A weird dream I’ve had for around 20 years is a machine-learning algorithm that reads a paper for you and places it in the context of other papers. It doesn’t try to judge importance or correctness, but it merely tells you which ideas the paper genuinely depends on (as opposed to which citations are actually included, and among those which are just for political reasons) and which other papers genuinely depends on it, and what the main result of the paper is. This would help a lot to find relevant literature, and indeed to indicate at a glance how important a paper might actually be in a much more accurate way than a citation count or even, perhaps, than a MathSciNet or Zentralblatt review.

In any event, I find it weird that citations which are just “here is somebody else (famous) who used vaguely similar keywords” are weighted exactly the same as “my main theorem is based on this idea in an essential way”.

]]>Geometric Structures on 3-manifolds: https://www.math.ias.edu/wgso3m

and “Flows, foliations and contact structures” will be during the week of December 7-11, 2015 (there will eventually be a link here https://www.math.ias.edu/sp/geometricstructures). ]]>

We have published the first part “homological nature of entropy”

http://www.mdpi.com/1099-4300/17/5/3253

There are many different nice approaches, and there is an amazing current convergence of some of them. We organize a little sattelite session with different approaches to help the “entire field” to appear, here is the speech:

Geometric Science of Information, session Topology and information. GSI-2015 web Site 2015 session of GSI-2015, École Polytechnique, Palaiseau, France, oct. 2015 (with D. Bennequin). École Polytechnique, Palaiseau, France, oct. 2015. (with D. Bennequin).

http://www.gsi2015.org/

In the context of the 2nd conference on Geometric Science of Information, we organise a session dedicated to “Information and topology”, that will review the results on information functions in algebraic geometry and topology, and discuss their potential development in algebraic topology, information theory, thermodynamic and applications. The conference Geometric Science of Information will take place at Ecole Polytechnique, Paris-Saclay (France) from October 28th to October 30th 2015. Please, do not hesitate to forward this announcement and the call for paper to your colleagues that could be interested in this session or in the other topics covered by the conference.

Informative motivation of the session:

The discovery of entropy and thermodynamic has accompanied the industrial revolution, the development of its engines and mechanical machine. The formalisation of information has in turn accompanied the numeric revolution with its digital computer machines. The uncover of Information’s functional equation in the context of motives, of new information inequalities, and the axiomatization of information within category-homology theory, unravel new aspects on what is Information-entropy. This workshop reviews these first steps and new developments, toward an effective thermodynamic and information theory, following those algebraic geometry and topology lines.

The topics will be operad, cohomology, polylogarithms, motives, and entropic inequalities, in the perspective of thermodynamic and information theory.

It would be nice if you could come. here are some ref. on the subject I could find:

Baez, J.; Fritz, T. & Leinster, T. A Characterization of Entropy in Terms of Information Loss Entropy, 2011, 13, 1945-1957 PDF

Baez J. C.; Fritz, T. A Bayesian characterization of relative entropy. Theory and Applications of Categories, 2014, Vol. 29, No. 16, p. 422-456. PDF

Baudot, P. & Bennequin, D. Topological forms of information. AIP Conf. Proc., 2015, 1641, 213-221 PDF

Baudot P., & Bennequin D. The homological nature of entropy. Entropy, 2015, 17, 1-66; doi:10.3390 PDF

Bloch S.; Esnault, H. The Additive Dilogarithm, Documenta Mathematica Extra Volume : in Kazuya Kato’s Fiftieth Birthday., 2003, 131-155. PDF

Burgos Gil J.I., Philippon P. , Sombra M., Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque 360. PDF

Cathelineau, J. Sur l’homologie de sl2 a coefficients dans l’action adjointe, Math. Scand., 1988, 63, 51-86 PDF

Connes, A. & Marcolli, M. Noncommutative Geometry, Quantum Fields and Motives, Colloquium Publications American Mathematical Society, 2008, 55. PDF

Elbaz-Vincent, P. & Gangl, H. On poly(ana)logs I., Compositio Mathematica, 2002, 130(2), 161-214. PDF

Gmeiner P., Information-Theoretic Cheeger Inequalities, Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik.

Gromov, M. In a Search for a Structure, Part 1: On Entropy, unpublished manuscript, 2013. PDF

Gromov, M. Symmetry, probability, entropy. Entropy 2015. PDF

Gromov, M. Morse Spectra, Homology Measures, Spaces of Cycles and Parametric Packing Problems, april 2015. PDF

Kontsevitch, M. The 1+1/2 logarithm. Unpublished note, Reproduced in Elbaz-Vincent & Gangl, 2002, 1995 PDF

Marcolli, M. & Thorngren, R. Thermodynamic Semirings, arXiv 10.4171/JNCG/159, 2011, Vol. abs/1108.2874 PDF

Marcolli, M. & Tedeschi, R. Entropy algebras and Birkhoff factorization, arXiv, 2014, Vol. abs/1108.2874 PDF

Hope to see you and discuss the topic

]]>I think that my objection might be disjoint: My main problem is that 3-manifolds aren’t popping up all over mathematics and in the sciences, in diverse contexts. That is what I would expect of a “mathematical primitive”- I would expect it to make veiled appearances all over the place. Perhaps Monsterous Moonshine and its cousins “justify” sporatic simple groups. They are clearly more than just artifacts of suboptimal definitions. But I don’t see a parallel for 3-manifolds- the only place they seem to appear is inside geometric topology.

To sharpen the philosophical objection further, there are several related objects which DO appear in other contexts, but they’re no longer “just” 3-manifolds. For example, there is a whole industry devoted to the analogy between 3-manifolds and number fields, which would remove my objection, except that the 3-manifolds are equipped with a flow. I found the answers here, and linked references, quite illuminating in this respect.

So my personal speculation would be that 3-manifolds could perhaps be thought of as a partial manifestation of some more “fundamental” structure, such as some sort of spacial algebra of the sort that Habiro and Asaeda have looked at… or maybe something completely different. I really don’t know… Dror Bar-Natan may have vaguely similar doubts when during talks he says phrases like, “I don’t know what a 3-manifold is”.

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