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Answer by unknown (yahoo) for Is rigour just a ritual that most mathematicians wish to get rid of if they could?

2013-04-21T21:25:54Z

Answer by unknown (yahoo) for Philosophy behind Mochizuki's work on the ABC conjecture

2013-02-22T18:21:30Z

NEW !! (2013-02-21)

A Panoramic Overview of Inter-universal Teichmüller Theory By Shinichi Mochizuki

http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf

Answer by unknown (yahoo) for Trichotomies in mathematics

2013-02-04T12:29:05Z

Previous MO questions:

rational/trigonometric/elliptic trichotomy http://mathoverflow.net/questions/58040/groups-quantum-groups-and-fill-in-the-blank
trichotomy of interrelated model structures: h-model, q-model, m-model http://mathoverflow.net/questions/86942/is-the-category-of-metric-spaces-and-continuous-maps-quillen-equivalent-to-top
"such "log-exp functions" are either eventually positive, eventually zero, or eventually negative. ... It guarantees that the germs at infinity of such functions do indeed form a field K." http://mathoverflow.net/questions/45284/examples-of-sequences-whose-asymptotics-cant-be-described-by-elementary-function
a function of a complex variable with an algebraic addition theorem must be: 1) A rational function, 2) A rational function of e^px, or 3) A rational function of the Weierstrass elliptic function and its derivative. http://mathoverflow.net/questions/96452/trig-functions-based-on-convex-curves
"Every finitely generated infinite profinite group has a just infinite quotient. There is a trichotomy due to Wilson (and refined by Grigorchuk) describing what they can look like." http://mathoverflow.net/questions/49591/what-is-the-virtue-of-profinite-groups-as-mathematical-objects/68895
"there is a trichotomy of curves given by g=0, g=1, and g≥2. If you look at topological, geometric, arithmetic properties of these curves, their properties align very strongly with these classes." http://mathoverflow.net/questions/56011/why-should-i-believe-the-mordell-conjecture
Kodaira dimension. κ(Y)<0, κ(Y)=0, κ(Y)=dimY. http://mathoverflow.net/questions/81913/how-frequent-are-smooth-projective-varieties-with-anti-ample-canonical-bundle
"Rank and period of primes in the Fibonacci sequence; a trichotomy," Fib. Quart., 45 (No. 1, 2007), 56-63). http://mathoverflow.net/questions/84797/can-the-difference-of-two-distinct-fibonacci-numbers-be-a-square-infinitely-often

M.SE:

"The set-theoretic setup of Categories for the working mathematician is somewhat subtle. ... There is therefore a trichotomy of small sets, large sets, and proper classes. This is not the usual practice: we normally think of all sets as being small." http://math.stackexchange.com/questions/201062/confusion-over-the-use-of-universes-in-category-theory
"There are three distinct aspects of schemes that each have their own purpose" http://math.stackexchange.com/questions/99605/why-study-schemes/99615

TCS.SE:

"one of the most amazing facts about logic is that consistency strength boils down to the question "what is the fastest-growing function you can prove total in this logic?" As a result, the consistency of many classes of logics can be linearly ordered! If you have an ordinal notation capable of describing the fastest growing functions your two logics can show total, then you know by trichotomy that either one can prove the consistency of the other, or they are equiconsistent." http://cstheory.stackexchange.com/questions/4816/axioms-necessary-for-theoretical-computer-science/4821

A frequently cited paper: "A trichotomy theorem in natural models of AD+", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227-258.

Answer by unknown (yahoo) for How are Modal Logic and Graph Theory related?

2013-02-03T22:00:44Z

On page 724 the book "Handbook of Modal Logic" contains the phrase "modal logics are merely sublogics of appropriate monadic second-order logic" therefore you might be interested in the book "Graph Structure and Monadic Second-Order Logic" by Bruno Courcelle and Joost Engelfriet.

Answer by unknown (yahoo) for A General Framework for Ramsey Theory ?

2013-01-13T20:45:32Z

You might be interested in the 2010 book "Introduction to Ramsey Spaces" by Stevo Todorcevic

"Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite."

Answer by unknown (yahoo) for How would set theory research be affected by using ETCS instead of ZFC?

2013-01-06T21:59:10Z

Some quotes:

"I hope the math community has reached the point of realizing that we really need not one foundation of mathematics, but many, together with clearly described relations between them. Indeed at this point the word ‘foundation’ is perhaps less helpful than something else… like maybe ‘entrance’." , John Baez http://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html#c042716

"There are some problems with this sort of compromise where material and structural set theories cohabitate as equal alternative foundations. ... this kind of cohabitation model is actually not the best possible way to accommodate both the material and the structural perspective at the foundational level. ... what we end up doing is moving all the annoyances from both sides into the translation between the two ... it would be much more satisfactory to have a greater overarching foundation similar to Cantor’s universe that comprises both the material and structural views.", François G. Dorais http://dorais.org/archives/1135

"if you care about internalizing mathematics in nonstandard models — not only in order to prove formal independence results, as set theorists do, but because you care about the nonstandard models in their own right and you want to use internalization to prove things about them — then ETCS is your friend and ZFC is your enemy. Structural set theory — and its sister, type theory — is very well adapted to internalizing in many different contexts" , Mike Shulman http://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html#c042933

"with ZFC is that it allows things like ⟨0,1⟩⊆3 which make no sense mathematically. I can assure you, though, that after you write one piece of code in x86 assembler which rewrites itself as it runs that manipulating all objects as binary and manipulating all objects as sets make perfect sense. This is what annoys me, in some sense, that people have an issue that in a system which allows you to encode everything as sets you can write some weird junk. Equivalently, this would be baffling to those people that if you write a piece of code in Common Lisp then the code itself is just a list and the program can manipulate itself (easier than to do that in assembler, I have to admit).", Asaf Karagila http://boolesrings.org/asafk/2013/on-leinsters-rethinking-set-theory/#comment-789

"I don’t like the hard line that certain things are meaningless, simply because this depends on the setting. I don’t object to the idea of saying that an expression is, for your current purposes, well-typed, since this just means you say nothing about non-well-typed expressions. I would say not “meaningless”, but rather “does not have an intended meaning”. There are expressions, such as 0∈1 that do not have an intended meaning. If they have a meaning in a certain setting, then so be it.", Walt http://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html#c042992

"ETCS is one, but apparently not one that is very congenial to many traditional set theorists, for a variety of reasons, maybe one being that many set theorists “don’t like categories” (well, okay, but again I think that’s a pity). Maybe some set theorists don’t like the privileging of functions over elements (although one take is that functions are generalized elements).", Todd Trimble, http://boolesrings.org/asafk/2013/on-leinsters-rethinking-set-theory/#comment-798

"The same arrows aimed at ZFC can be easily turned at ETCS. It is equally easy to ask about “senseless things” which happen in ETCS but not in general mathematics. And this is not a critique against ETCS, but rather against the critique aimed at ZFC.", Asaf Karagila http://boolesrings.org/asafk/2013/on-leinsters-rethinking-set-theory/#comment-815

"the issue is that first-order logic is not “type safe”, and I would suspect that people who have had exposure to first-order logic take this for granted, while those who have not may find it surprising. The reason is that syntax, in first-order logic, is defined independently of semantics, so that whether a formula is well-formed (syntactically correct) is independent of what the formula means. ... This issue of “type safety” arises also in programming languages. Some languages, such as C, have little run-time type checking. I can take a 64 bit integer and treat it as an array of eight 8-bit characters – with implementation-defined results. ... In working mathematics we have humans to check that the formulas we actually write make sense – nobody accidentally writes 1∈sin in an analysis paper – so it is not clear that there is any real advantage to stronger “type checking” in the foundations.", Carl Mummert http://boolesrings.org/asafk/2013/on-leinsters-rethinking-set-theory/#comment-794

Answer by unknown (yahoo) for Math Zeitgeist 2012

2013-01-01T13:40:10Z

Ian Agol proved the last four remaining of Thurston's questions about three dimensional manifolds.

An exposition is given by Erica Klarreich at the Simons Foundation: https://simonsfoundation.org/features/science-news/mathematics-and-physical-science/getting-into-shapes-from-hyperbolic-geometry-to-cube-complexes-and-back/

Agol's paper: http://arxiv.org/abs/1204.2810

Thurston's questions: http://www.ams.org/journals/bull/1982-06-03/S0273-0979-1982-15003-0/

There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem Gary McGuire, Bastian Tugemann, Gilles Civario, http://arxiv.org/abs/1201.0749

AMS 2012 Math Digest

http://www.ams.org/news/math-in-the-media/mathdigest-md-2012-toc

The Best Writing on Mathematics 2012, Edited by Mircea Pitici

http://press.princeton.edu/titles/9821.html

"The Year of the MOOC

Massive Open Online Courses. MOOCs. This was, without a doubt, the most important and talked-about trend in education technology this year."

http://www.insidehighered.com/blogs/hack-higher-education/top-ed-tech-trends-2012-moocs

2012 Complexity Year in Review

http://blog.computationalcomplexity.org/2012/12/2012-complexity-year-in-review.html

Top ten algorithms preprints of 2012

http://11011110.livejournal.com/260838.html

Answer by unknown (yahoo) for New grand projects in contemporary math

2012-12-31T09:24:04Z

Hyperfields

"Krasner, Marshall, Connes and Consani and the author came to hyperfields for different reasons, motivated by different mathematical problems, but we came to the same conclusion: the hyperrings and hyperfields are great, very useful and very underdeveloped in the mathematical literature. Probably, the main obstacle for hyperfields to become a mainstream notion is that a multivalued operation does not fit to the tradition of set-theoretic terminology, which forces to avoid multivalued maps at any cost. I believe the taboo on multivalued maps has no real ground, and eventually will be removed. Hyperfields, as well as multigroups, hyperrings and multirings, are legitimate algebraic objects related in many ways to the classical core of mathematics. They provide elegant terminological and conceptual opportunities. In this paper I try to present new evidences for this."

Hyperfields For Tropical Geometry I. Hyperfields And Dequantization, Oleg Viro, http://arxiv.org/abs/1006.3034

The hyperring of adèle classes, Alain Connes, Caterina Consani, http://arxiv.org/abs/1001.4260

Answer by unknown (yahoo) for New grand projects in contemporary math

2012-12-30T21:24:56Z

Large Networks and Graph Limits - new AMS book by László Lovász

"The theory has rich connections with other approaches to the study of large networks, such as ``property testing'' in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization)."

"This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future." --Persi Diaconis, Stanford University

"This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor." --Noga Alon, Tel Aviv University, Israel

"Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory." --Terence Tao, University of California, Los Angeles, CA

"Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike." --Bela Bollobas, Cambridge University, UK

Answer by unknown (yahoo) for New grand projects in contemporary math

2012-12-30T21:45:20Z

Tropical mathematics

The algebraic geometry, analysis and other mathematics over the tropical semiring instead of the real numbers.

Comment by

2013-04-29T21:44:11Z

I think the tag compressed-sensing could be added to this post. nuit-blanche.blogspot.co.uk/2013/04/… goo.gl/xl3EQ

Comment by

2013-02-12T04:21:15Z

community wikified

Comment by

2013-02-09T05:43:16Z

mathoverflow.net/questions/22927/…

Comment by

2013-02-09T05:41:12Z

mathoverflow.net/questions/13029/…

Comment by

2013-02-09T05:40:06Z

mathoverflow.net/questions/40178/…

Comment by

2013-02-08T17:48:55Z

For a Riemannian signature Einstein AH structure on a compact n-manifold there holds one of the following mutually exclusive possiblities: 1) It is proper and exact with parallel scalar curvature. 2) Its scalar curvature is identically zero and it is closed. 3) It is not closed, and its scalar curvature is not parallel. arxiv.org/pdf/1203.2575

Comment by

2013-02-08T17:43:15Z

Structure of Kac–Moody algebras: finite type, affine type, and wild type.

Comment by

2013-02-08T17:06:13Z

Tautology/contingency/absurdity for propositional forms and the analogous valid/satisfiable/unsatisfiable. books.google.com/…

Comment by

2013-02-08T17:04:00Z

An infinite locally finte homogenneous geometry is isomorphic to 1) trivial geometry, 2) projective geometry over a finite field, or 3) affine geometry over a finite field. books.google.com/…

Comment by

2013-02-06T13:28:26Z

Or the minimum function. Or any subring of $\mathbb{R}$ such as $\mathbb{Q}$, $\mathbb{Z}$, $\mathbb{Q(\sqrt{2})}$.

Comment by

2013-02-05T17:13:12Z

Related question: mathoverflow.net/questions/110378/…

Comment by

2013-02-04T17:54:09Z

Underdetermined, determined, overdetermined.

Comment by

2013-02-04T14:26:01Z

SubX, X, superX.

Comment by

2013-02-04T14:24:17Z

Clockwise, anti-clockwise, stationary.

Comment by

2013-02-04T14:23:41Z

One, both, neither.