Category Archives: Research work

General Mathematics, Post Doc Poland, Research work

G-torsors

Leave a comment

Let me quickly define a G-torsor.

Definition A G-torsor is a non-empty set X on which a group G acts freely and transitively.

A little more explicitly, X is a G-torsor if X is a nonempty set that is equipped with a map X × G → X such that
1) x·1 = x
2) x·(gh) = (x·g)·h

for all x in X, and h,g in G such that

(x,g) → (x, g·x)

is an isomorphism of sets.

Note that here we have picked the right action of G on X.

Remark One can modify these definitions to include categories other than sets, for example topological groups and spaces or even Lie groups and spaces.

Note that as we have an isomorphism, as sets, between X and G, they are equivalent objects. However, the subtlety is that there is no preferred identity point in X.

Ethos A G-torsor is a group that has lost its identity.

Once you have picked an identity element in X, you get an isomorphism as groups between X and G. This means that X and G are isomorphic as groups, but not canonically, a choice is needed.

What is the point of all this?
So it seems at first glance that torsors are very abstract objects far too complicated to be of much use to anyone. That is, until you realise that you have been using torsors without knowing it.

A good example of the use of a torsor is the potential difference in electromagnetism. When you measure a voltage, you in fact measure the difference of some voltage relative to some other fixed voltage. In practice one takes the ground to be zero, but this is a choice. Other values would work just as well. You can think of voltages as being elements of a torsor as there is no fixed identity voltage to measure against.

Energy in classical physics is very similar. The energy of a specified isolated system only really makes sense when one has set the “zero point energy”. One can only really measure energy differences relative to the “zero point energy”. This is why one can arbitrarily shift energies without effecting the physics. Actually, this is important when looking at the notion of energy in quantum field theory, but that is another story. Anyway, energies can be viewed as being elements of a torsor, you have no fixed “zero point energy” to measure all other energies against.

Physics is littered with similar examples.

A counter example would be temperature. We have a zero point temperature, that is absolute zero fixed for us.

Mathematics of course has lots of its own examples of torsors.

Consider a vector space V we can take G to be the general linear group GL(V) and X to be the set of all ordered bases of V. The group G acts transitively on X since any basis can be transformed via G to any other basis. In essence, one can take a specified basis and transform it into any other basis. Thus, one can consider all other bases as transformed versions of the initial basis. However, there is no natural choice of this “identity frame”. The set of bases do not form a group, but rather a torsor.

I will let the John Baez explain further here.

General Mathematics, Research work

Compatible homological vector fields

Leave a comment

In an earlier post, here, I showed that the homological condition on an odd vector field \(Q \in Vect(M)\), on a supermanifold \(M\), that is \(2Q^{2}= [Q,Q]=0\), is precisely the condition that \(\gamma^{*}x^{A} = x^{A}(\tau)\), where \(\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)\), be an integral curve of \(Q\).

A very natural question to answer is what is the geometric interpretation of a pair of mutually commuting homological vector fields?

Suppose we have two odd vector fields \(Q_{1}\) and \(Q_{2}\) on a supermanifold \(M\). Then we insist that any linear combination of the two also be a homological vector field, say \(Q = a Q_{1} + b Q_{2}\), where \(a,b \in \mathbb{R}\). It is easy to verify that this forces the conditions

\([Q_{1}, Q_{1}]= 0 \), $latex[Q_{2}, Q_{2}]= 0 $ and \([Q_{1}, Q_{2}]= 0 \).

That is, both our original odd vector fields must be homological and they mutually commute. Such a pair of homological vector fields are said to be compatible. So far this is all algebraic.

Applications of pairs, and indeed larger sets of compatible vector fields, include the description of n-fold Lie algebroids [1,3] and Q-algebroids [2].

The geometric interpretation
Based on the earlier discussion about integrability of odd flows, a pair of compatible homological vector fields should have something to do with an odd flow. We would like to interpret the compatibility of a pair of homological vector fields as the integrability of the flow of \(\tau = \tau_{1} + \tau_{2}\). Indeed this is the case;

Consider \(\gamma^{*}_{\tau_{1} + \tau_{2}}(x^{A}) = x^{A}(\tau_{1} + \tau_{2}) = x^{A}(\tau_{1}, \tau_{2})\), remembering that we define the flow via a Taylor expansion in the “odd time”. Expanding this out we get

\( x^{A}(\tau_{1}, \tau_{2}) = x^{A} + \tau_{1}\psi_{1}^{A} + \tau_{2}\psi_{2}^{A} + \tau_{1} \tau_{2}X^{A}\).

Now we examine the flow equations with respect to each “odd time”. We do not assume any conditions on the odd vector fields \(Q_{1}\) and \(Q_{2}\) at this stage.

\(\frac{\partial x^{A}}{\partial \tau_{1}} = \psi_{1}^{A} + \tau_{2}X^{A} = Q_{1}^{A}(x(\tau_{1}, \tau_{2}))\)
\(= Q_{1}^{A}(x) + \tau_{1}\psi^{B}_{1} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}} + \tau_{2}\psi^{B}_{2} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}} + \tau_{1}\tau_{2}X^{B} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}}\),

and
\(\frac{\partial x^{A}}{\partial \tau_{2}} = \psi_{2}^{A} {-} \tau_{1}X^{A} = Q_{2}^{A}(x(\tau_{1}, \tau_{2}))\)
\(= Q_{2}^{A}(x) + \tau_{1}\psi^{B}_{1} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}} + \tau_{2}\psi^{B}_{2} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}} + \tau_{1}\tau_{2}X^{B} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}}\).

Then equating coefficients in order of \(\tau_{1}\) and \(\tau_{2}\) we arrive at three types of equations

i) \(\psi_{1}^{A} = Q_{1}^{A}\), \(\psi^{B}_{1} \frac{\partial Q_{1}^{A}}{\partial x^{B}}=0\) and \(\psi_{2}^{A} = Q_{2}^{A}\), \(\psi^{B}_{2} \frac{\partial Q_{2}^{A}}{\partial x^{B}}=0\).

ii) \(X^{A} = \psi^{B}_{2} \frac{\partial Q_{1}^{A}}{\partial x^{B}}\) and \(X^{A} = {-}\psi^{B}_{1} \frac{\partial Q_{2}^{A}}{\partial x^{B}}\).

iii) \(X^{B} \frac{\partial Q_{1}^{A}}{\partial x^{B}} =0\) and \(X^{B} \frac{\partial Q_{2}^{A}}{\partial x^{B}} =0\).

It is now easy to see that;

i) implies that \([Q_{1}, Q_{1}] =0 \) and \([Q_{2}, Q_{2}] =0 \) meaning we have a pair of homological vector fields.

ii) implies that \([Q_{1}, Q_{2}]=0\), that is they are mutually commuting, or in other words compatible.

iii) is rather redundant and follows from the first two conditions.

Thus our geometric interpretation was right.

References
[1] Janusz Grabowski and Mikolaj Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59(2009), 1285-1305.

[2]Rajan Amit Mehta, Q-algebroids and their cohomology, Journal of Symplectic Geometry 7 (2009), no. 3, 263-293.

[3] Theodore Th. Voronov, Q-Manifolds and Mackenzie Theory, Commun. Math. Phys. 2012; 315:279-310.

General Mathematics, Research work

Odd curves and homological vector fields

Leave a comment

On a supermanifold one has not just even vector fields but also odd vector fields. Importantly, the Lie bracket of an odd vector field with itself does not automatically vanish.

This is in stark contrast to the even vector fields on a supermanifold and indeed all vector fields on a classical manifold. Odd vector fields that self-commute under Lie bracket are known as homological vector fields and a supermanifold equipped with such a vector field is known as a Q-manifold.

In the literature one is often interested in homological vector fields from an algebraic perspective. Indeed, the nomenclature “homological” refers to the fact that on a Q-manifold one has a cochain complex on the algebra of functions on the supermanifold. You should have in mind the de Rham differential and the differential forms on a manifold in mind here.

In fact, if we think of differential forms as functions on the supermanifold \(\Pi TM\), then the pair \((\Pi TM, d)\) is a Q-manifold.

But can we understand the geometric meaning of a homological vector field?

Odd curves and maps between supermanifolds
Consider a map \(\gamma : \mathbb{R}^{0|1} \longrightarrow M\), for any supermanifold \(M\). We need to be a little careful here as we take $latex\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)$, that is we include odd maps here. Informally, we will use odd parameters at our free disposal. More formally, we need the inner homs, which requires the use of the functor of points, but we will skip all that.

Let us employ local coordinates $latex(x^{A})$ on \(M\) and \(\tau \) on \(\mathbb{R}^{0|1}\). Then

\(\gamma^{*}(x^{A}) = x^{A}(\tau) = x^{A} + \tau \; \; v^{A}\),

where $latex\widetilde{v^{A}} = \widetilde{A}+1$. This is why we need to include odd variables in our description. Note that as \(\tau\) is odd, functions of this variable can be at most linear.

Aside One can now show that \(\Pi TM = \underline{Map}(\mathbb{R}^{0|1}, M)\). Basically we have local coordinates \((x^{A}, v^{A})\) noting the shift in parity of the second factor. One can show we have the right transformation rules here directly.

Odd Flows
Now consider the flow on odd vector field, that is the differential equation

\(\frac{d x^{A}(\tau)}{d \tau} = X^{A}(x(\tau)) \),

where in local coordinates \(X = X^{A}(x)\frac{\partial}{\partial x^{A}}\).

From our previous considerations the flow equation becomes

\(v^{A} = X^{A}(x) + \tau v^{B} \frac{\partial X^{A}}{\partial x^{B}}\).

Thus equating the coefficients in order of \(\tau\) shows that

\(X^{A}(x) = v^{A}\) and \(v^{B} \frac{\partial X^{A}}{\partial x^{B}}=0\).

Then we conclude that

\(X^{B} \frac{\partial X^{A}}{\partial x^{B}}=0\), which implies that \([X,X]=0\) and thus we have a homological vector field.

Conclusion
The homological condition is the necessary and sufficient condition for the integrability of an odd vector field. Note that in the classical case there are no integrability conditions on vector fields.

Post Doc Poland, Research work

Jacobi algebroids and quasi Q-manifolds revisited

Leave a comment

I have already spoken about Jacobi algebroids and quasi Q-manifolds in earliear posts here and here. Details can be found in [1].

In the paper [1] I show that a Jacobi algebroid, which is nothing more than a linear odd Jacobi bracket on a vector bundle, is equivalent to a weight one quasi Q-manifold structure.

A little more specifically, consider the supermanifold \(\Pi E \) build from a vector bundle \(E \rightarrow M\). The supermanifold \(\Pi E \) is equipped with natural coordinates \((x^{A}, \xi^{\alpha})\). Recall that \(\Pi\) is the parity reversion functor and that it shits the parity of the fibre coordinates. So, is we have fibre coordinate \((y^{\alpha})\) on \(E\) of parity \(\widetilde{y^{\alpha}} =\widetilde{\alpha}\), then \(\widetilde{\xi^{\alpha}}= \widetilde{\alpha}+1 \). The weight is assigned naturally as zero to the base coordinates and one to the fibre coordinates. The parity reversion functor does not act on the weights.

A Jacobi algebroid is then in one-to-one equivalence with an odd vector field on \(\Pi E\)

\(D = \xi^{\alpha}Q_{\alpha}^{A}(x) \frac{\partial}{\partial x^{A}} + \frac{1}{2} \xi^{\alpha}\xi^{\beta}Q_{\beta \alpha}^{\gamma}(x) \frac{\partial}{\partial \xi^{\gamma}}\),

and an odd function also on \(\Pi E\)

\(q = \xi^{\alpha}Q_{\alpha}(x)\),

both of weight one and satisfy

\(\left[D,D\right] = 2 q D\) and \(D(q)=0\).

A supermanifold with such a structure I call a quasi Q-manifold.

Back to Lie algebroids
There is a well established one-to-one correspondence between Jacobi algebroids and Lie algebroids in the presence of a one cocycle [2,3]. A Lie algerbroid in the presence of a one cocycle is understood as a \((\Pi E, Q, \phi)\), where \(Q\) is a homological vector field of weight one and \(\phi \) is a weight one (linear) function on \(\Pi E\). Now as we are in the category of supermanifold, we need to insist that the weight one function is odd. The structures here satisfy

\(Q^{2}=0\) and \(Q(\phi) =0\).

Now, given the initial data of a weight one quasi Q-manifold that encodes the Jacobi algebroid we can pass directly to a Lie algebroid in the presence of an odd one cocyle viz

\(Q = D {-} q \Delta\)
and set
\(\phi = q\),

where \(\Delta\) is the Euler vector field, which in local coordinates looks like

\(\Delta = \xi^{\alpha} \frac{\partial}{\partial \xi^{\alpha}}\).

So, now what about the bracket on sections of this Lie algebroid and the anchor?

By thinking of the sections of our vector bundle \(E\rightarrow M\) as weight minus one vector fields on \(\Pi E\), we can use the derived bracket formalism. In particular

\(u = u^{\alpha}(x)s_{\alpha} \longrightarrow i_{u} = (-1)^{\widetilde{u}}u^{\alpha}(x)\frac{\partial}{\partial \xi^{\alpha}}\)

provides us with the appropriate identification. Then

\(a(u)(f) := \left[\left[Q, i_{u}\right],f \right] = [[D, i_{u}],f]\)

and

\(i_{[u,v]} := (-1)^{\widetilde{u}}[[Q, i_{u}], i_{v}] =(-1)^{\widetilde{u}} [[D,i_{u}], i_{v}] + i_{u}(q) i_{v} – (-1)^{\widetilde{u} \widetilde{v}}i_{v}(q)i_{u}\),

where \(u,v \in \Gamma(E)\) and \(f \in C^{\infty}(M)\).

The interested reader can now work out all the local expressions if they want, it is not hard to do so.

The final remark must be that similar formula appear in the existing literature on Jacobi algebroids for the Lie bracket. This I may try to unravel at some point.

References
[1]Andrew James Bruce, Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids, Extracta Math. 27(1) (2012), 91-123.

[2]J. Grabowski and G. Marmo, Jacobi structures revisited, J. Phys. A: Math. Gen., 34:10975–10990, 2001.

[3]D. Iglesias and J.C. Marrero. Generalized Lie bialgebroids and Jacobi structures, J. Geom. and Phys., 40, 176–199, 2001.

General Mathematics, Post Doc Poland, Research work

How mathematical proofs are obtained.

Leave a comment

Abstruse goose

The above cartoon from Abstruse Goose demonstrates the not always linear path of new discoveries in mathematics. How mathematics is discovered is not the same as how mathematics is presented.

This also reminds me of a very famous humorous quote:

mathematicians can prove only trivial theorems, because every theorem that is proved is trivial!

Richard Feynman in “Surely You’re Joking, Mr. Feynman!”: Adventures of a Curious Character, Richard Feynman, Ralph Leighton (contributor), Edward Hutchings (editor), 1985, W W Norton, ISBN 0-393-01921-7

Post Doc Poland, Research work

Warsztaty Geometrii Różniczkowej talk

Leave a comment

logo

I am giving a talk on Wednesday 6th of March as part of the Warsztaty Geometrii Różniczkowej (Workshop on Differential Geometry) here at IMPAN Warsaw. The title will be “Odd Jacobi manifolds and Jacobi algebroids”.

In the talk I will outline some of my work on odd Jacobi manifolds as well as their applications to odd contact geometry and Jacobi algebroids (a.k.a generalised Lie algebroids or Lie algebroids in the presence of a one-cocycle). The talk will be largely based on my paper “Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids” Extracta Math. 27(1) (2012), 91-123.

Link
Seminars

General Mathematics, Physics, Research work

International Conference on Mathematical Modeling in Physical Sciences

Leave a comment

modeling

The International Conference on Mathematical Modeling in Physical Sciences is to be held at Prague, Czech Republic during September 1-5, 2013. The conference aims to promote the knowledge and the development of high-quality research in mathematical fields that have to do with the applications of other scientific fields and the modern technological trends that appear in them, these fields being those of Physics, Chemistry, Biology, Medicine, Economics, Sociology, Environmental sciences etc.

Topics
ICMSQUARE-2013 topics encompass, but are not restricted to, the following areas:

  • mathematical modeling in Fundamental Physics
  • evolutionary computation
  • complex physical and technical systems
  • software and computer complexes for experimental data processing
  • qualitative modeling including fuzzy and iterative approaches to modeling
  • nonlinear problems
  • computational chemistry, biology, and biophysics
  • new generation computing tools, distributed scientific computing
  • computational modeling in engineering and science
  • multiscale modeling, multiphysics modeling
  • progress in discretization methods
  • financial mathematics and mathematics in economics etc.

Venue
The conference is to be held at Prague, Czech Republic during September 1-5, 2013 at the Conference Center of the Orea Hotel Pyramida 4*. The hotel is situated by Hradcany and Strahov, within walking distance from the Prague Castle, Loreta, Strahov monastery and the Lesser Town. The Conference Center of the Hotel Pyramida offers ten fully air-conditioned conference rooms. Together they can accommodate up to 1100 people.

Registration and submission
All the actions related to the IC-MSQUARE 2013 (paper submission, registration etc) may be completed via the Conference website at http://www.icmsquare.net.

Important dates:

Tuesday, 30 April 2013, Abstract submission deadline
Wednesday, 15 May 2013, Notification of acceptance
Friday, 31 May 2013, Early registration deadline
Wednesday, 31 July 2013, Full paper review submission deadline
Sunday, 1 September 2013, Conference opening

Organizing committee

Prof Theodosios Christodoulakis
Dr Elias Vagenas
Prof Dimitrios Vlachos

General Mathematics, Physics, Research work

The Myth of Academic Excellence?

1 Comment

Walter F.Wreszinski has written a short article called “The Myth of Academic Excellence and Scientific Curiosity” in the January 2013 edition of the News Bulletin of the International Association of Mathematical Physics [Wre]. It makes for some thought provoking reading…

The journals which we will call top A, of high “impact factor” (a rather controversial number analysed in detail in [Bin]), boast of high rejection indices, which reach 95 per cent, encouraging referees to recommend refusal in the almost totality of cases in order to justify this so important “measure of quality.” Thus, a major objective of these journals became the search for justifications to substantiate refusal, resulting in a one sided, and therefore unfair and unrealistic, view of the refereeing process.

Another interesting sentiment is quoted below.

We see that the main obstacle to the truth becoming known was the false concept of excellence associated to peer-review in top-A journals. A rejection to publish there leads to publication in a “non top A,” but this is equivalent to impart a label: the author did not succeed in publishing in top A, therefore the contribution is not first rate.

About Walter F.Wreszinski
Walter F.Wreszinski is a native of Rio de Janeiro (Brazil) and obtained his Ph.D. in 1973 at the Seminar für Theoretische Physik der ETH, Zürich, in the field of mathematical physics, having Prof. Dr. K. Hepp as thesis advisor. Since 1990 he has been full professor at the Departamento de Física Matemática, Instituto de Física, USP (University of São Paulo). His main research interests are mathematical statistical mechanics and quantum field theory.

Reference
[Bin] Mathias Binswanger, Sinnlose Wettbewerbe. Herder, 2010.

[Wre] Walter F.Wreszinski, The Myth of Academic Excellence and Scientic Curiosity, IAMP News Bulletin, January 2013 (pdf)

___________________________________________________

Note added 06/02/2013: Wreszinski’s article also appeared in the Brazilian Journal of Physics December 2012