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”.

]]>For me, what would “convince” me that 3-manifolds “exist” in some philosophical sense would be if one could find a strong tie-in with the main body of mathematics whose “existence” is somehow well-established. For links we sort-of have this via quantum invariants, which provide insights into number theory (multiple zetas etc), field with one element, Lie theory etc. and also somehow reflect quite concrete physical objects which one could almost hold in one’s hand. I don’t know of a parallel sense in which 3-manifolds tie in with the rest of mathematics except as examples; for example, 3-manifold quantum invariants tend to come from link invariants, and the tie-ins with the rest of math happen already on the level of links (e.g. the MOO invariant, as discussed in the post).

]]>But I think this is exactly the kind of thing that makes mathematics entertaining. One studies PDEs not because one hopes to have a single unifying perspective on the subject that answers every question efficiently. You tend to study particular families that are useful. 3-manifolds are of that variety of question. They’re a useful family, rather than a generality. They’re also not so small that one can quickly bludgeon the subject to death.

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New Ideas in Low Dimensional Topology

Edited by: Louis Kauffman & Vassily Olegovich Manturov