seemingly unrelated concepts which you might never have heard about before, so let me start by explaining the first one.

So, what is an “unparticle”? In particle physics it has recently been suggested by Howard Georgi, that there exists “stuff” which cannot be thought of as particles:

Unparticle Physics 

but nevertheless could be observed at the LHC accelerator in CERN, due to start in 2008. He calls this stuff “unparticles”. This is an intriguing and controversial idea, since our world seems to be well-described in terms of particles.

The idea of unparticles comes from the principle of scale invariance, meaning that the physics of a system remains the same regardless of a change of length (or equivalently energy). Such a scale transformation looks like x -> x’ = (e^s) x. A theory of particles can only be scale invariant if the particles have zero mass and charge: A scale transformation multiplies the mass with a rescaling factor raised to the mass dimension. The standard model of particle physics is surely not scale invariant; the photon, for example, is massless, but its charge is non-zero. However, it is possible that there is another sector of the standard model, the “unparticles”, which interacts so weakly with the known particles of the standard model that they have not been observed; and which is exactly scale-invariant. It is difficult to describe the detailed physics of such a sector, but important characteristics at low energy can be derived from scale invariance. One important consequence is that unparticle stuff will look in the detector like a non-integer number of invisible particles. For example, it could happen that 3/7 particles were missing in the detector. Such an observation would be a very clear sign of something interesting going on!

While you might need a 2 billion EUR detector like LHC to discover unparticles, “unpolitics” is easy to recognize. But, what is “unpolitics”?

While following the general election in Denmark in Nov this year, I thought that a new term, unpolitics, should apply to one of the parties, called New Alliance (Ny Alliance). However, I later realized that such a term already existed, but used as meaning “apolitical”, or “not being concerned with politics”. This is not exactly how I am going to define it. 

 Ny Alliance (New Alliance) is a danish political party which was founded in May 2007 by Naser Khader and two others. Naser was a member of the Social Liberals Party, but wanted to counter the influence of the right-wing and xenophobic Danish People’s Party. At first, this project gave New Alliance a lot of momentum, and early opinion polls indicated that they could secure 12 out of 179 seats in the Parliament. In the November election they only managed to get 5 seats, and a times it was uncertain if they would be able to be represented at all. Why was this so? One of the main reason, I think, is that New Alliance is a typical representative of what I will call “unpolitics”.

Unpolitics is “stuff” in the world of politics, which is represented by political persons, but which not really can be thought of as politics. In unpolitics, the most important elements are often popular persons, but with no, or just very few, really new ideas. One idea of New Alliance was to reduce the income tax to 40%; a member even suggested that the 40% could be experimentally implemented on Denmark’s third-largest island. This proposal was quickly abandoned. Another idea is free food for school children. New Alliance has been notoriously slow in formulating a detailed party program. When asked about concrete political questions, the typical answer was that such an answer could not be given, since they represented a “new” approach towards danish politics. Their main reason of existence just being to counter the influence of another party. In reality this did not happen. 

Therefore, a possible signature of unpolitics is this. Unpolitics is scale invariant: at every scale – large and small – you don’t find any “stuff” of politics, just popular persons.

References: Howard Georgi’s two papers on unparticles, hep-ph/0703260, and 0704.2457 [hep-ph]

The Nobel Prize in Physics 2007

October 9, 2007

And the prize goes to…. 

Albert Fert (France), and Peter Grünberg (Germany), “for the discovery of Giant Magnetoresistance” (GMR). More details about the physics of GMR over at Clifford Johnson’s blog.

Via: nobelprize.org

String Theory: Crash Course

April 21, 2007

A Tool for Living in the 21st Century:

The excellent Seed magazine offers a cribsheet [PDF] which includes a basic introduction to string theory.

Prof. Clifford Johnson over at Asymptotia.com was an adviser.

Other interesting cribsheets are one on stem cells, and one on climate change.

Global Temperature, Global Warming?

March 17, 2007

What is global warming? Most people would answer this seemingly simple question with something like the following (see the article at wikipedia.org):

Global warming is the observed increase in the average temperature of the Earth’s near-surface air and oceans in recent decades and its projected continuation. […]

One would think that all scientist agree on this definition. However, actually they don’t.

Some scientist would say that it does not even make sense. The June 2007 issue of the Journal of Non-equilibrium Thermodynamics includes a paper of Christopher Essex (U. of Western Ontario), Ross McKitrick (U. of Guelph) and Bjarne Andresen (Niels Bohr Institute), with the interesting title:

Does a Global Temperature Exist? [PDF]

In this paper it is argued that the concept of a “global temperature” is thermodynamically as well as mathematically meaningless. First of all, you cannot just add local temperatures on the Earth and then take the average to define a single “global” temperature of the Earth. Secondly, the average is not canonically defined. For example, taking a box of air with temperature 0 degrees and an identical one with temperature 100 degrees would lead to an arithmetic average of 50 degrees (add the two numbers and divide by two). However, the geometric average in this case, obtained by multiplying the two numbers (in degree Kelvin) and taking the square root is 46 degrees. Thus claims of distaster – or not – maybe a consequence of the averaging method used.

So, what is Global Warming? Can it be defined in a sound way, both from a physics and mathematics viewpoint?

Eli Rabett over at Rabett Run thinks that this paper is “a bowl of steaming crap”; I guess Lubos Motl thinks otherwise.

Update: The climate-friends at RealClimate.org thinks that this paper is irrelevant.