Profile cover photo
Profile photo
Luis Minero
254 followers
254 followers
About
Posts

Post has shared content
Add a comment...

Post has shared content
Mathematics and public engagement. A parable.

There was a wise old woman who lived in a forest. She loved the forest, and she learned to live in harmony with it.

Over time, the people of the village discovered paper, and all the ways it could be used. They asked the old woman if she knew how to make paper, and she told them that trees were good for making paper. The villagers were very happy about the paper, and they learned to use it in many clever ways. Cardboard, and tissue paper, and construction paper. Over time, virtually every aspect of the villagers technology used paper products in some fundamental way.

One day the old woman was visiting the village and she said, "I know that paper is useful, but I know about a lot of other things in the forest that you might enjoy, and that might be useful for you in other ways." The villagers were very interested in this, and began asking her whether there were other plants in the forest that can make paper, or other types of paper that can be made from trees. She said "No, no, I'm not talking about paper -- I'm talking about the other things in the forest." The villagers stared at her blankly. She said, "for example, there are all sorts of wonderful creatures that live in the trees, with feathers, or fur, and all different sizes."

The villagers, visibly confused, said "so are the animals good for making paper?"


Image by Donar Reiskoffer: https://commons.m.wikimedia.org/wiki/File:Brussels_Zonienwoud.jpg
Photo
Add a comment...

Post has shared content
"Tras una fuerte campaña de erradicación, organizada y financiada por el gobierno estadounidense, el lobo mexicano fue exterminado en casi toda su distribución histórica durante la primera mitad del siglo pasado. Hacia los años sesenta, al norte de la frontera no se encontraba ni un ejemplar en vida libre y se sabía solamente de unos cuantos en las montañas del noroeste mexicano": Pamela Maciel Cabañas

https://www.revistadelauniversidad.mx/articles/e3ab271b-74cf-43d7-b351-bf8cc8450ee4/el-lobo-mexicano
Photo
Add a comment...

Post has shared content
Whoever thought we'd see such a convergence of the Euler characteristic with sport, traffic engineering, and politics?

The official UK road signs directing traffic to football stadiums depict a mathematically impossible shape — a ball made only of hexagons. Matt Parker wants to change that. His video uses the Euler characteristic to explain why it's impossible. If you're a UK citizen or resident, you can help by signing his petition.

Via +The Aperiodical at https://plus.google.com/+Aperiodical/posts/8sHUs7bnz5S
Add a comment...

Post has shared content
Applied Category Theory 2018

We're having a conference on applied category theory! I'm excited about this, because I'm trying to lure more people into applying beautiful math to real-world problems, especially ecological problems. To help do this, there will also be a school for grad students. It's called the Adjoint Research School - which is a pun - and in it 16 lucky students will discuss papers and work on problems for 16 weeks before the actual meeting.

Applied Category Theory (ACT 2018). School 23–27 April 2018 and conference 30 April–4 May 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).

The plenary speakers will be:

• Samson Abramsky (Oxford)
• John Baez (UC Riverside)
• Kathryn Hess (EPFL)
• Mehrnoosh Sadrzadeh (Queen Mary)
• David Spivak (MIT)

There will be a lot more to say as this progresses, but for now let me just quote from the conference website, which is here:

http://www.appliedcategorytheory.org/

Applied Category Theory (ACT 2018) is a five-day workshop on applied category theory running from April 30 to May 4 at the Lorentz Center in Leiden, the Netherlands.

Towards an Integrative Science: in this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one scientific discipline can be reused in another. The aim of the workshop is to (1) explore the use of category theory within and across different disciplines, (2) create a more cohesive and collaborative ACT community, especially among early-stage researchers, and (3) accelerate research by outlining common goals and open problems for the field.

While the workshop will host discussions on a wide range of applications of category theory, there will be four special tracks on exciting new developments in the field:

1. Dynamical systems and networks
2. Systems biology
3. Cognition and AI
4. Causality

Please note that the workshop is heavily geared toward discussion and is not currently accepting paper or poster submissions.

Accompanying the workshop will be an Adjoint Research School for early-career researchers. This will comprise a 16 week online seminar, followed by a 4 day research meeting at the Lorentz Center in the week prior to ACT 2018. Applications to the school will open prior to October 1, and are due November 1. Admissions will be notified by November 15.

Sincerely,
The organizers

Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford)

The proposal: Towards an Integrative Science

Category theory was developed in the 1940s to translate ideas from one field of mathematics, e.g. topology, to another field of mathematics, e.g. algebra. More recently, category theory has become an unexpectedly useful and economical tool for modeling a range of different disciplines, including programming language theory [10], quantum mechanics [2], systems biology [12], complex networks [5], database theory [7], and dynamical systems [14].

A category consists of a collection of objects together with a collection of maps between those objects, satisfying certain rules. Topologists and geometers use category theory to describe the passage from one mathematical structure to another, while category theorists are also interested in categories for their own sake. In computer science and physics, many types of categories (e.g. topoi or monoidal categories) are used to give a formal semantics of domain-specific phenomena (e.g. automata [3], or regular languages [11], or quantum protocols [2]). In the applied category theory community, a long-articulated vision understands categories as mathematical workspaces for the experimental sciences, similar to how they are used in topology and geometry [13]. This has proved true in certain fields, including computer science and mathematical physics, and we believe that these results can be extended in an exciting direction: we believe that category theory has the potential to bridge specific different fields, and moreover that developments in such fields (e.g. automata) can be transferred successfully into other fields (e.g. systems biology) through category theory. Already, for example, the categorical modeling of quantum processes has helped solve an important open problem in natural language processing [9].

In this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one discipline can be reused in another. Tangibly and in the short-term, we will bring together people from different disciplines in order to write an expository survey paper that grounds the varied research in applied category theory and lays out the parameters of the research program.

In formulating this research program, we are motivated by recent successes where category theory was used to model a wide range of phenomena across many disciplines, e.g. open dynamical systems (including open Markov processes and open chemical reaction networks), entropy and relative entropy [6], and descriptions of computer hardware [8]. Several talks will address some of these new developments. But we are also motivated by an open problem in applied category theory, one which was observed at the most recent workshop in applied category theory (Dagstuhl, Germany, in 2015): “a weakness of semantics/CT is that the definitions play a key role. Having the right definitions makes the theorems trivial, which is the opposite of hard subjects where they have combinatorial proofs of theorems (and simple definitions). […] In general, the audience agrees that people see category theorists only as reconstructing the things they knew already, and that is a disadvantage, because we do not give them a good reason to care enough” [1, pg. 61].

In this workshop, we wish to articulate a natural response to the above: instead of treating the reconstruction as a weakness, we should treat the use of categorical concepts as a natural part of transferring and integrating knowledge across disciplines. The restructuring employed in applied category theory cuts through jargon, helping to elucidate common themes across disciplines. Indeed, the drive for a common language and comparison of similar structures in algebra and topology is what led to the development category theory in the first place, and recent hints show that this approach is not only useful between mathematical disciplines, but between scientific ones as well. For example, the ‘Rosetta Stone’ of Baez and Stay demonstrates how symmetric monoidal closed categories capture the common structure between logic, computation, and physics [4].


[1] Samson Abramsky, John C. Baez, Fabio Gadducci, and Viktor Winschel. Categorical methods at the crossroads. Report from Dagstuhl Perspectives Workshop 14182, 2014.

[2] Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam, 2009.

[3] Michael A. Arbib and Ernest G. Manes. A categorist’s view of automata and systems. In Ernest G. Manes, editor, Category Theory Applied to Computation and Control. Springer, Berlin, 2005.

[4] John C. Baez and Mike Stay. Physics, topology, logic and computation: a Rosetta Stone. In Bob Coecke, editor, New Structures for Physics. Springer, Berlin, 2011.

[5] John C. Baez and Brendan Fong. A compositional framework for passive linear networks. arXiv e-prints, 2015.

[6] John C. Baez, Tobias Fritz, and Tom Leinster. A characterization of entropy in terms of information loss. Entropy, 13(11):1945–1957, 2011.

[7] Michael Fleming, Ryan Gunther, and Robert Rosebrugh. A database of categories. Journal of Symbolic Computing, 35(2):127–135, 2003.

[8] Dan R. Ghica and Achim Jung. Categorical semantics of digital circuits. In Ruzica Piskac and Muralidhar Talupur, editors, Proceedings of the 16th Conference on Formal Methods in Computer-Aided Design. Springer, Berlin, 2016.

[9] Dimitri Kartsaklis, Mehrnoosh Sadrzadeh, Stephen Pulman, and Bob Coecke. Reasoning about meaning in natural language with compact closed categories and Frobenius algebras. In Logic and Algebraic Structures in Quantum Computing and Information. Cambridge University Press, Cambridge, 2013.

[10] Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55–92, 1991.

[11] Nicholas Pippenger. Regular languages and Stone duality. Theory of Computing Systems 30(2):121–134, 1997.

[12] Robert Rosen. The representation of biological systems from the standpoint of the theory of categories. Bulletin of Mathematical Biophysics, 20(4):317–341, 1958.

[13] David I. Spivak. Category Theory for Scientists. MIT Press, Cambridge MA, 2014.

[14] David I. Spivak, Christina Vasilakopoulou, and Patrick Schultz. Dynamical systems and sheaves. arXiv e-prints, 2016.
Animated Photo
Add a comment...

Post has attachment
Add a comment...

Post has shared content
How (not) to write mathematics

Some tips from the mathematician John Milne:

If you write clearly, then your readers may understand your mathematics and conclude that it isn't profound. Worse, a referee may find your errors. Here are some tips for avoiding these awful possibilities.

1. Never explain why you need all those weird conditions, or what they mean. For example, simply begin your paper with two pages of notations and conditions without explaining that they mean that the varieties you are considering have zero-dimensional boundary. In fact, never explain what you are doing, or why you are doing it. The best-written paper is one in which the reader will not discover what you have proved until he has read the whole paper, if then.

2. Refer to another obscure paper for all the basic (nonstandard) definitions you use, or never explain them at all. This almost guarantees that no one will understand what you are talking about (and makes it easier to use the next tip). In particular, never explain your sign conventions --- if you do, someone may be able to prove that your signs are wrong.

3. When having difficulties proving a theorem, try the method of "variation of definition"---this involves implicitly using more that one definition for a term in the course of a single proof.

4. Use c, a, b respectively to denote elements of sets A, B, C.

5. When using a result in a proof, don't state the result or give a reference. In fact, try to conceal that you are even making use of a nontrivial result.

6. If, in a moment of weakness, you do refer to a paper or book for a result, never say where in the paper or book the result can be found. In addition to making it difficult for the reader to find the result, this makes it almost impossible for anyone to prove that the result isn't actually there. Alternatively, instead of referring to the correct paper for a result, refer to an earlier paper, which contains only a weaker result.

7. Especially in long articles or books, number your theorems, propositions, corollaries, definitions, remarks, etc. separately. That way, no reader will have the patience to track down your internal references.

8. Write A==>B==>C==>D when you mean (A==>B)==>(C==>D), or (A==>(B==>C))==>D, or.... Similarly, write "If A, B, C" when you mean "If A, then B and C" or "If A and B, then C", or .... Also, always muddle your quantifiers.

9. Begin and end sentences with symbols wherever possible. Since periods are almost invisible (and may be mistaken for a mathematical symbol), most readers won't even notice that you've started a new sentence. Also, where possible, attach superscripts signalling footnotes to mathematical symbols rather than words.

10. Write "so that" when you mean "such that" and "which" when you mean "that". Always prefer the ambiguous expression to the unambiguous and the imprecise to the precise. It is the readers task to determine what you mean; it is not yours to express it.

11. If all else fails, write in German.

These helpful tips are from his webpage:

http://www.jmilne.org/math/tips.html

He has some footnotes, including this for item 11:

The point is that most mathematicians find it very difficult to read mathematics in German, and so, by writing in German, you can ensure that your work is inaccessible to most mathematicians, even though, of course, German is a perfectly good language for expressing mathematics.

Hmm. He could have chosen Basque, but he chose German.
Photo
Add a comment...

Post has attachment

Post has shared content
Add a comment...

Post has shared content
Add a comment...
Wait while more posts are being loaded