David Roberts

Recommendations

[1] Recommendation for What is good mathematics? (Oct. 21, 2013): Show discussion.

An old essay by +Terence Tao

http://arxiv.org/abs/math.HO/0702396

which attempts to frame and then give some thoughts, driven by a particular example, on the titular question. In particular, Terry opens with a (non-exhaustive!) disambiguation of the question into 21 different qualities of mathematics that might be singled out as being 'good': problem solving, technique, theory, insight, discovery, application, exposition, pedagogy, vision, taste, public relations, meta-mathematics, rigorous maths, beautiful maths, elegant maths, creative maths, useful maths, strong maths, deep maths, intuitive maths, definitive maths.

He also notes that the essay (and the list above) only focuses on research mathematics, excluding textbooks, mathematical education and mathematics from other closely scientific disciplines.

The particular example that is covered is Szemerédi's theorem,

http://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem

showing the existence of arbitrarily long arithmetic progressions in subsets of positive density of the integers. On a completely unrelated note, I found on the Wikipedia page this upper bound established by +Timothy Gowers (Geom. Funct. Anal. 11 (3): 465–588, (2001) http://dx.doi.org/10.1007%2Fs00039-001-0332-9) for the finitary case:

N \leq 2^(2^(d^(-2^(2^(k+9)))))

(here d \in (0,1/2] and N is the minimum size of interval [1,N] in the integers for the existence of a k-term arithmetic progression in every subset of [1,N] of size at least dN).

Aside from the sociological interest, this paper is a great overview of the sorts of mathematics that surrounds Szemerédi's theorem and related results.

#spnetwork arXiv:math.HO/0702396 #mustread
Philip Thrift replies (Oct. 21, 2013): I can only think to add: Good computer-based mathematics — when a program proves a theorem, discovers a new theorem, etc.

David Roberts replies (Oct. 21, 2013): Or a clean formalisation of an important proof (cf the relatively recent Coq proof of the Odd-order Theorem).

Discussion

[1] Comment on Large gaps between consecutive prime numbers RE: topics: #numbertheory #primegaps (Dec. 11, 2014): Show discussion.

Recent progress on a conjecture of Erdős

Erdős offered prizes, out of his own money, for the solution of various conjectures he made. These prizes reflected the difficulty or importance of the problem, as he saw it, and ranged from a purely symbolic amount (less than $100), to the one discussed in the following article: $10 000. This reflected the difficulty of the problem, even though afterwards Erdős afterwards admitted such a large sum was "a little rash" (he famously had no fixed address, and lived out of a suitcase). Ron Graham, Erdős collaborator and guest in several +Numberphile videos, will pick up the tab.

The article discusses two papers:

+Terence Tao, Kevin Ford, +Ben Green , Sergei Konyagin, Large gaps between consecutive prime numbers, http://arxiv.org/abs/1408.4505v1

James Maynard, Large gaps between primes, http://arxiv.org/abs/1408.5110v1

and hints that a third, joint paper between all these authors will be released shortly improving on the results in the above papers. The key idea is that one can give an estimate on the lower bound on how big gaps between prime numbers can become, as they get bigger and bigger. This is in contrast with the #primegaps PolyMath project (also involving Tao and Maynard, among many others), which wanted to show there were infinitely many primes with gap smaller than some fixed small number.

The photo of Erdős and Tao shown below was taken at the University of Adelaide in the 1980s (and comes from http://commons.wikimedia.org/wiki/File:Paul_Erdos_with_Terence_Tao.jpg, though shared originally by Tao on G+)

#spnetwork arXiv:1408.4505 arXiv:1408.5110 #numbertheory (TeX $ CONVERSION FAILED: unbalanced $)
[2] Comment on Tensor categorical foundations of algebraic geometry RE: topics: #algebraicgeometry #arXiv #categorytheory (Oct. 9, 2014): Show discussion.

I'm very excited to see this come out. I've been watching Brandenburg's questions on MathOverflow over the past few years with interest. The idea that arbitrary objects in algebraic geometry, not necessarily those that are spectra of commutative rings, are always spectra of certain tensor categories is very cool, and provides an avenue to discuss schemes over the fictional 'field with one element', namely tensor categories that don't seem to arise from schemes, but which should be considered as the category of quasicoherent sheaves on some geometric object.

Some of the ideas here no doubt intersect somewhat with the unpublished work of Jim Dolan on algebraic geometry from a 2-category-theoretic point of view.

http://arxiv.org/abs/1410.1716

Title: Tensor categorical foundations of algebraic geometry
Author: Martin Brandenburg

Abstract: Tannaka duality and its extensions by Lurie, Schäppi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocomplete tensor category. We then discuss tensor categorical globalizations of affine morphisms, projective morphisms, immersions, classical projective embeddings (Segre, Pl\"ucker, Veronese), blow-ups, fiber products, classifying stacks and finally tangent bundles. It turns out that the universal properties of several moduli spaces or stacks translate to the corresponding tensor categories.

#arXiv #spnetwork arXiv:1410.1716 #algebraicgeometry #categorytheory
John Baez replies (Oct. 9, 2014): I'm glad you mentioned Jim's work on this sort of thing. Btw, I believe he's going to Macquarie to work with the category theorists there.

David Roberts replies (Oct. 9, 2014): +John Baez yes, I think he told me that a while back. I haven't heard he's got there yet.
[3] Comment on The 2-type model structure on the category of bicategories RE: topics: #arXiv (Oct. 3, 2014): Show discussion.

This looks like a nice paper

http://arxiv.org/abs/1410.0248

Title: The 2-type model structure on the category of bicategories
Author: Kohei Tanaka

Abstract: We define a model structure on the category of bicategories closely related to homotopy 2-types. The fibrant objects are bigroupoids. We state that the fibrations satisfy the product formula with respect to the Euler characteristic of bicategories.

#arXiv #spnetwork arXiv:1410.0248
[4] Comment on The "bounded gaps between primes" Polymath project - a retrospective RE: topics: #primegaps #arXiv (Oct. 1, 2014): Show discussion.

Yours truly has a small part in this. #primegaps

http://arxiv.org/abs/1409.8361

Title: The "bounded gaps between primes" Polymath project - a retrospective
Author: D.H.J. Polymath

#arXiv #spnetwork arXiv:1409.8361
[5] Comment on Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors RE: topics: #arXiv (Sep. 22, 2014): Show discussion.

A small delay in being released, but here it is.

http://arxiv.org/abs/1409.5723

Title: Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors
Authors: +Domenico Fiorenza , Alessandro Valentino

Abstract: We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level m, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for n-characters on ∞-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for n+1-dimensional TQFTs produce n-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.

#arXiv #spnetwork arXiv:1409.5723
John Baez replies (Sep. 22, 2014): Great! I'll try to get +Jamie Vicary, +Bruce Bartlett and the rest of that gang to read this....

Urs Schreiber replies (Sep. 22, 2014): Good that this is finally out. We already used this in a bunch of places. Such as in [1] to make precise how exactly higher WZW-type models are boundaries of higher CS-models, in [2] to show how particle mechanics is the boundary of the 2d Poisson-Chern-Simons model and in [3] to describe the quantization of these boundaries by pull-push in generalized cohomology (and in particular to reproduce and generalize ordinary quantum mechanics this way).

[1]
http://ncatlab.org/schreiber/show/The+brane+bouquet

[2]
http://ncatlab.org/schreiber/show/master+thesis+Bongers
http://ncatlab.org/schreiber/show/master+thesis+Nuiten

[3]
http://ncatlab.org/schreiber/show/master+thesis+Nuiten
http://ncatlab.org/schreiber/show/Quantization+via+Linear+homotopy+types

Urs Schreiber replies (Sep. 22, 2014): Here is a little history of the development of this idea: https://plus.google.com/+UrsSchreiber/posts/7tyJuUjynVF
[6] Comment on A mathematical theory of resources RE: topics: #arXiv (Sep. 22, 2014): Show discussion.

http://arxiv.org/abs/1409.5531

Title: A mathematical theory of resources
Authors: +Bob Coecke , Tobias Fritz, +Robert Spekkens

Abstract: In many different fields of science, it is useful to characterize physical states and processes as resources. Chemistry, thermodynamics, Shannon's theory of communication channels, and the theory of quantum entanglement are prominent examples. Questions addressed by a theory of resources include: Which resources can be converted into which other ones? What is the rate at which arbitrarily many copies of one resource can be converted into arbitrarily many copies of another? Can a catalyst help in making an impossible transformation possible? How does one quantify the resource? Here, we propose a general mathematical definition of what constitutes a resource theory. We prove some general theorems about how resource theories can be constructed from theories of processes wherein there is a special class of processes that are implementable at no cost and which define the means by which the costly states and processes can be interconverted one to another. We outline how various existing resource theories fit into our framework. Our abstract characterization of resource theories is a first step in a larger project of identifying universal features and principles of resource theories. In this vein, we identify a few general results concerning resource convertibility.

#arXiv #spnetwork arXiv:1409.5531
John Baez replies (Sep. 22, 2014): Yay! They've been working on this for quite a while.

Jean-Luc Delatre replies (Sep. 22, 2014): Great! Jevons paradox will strike hard and fast!

Rongmin Lu replies (Sep. 23, 2014): Indeed. I wonder if this can be used to model/study the Jevons effect: http://en.wikipedia.org/wiki/Jevons_paradox

Allen Knutson replies (Sep. 24, 2014): I read this abstract and think about incremental games (Candy Box, A Dark Room, Kittens Game, etc.).
[7] Comment on Division Algebras and Supersymmetry IV RE: topics: #arXiv (Sep. 16, 2014): Show discussion.

Fantastic! I've been waiting to see this.

http://arxiv.org/abs/1409.4361

Title: Division Algebras and Supersymmetry IV
Author: +John Huerta

Abstract: Recent work applying higher gauge theory to the superstring has indicated the presence of 'higher symmetry', and the same methods work for the super-2-brane. In the previous paper in this series, we used a geometric technique to construct a 'Lie 2-supergroup' extending the Poincare supergroup in precisely those spacetime dimensions where the classical Green-Schwarz superstring makes sense: 3, 4, 6 and 10. In this paper, we use the same technique to construct a 'Lie 3-supergroup' extending the Poincare supergroup in precisely those spacetime dimensions where the classical Green-Schwarz super-2-brane makes sense: 4, 5, 7 and 11. Because the geometric tools are identical, our focus here is on the precise definition of a Lie 3-supergroup.

#arXiv #spnetwork arXiv:1409.4361
Mike Stay replies (Sep. 16, 2014): Congratulations +John Huerta!
[8] Comment on A cohomological framework for homotopy moment maps RE: topics: #arXiv (Sep. 11, 2014): Show discussion.

Some recent writing using a baby example of this paper: http://www.math.columbia.edu/~woit/wordpress/?p=7146

http://arxiv.org/abs/1409.3142

Title: A cohomological framework for homotopy moment maps
Authors: Yael Fregier, +Camille Laurent-Gengoux , Marco Zambon

Abstract: Given a Lie group acting on a manifold M preserving a closed n+1-form ω, the notion of homotopy moment map for this action was introduced in [FRZ], in terms of L∞-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of the results of [FRZ].

#arXiv #spnetwork arXiv:1409.3142
Urs Schreiber replies (Sep. 11, 2014): The fully general homotopy-theoretic context for homotopy moment maps we have in http://arxiv.org/abs/1304.0236 .
[9] Comment on Quasistrict symmetric monoidal 2-categories via wire diagrams RE: topics: #arXiv #categorytheory (Sep. 9, 2014): Show discussion.

Hurrah, more higher-dimensional diagrammatic calculus!

http://arxiv.org/abs/1409.2148

Title: Quasistrict symmetric monoidal 2-categories via wire diagrams
Author: +Bruce Bartlett

Abstract: In this paper we give an expository account of quasistrict symmetric monoidal 2-categories, as introduced by Schommer-Pries. We reformulate the definition using a graphical calculus called wire diagrams, which facilitates computations and emphasizes the central role played by the interchangor coherence isomorphisms.

#arXiv #spnetwork arXiv:1409.2148 #categorytheory
[10] Comment on Comparison of models for $(\infty, n)$-categories, II RE: topics: #arXiv (Jun. 18, 2014): Show discussion.

One definition of a weak (n+1)-category is a category (weakly) enriched in weak n-categories. However, if one already has a definition of a weak n-category for all n, it is natural to compare this existing definition with the definition via enrichment. The paper below does just this for the model of n-categories, or rather (\infty,n)-categories (that is, there are morphisms of all dimensions and those above dimension n are invertible), given by Θ_n-spaces. One way of comparing such definitions is to show they are the objects of Quillen equivalent model categories, but in general one needs to, as in the paper, construct a zig-zag of Quillen equivalences.

This is really cool, as it addresses a case of what is known in the category theory community as the Comparison Problem: given two definition of n-categories, how do we show they are in some sense 'the same', where 'the same' means the equivalence of the (n+1)-categories of the two models of n-categories.

http://arxiv.org/abs/1406.4182

Title: Comparison of models for (∞,n)-categories, II
Author: Julia E. Bergner, Charles Rezk

Abstract: In this paper we complete a chain of explicit Quillen equivalences between the model category for Θ_{n+1}-spaces and the model category of small categories enriched in Θ_n-spaces.

#arXiv #spnetwork ArXiv:1406.4182
[11] Comment on Internal and local homotopy theory RE: topics: #arXiv #categorytheory (May 1, 2014): Show discussion.

A very nice article on what is essentially higher topos theory, by considering Kan simplicial objects in regular categories, rather than simplicial (pre)sheaves. One result is vast generalisation of an observation I once made (and was presumably not original to me), that groupoids internal to a regular category (or more generally a subcanonical unary site) form a category of fibrant objects in the sense of Brown, with Kan simplicial objects instead of groupoids. However, the article goes much further in developing the homotopy theory past this initial observation.

Title: Internal and local homotopy theory
Author: +Zhen Lin Low

Abstract: There is a well-established homotopy theory of simplicial objects in a Grothendieck topos, and folklore says that the weak equivalences are axiomatisable in the geometric fragment of L_{ω_1,ω}. We show that it is in fact a theory of presheaf type, i.e. classified by a presheaf topos. As a corollary, we obtain a new proof of the fact that the local Kan fibrations of simplicial presheaves that are local weak homotopy equivalences are precisely the morphisms with the expected local lifting property.

#arXiv #spnetwork arXiv:1404.7788 #categorytheory
[12] Comment on A pathological o-minimal quotient RE: topics: #arXiv #tametopology (Apr. 15, 2014): Show discussion.

This paper is interesting, but the title is a little misleading for those not familiar with o-minimal structures. The 'pathological' quotient here is an 'interpretable set', which is essentially some sort of space that is 'locally a definable set'. One can safely think of this as a sort of real affine variety, though this is a bit of an exaggeration. The 'pathology' is that this interpretable set is not itself a definable set in some M^k for M an o-minimal structure. To me, this screams out that one should be using some sort of sheaves on definable sets (perhaps with nice properties) to capture these more general examples. And the example as constructed in the paper is not crazily awful, being built out of RP^1 and fractional linear transformations.

Title: A pathological o-minimal quotient
Author: Will Johnson

Abstract: We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure M whose elementary diagram does not eliminate imaginaries. We also give a positive answer to a related question, showing that any imaginary in an o-minimal structure is interdefinable over an independent set of parameters with a tuple of real elements. This can be interpreted as saying that interpretable sets look "locally" like definable sets, in a sense which can be made precise.

http://arxiv.org/abs/1404.3175 #spnetwork arXiv:1404.3175 #arXiv #tametopology
[13] Comment on The heart of a combinatorial model category RE: topics: #arXiv (Feb. 27, 2014): Show discussion.

A very nice result from +Zhen Lin Low ! Combinatorial model categories, in a precise way, are those that are 'nice' presentations for (∞,1)-categories, so finding ways to get them from other model categories is extremely useful.

The point of the article is to make precise a comment by Dugger:

...for a combinatorial category the interesting part of the homotopy theory is all concentrated within some small subcategory—beyond sufficiently large cardinals the homotopy theory is some how “formal”.

This is useful, because it removes worries that results about homotopy theory are dependent on universes, or possibly change under enlargement of universe.

At a very basic level, the homotopy theory of spaces, which can be presented by the Quillen model structure on simplicial sets, falls under this result. A priori one might worry that since this was originally defined in terms of the category of all topological spaces (or perhaps all CW complexes) serious size issues come into play; after all, this was one reason Quillen introduced model categories: to localise large categories such as these.

Other examples given are the category of unbounded chain complexes of R-modules, and the category of symmetric spectra.

http://arxiv.org/abs/1402.6659

Title: The heart of a combinatorial model category
Author: Zhen Lin Low

Abstract: We show that small model categories satisfying certain size conditions can be completed to yield a combinatorial model category, and conversely, that every combinatorial model category arises in this way. We also show that these constructions preserve right properness and compatibility with simplicial enrichment.

#arXiv
#spnetwork arXiv:1402.6659
Timothy Porter replies (Feb. 27, 2014): Thanks for reminding me of Zhen Lin's work
[14] Comment on Small gaps between primes RE: topics: #numbertheory #primegap #arXiv (Nov. 20, 2013): Show discussion.

Maynard's paper on the #primegap problem. Much as Wiles' proof of Fermat's last theorem was by proving a large part of the Taniyama-Shimura-Weil conjecture, Maynard proves that, for each m, there are infinitely many consecutive (m+1)-tuples of primes which are contained within an interval of fixed finite computable width (said width depending on m). Zhang's theorem is "merely" the case m=1. And Maynard proves this by proving the prime k-tuples conjecture holds at least some of the time, for every k. The prime k-tuples conjecture is a massive conjecture in number theory, so it's very exciting he's made even what seems like partial progress.

http://arxiv.org/abs/1311.4600

Title: Small gaps between primes
Author: James Maynard

Abstract: We introduce a refinement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each k, the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples. In particular, \liminf_{n} (p_{n+m}-p_n)<\infty for any integer m. We also show that \liminf(p_{n+1}-p_n)\le 600, and, if we assume the Elliott-Halberstam conjecture, that \liminf_n(p_{n+1}-p_n)\le 12 and \liminf_n (p_{n+2}-p_n)\le 600.

#arXiv #spnetwork arXiv: 1311.4600 #numbertheory
Rongmin Lu replies (Nov. 20, 2013): Did you mean "(m+1)-tuples"?

David Roberts replies (Nov. 20, 2013): Yes, thanks. Fixed.
[15] Comment on Universes for category theory RE: topics: #CT2013 #categorytheory (Nov. 19, 2013): Show discussion.

An interesting observation from +Zhen Lin Low :

"Does the theory of accessible extension still work if we replace 'universe' with 'weak universe' everywhere?

* In Mac Lane set theory, if U is a weak universe, then U is a model of Zermelo set theory with (global) choice, so the category of U-sets is a model of ETCS.

* Moreover, in ordinary ZFC, every set is a member of some weak universe: indeed, for every limit ordinal alpha > omega, the set V_alpha is a weak universe.

* If things still work in this context, it would afford an adequate framework for applying category-theoretic methods to study category theory, without needing any large cardinals."

Here a weak universe is a (material) set U such that:

- U is downward closed in the \in-relation,
- U contains all pairs of elements in U,
- U contains the power sets of its elements,
- for x \in U, \bigcup_{y\in x} y \in U (i.e. U contains the unions of elements of its elements)
- \omega \in U

The theory of accessible extensions is detailed in Zhen Lin's slides [1] from #CT2013 and covered in more detail in http://arxiv.org/abs/1304.5227. Essentially, it concerns itself with the preservation of universal properties on universe enlargement, or alternatively with the problem of limits, colimits, Kan extensions etc depending on the choice of universe. The key idea is to find when categories and structures on them are determined by 'small' amounts of data ('small' meaning something like a regular cardinal!) and this 'small' data is not unduly affected by moving to a larger ambient universe.

#spnetwork arXiv:1304.5227 #categorytheory

[1] https://www.dpmms.cam.ac.uk/~zll22/slides/2013-07-11-AccessibleAdjoints.pdf
[16] Comment on Multiplicative differential algebraic K-theory and applications RE: topics: #arXiv (Nov. 7, 2013): Show discussion.

A quote: "In Section 5 we explain that differential algebraic K-theory is a natural home for the absolute height for number rings. We show how the height is related with natural constructions with line bundles."

Looks very nice, but I'm worried that the diagram of localisations at the bottom of page 4 should only commute up to a (non-invertible) 2-arrow. I'm sure it all works out, but they do not justify this step. Possibly it's in Lurie's Higher Topos Theory, even if not due to him.

Intriguingly, they talk about infinity-sheaves on a site Mfld x Sch_Z, which I've not seen before.

http://arxiv.org/abs/1311.1421

Title: Multiplicative differential algebraic K-theory and applications
Authors: Ulrich Bunke, Georg Tamme

Abstract: We construct a version of Beilinson's regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic K-theory. We use this theory to give an interpretation of Bloch's construction of K_3-classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic K-theory of number rings.

#arXiv #spnetwork arXiv:1311.1421
David Corfield replies (Nov. 7, 2013): Any cohesion to be seen there, or should we wait for [BNV] Differential cohomology
theories as sheaves of spectra?

David Roberts replies (Nov. 7, 2013): +David Corfield Hmm, I didn't think of that. What I did think of was Urs' recent comment at the nForum about oo-sheaves on Diff with values in oo-sheaves on Sch (or similar). Clearly that is closely related to what is going on here. It also reminded me of conversations you've had with Urs about the algebro-geometric analogues of differential cohomology.

David Roberts replies (Nov. 7, 2013): (To forestall a suggestion from Urs to add this to the nLab, I'm not quite sure where and in what context to reference it.)

Urs Schreiber replies (Nov. 7, 2013): http://ncatlab.org/nlab/show/differential+algebraic+K-theory

Urs Schreiber replies (Nov. 7, 2013): Yes, what they, somewhat implicitly do, is consider oo-stacks on MfdxSch as being cohesive over oo-stacks on Sch. That's the context on which we talked about this on the nForum. I have just added a few more lines of comment here: http://ncatlab.org/nlab/show/differential+algebraic+K-theory

David Roberts replies (Nov. 7, 2013): Thanks, Urs.

Urs Schreiber replies (Nov. 7, 2013): Somebody should write all this out more explicitly. With Uli Bunke's recent work, they are opening a door to a universe of cohesion. If you see an army of graduate students willing to work on this, let me know.

Adeel Ahmad Khan replies (Nov. 7, 2013): I have to say, this makes me regret a little not going to Regensburg! Though I certainly can't complain about what I'm working on with Marc (derived algebraic cobordism).
[17] Comment on Homotopy coherent adjunctions and the formal theory of monads RE: topics: #arXiv #categorytheory (Nov. 1, 2013): Show discussion.

Some serious quasi-category theory here, present from a 2-categorical perspective - which is good because it is reducing, by one measure, ∞ down to 2. +Emily Riehl spoke about this at CT2013 in July (very nice talk, Emily!) and I would see her and +Dominic Verity catching what spare time they could in the café at Macquarie no doubt working on this.

http://arxiv.org/abs/1306.5144
http://arxiv.org/abs/1310.8279

Authors: Emily Riehl, Dominic Verity

Title: The 2-category theory of quasi-categories (1306.5144)
Abstract: In this paper we redevelop the foundations of the category theory of quasi-categories (also called infinity-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate form as absolute lifting diagrams in the 2-category, which we show are determined pointwise by the existence of certain initial or terminal vertices, allowing for the easy production of examples.
All the quasi-categorical notions introduced here are equivalent to the established ones but our proofs are independent and more "formal". In particular, these results generalise immediately to model categories enriched over quasi-categories.

Title: Homotopy coherent adjunctions and the formal theory of monads (1310.8279)
Abstract: In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterize its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes.
We extract several simplicial functors from the free homotopy coherent adjunction and show that quasi-categories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. We show that each vertex in the quasi-category of algebras for a homotopy coherent adjunction is a codescent object of a canonical diagram of free algebras. To conclude, we prove the quasi-categorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi-categories. Our proofs reveal that a mild variant of Beck's argument is "all in the weights" - much of it independent of the quasi-categorical context.

#arXiv
#spnetwork arXiv:1310.8279 #categorytheory
Emily Riehl replies (Nov. 1, 2013): Thanks David. Dom and I welcome questions and comments.

Urs Schreiber replies (Nov. 1, 2013): Hey david, could you add a pointer to the references section in the nlab article? Thanks!

David Roberts replies (Nov. 1, 2013): Done!

jesse mckeown replies (Nov. 5, 2013): #spnetwork

David Roberts replies (Nov. 5, 2013): +jesse mckeown - does it work when talking about more than one paper?

Urs Schreiber replies (Nov. 5, 2013): Thanks, David!

jesse mckeown replies (Nov. 5, 2013): not sure what you mean by "work". At least, for instance, https://selectedpapers.net/posts/z132xpghwxnqipdyh04chvywcofjhhzhwbk it notices when a second paper has been mentioned.

David Roberts replies (Nov. 6, 2013): I've edited to link to the second paper, at least.
[18] Comment on Graham's Number is Less Than 2^^^6 RE: topics: #combinatorics #RamseyTheory (Oct. 22, 2013): Show discussion.

The title of this paper is misleading. The number most often referred to as 'Graham's number', sometimes denoted by g_64, was originally mentioned in a 1977 article by Martin Gardner in Scientific American. It arose as an upper bound on a particular problem from Ramsey theory, a subfield of combinatorics [1]. +John Baez has some history of this [2], most notably the fact that g_64 doesn't appear in the joint paper Graham eventually published with Rothschild, but the smaller bound F^7(12), where F(n) = 2↑...n...↑3 (see [3] for Knuth's up-arrow notation). The paper by Lavrov, Lee and Mackey establishes a much smaller upper bound on the Ramsey-theoretic problem, and it is the exact solution to this problem that they are calling 'Graham's number'. In fact the bound given in the title, 2↑↑↑6, is a simplification, and not the smallest bound arrived at in the paper, which is 2↑↑2↑↑2↑↑9 (despite the apparent increase in complexity, this number is much smaller than 2↑↑↑6). In terms of the function F of Graham and Rothschild, the LLM bound is between F(4) and F(5).

[1] http://en.wikipedia.org/wiki/Ramsey_theory
[2] https://plus.google.com/117663015413546257905/posts/KJTgfjkTZCQ
[3] http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

#spnetwork arXiv:1304.6910 #RamseyTheory #combinatorics

http://arxiv.org/abs/1304.6910
[19] Comment on Six model structures for DG-modules over DGAs: Model category theory in homological action RE: topics: #arXiv (Oct. 7, 2013): Show discussion.

There was some discussion surrounding ideas from this preprint here: http://golem.ph.utexas.edu/category/2013/09/mapping_cocylinder_factorizations_via_the_small_object_argument.html

http://arxiv.org/abs/1310.1159

Title: Six model structures for DG-modules over DGAs: Model category theory in homological action
Authors: Tobias Barthel, J.P. May, +Emily Riehl

Abstract: In Part 1, we describe six projective-type model structures on the category of differential graded modules over a differential graded algebra A over a commutative ring R. When R is a field, the six collapse to three and are well-known, at least to folklore, but in the general case the new relative and mixed model structures offer interesting alternatives to the model structures in common use. The construction of some of these model structures requires two new variants of the small object argument, an enriched and an algebraic one, and we describe these more generally. In Part 2, we present a variety of theoretical and calculational cofibrant approximations in these model categories. The classical bar construction gives cofibrant approximations in the relative model structure, but generally not in the usual one. In the usual model structure, there are two quite different ways to lift cofibrant approximations from the level of homology modules over homology algebras, where they are classical projective resolutions, to the level of DG-modules over DG-algebras. The new theory makes model theoretic sense of earlier explicit calculations based on one of these constructions. A novel phenomenon we encounter is isomorphic cofibrant approximations with different combinatorial structure such that things proven in one avatar are not readily proven in the other.

#arXiv #spnetwork arXiv:1310.1159
[20] Comment on A Geometric Model for Odd Differential K-theory RE: topics: #KTheory (Sep. 12, 2013): Show discussion.

There is a small subtlety in the results of this paper which is worth amplifying. Recall that a differential extension of a cohomology theory is a refinement of a cohomology theory which incorporates extra smooth data. One version, and indeed the first, is the differential characters of Cheeger and Simons, which refines ordinary cohomology.

The paper at hand deals with the case of topological K-theory, and in particular the odd-graded part (recall that topological K-theory is 2-periodic, so we talk of its even and odd parts). This is where the subtlety arises: one can find many non-isomorphic differential refinement purely for the odd part of K-theory, whereas differential refinements of the even part of K-theory are unique up to isomorphism as differential refinements. However, if one has a differential refinement for K = K^even+K^odd, then this is unique up to isomorphism. What this means is that if one finds a model for the differential extension of K^1, it may not complete to a differential extension of K.

It is the odd part of this canonical refinement that paper calls 'odd differential K-theory'.

The authors construct isomorphisms from the model introduced in the paper to the odd part of other differential extensions of K-theory, and so show that the differential extension of K^1 given by Tradler, WIlson and Zeinalian is in fact the odd part of the full differential extension of K.

http://arxiv.org/abs/1309.2834

#spnetwork arXiv:1309.2834 #KTheory
John Baez replies (Sep. 12, 2013): Thanks for contributing to #spnetwork!

Is there a kind of classification of differential refinements of odd K-theory? Are they parametrized discretely, or is there a continuous 'moduli space' of them?
[21] Comment on Brute force searching, the typical set and Guesswork RE: topics: #entropy #informationTheory #cryptography (Aug. 15, 2013): Show discussion.

Some interesting work on cryptographic security, using notions of entropy which are different to Shannon's in that they don't just deal with the "average case". When trying to break a cryptosystem, a lucky guess can give enough information about the plaintext that can be leveraged to give more information - in this sense, not all random guesses give the same amount of information.

http://arxiv.org/abs/1301.6356

Press release here: http://web.mit.edu/newsoffice/2013/encryption-is-less-secure-than-we-thought-0814.html

#spnetwork #cryptography #informationTheory #entropy arXiv:1301.6356
[22] Comment on The Brauer group is not a derived invariant RE: topics: #geometry #derivedGeometry #arXiv #cohomology (Jun. 28, 2013): Show discussion.

Now this is very interesting: the Brauer group is not an invariant under derived equivalence of Calabi-Yau 3-folds. What the authors in fact show is that the group H_1+Br is an invariant, and then find a pair of derived equivalent 3-folds with different fundamental groups, and then deduce the Brauer groups must be different.

http://arxiv.org/abs/1306.6538

#arXiv #spnetwork arXiv:1306.6538 #geometry #derivedGeometry #cohomology
[23] Comment on Lagrangian structures on mapping stacks and semi-classical TFTs RE: topics: #geometry #TFTs #arXiv #mathematicalPhysics #geometricStacks (Jun. 17, 2013): Show discussion.

Ooh, this looks interesting...

http://arxiv.org/abs/1306.3235

Title: Lagrangian structures on mapping stacks and semi-classical TFTs
Author: Damien Calaque

Abstract: We extend a recent result of Pantev-Toen-Vaquie-Vezzosi, who constructed shifted symplectic structures on derived mapping stacks having a Calabi-Yau source and a shifted symplectic target. Their construction gives a clear conceptual framework for the so-called AKSZ formalism.
We extend the PTVV construction to derived mapping stacks with boundary conditions, which is required in most applications to quantum field theories (see e.g. the work of Cattaneo-Felder on the Poisson sigma model, and the recent work of Cattaneo-Mnev-Reshetikhin). We provide many examples of Lagrangian and symplectic structures that can be recovered in this way.
We finally give an application to topological field theories (TFTs). We expect that our approach will help to rigorously constuct a 2 dimensional TFT introduced by Moore and Tachikawa. A subsequent paper will be devoted to the construction of fully extended TFTs (in the sense of Baez-Dolan and Lurie) from mapping stacks.

#arXiv #spnetwork arXiv:1306.3235 #TFTs #mathematicalPhysics #geometry #geometricStacks
John Baez replies (Jun. 17, 2013): If you typed #spnetwork and a bit more, this would be on the selected papers network, and people interested in your arXiv links could find it there.

David Roberts replies (Jun. 17, 2013): I didn't want to clog up the system with an almost content-free comment. If I had something more to say about the paper I definitely would. But I can send every mention of any paper to spnetwork if there's demand for what are essentially glorified +1s

John Baez replies (Jun. 17, 2013): No, do whatever you like; just wanted to remind you of its existence.

As time passes there will be need for extensive filtering and sorting systems...

John Baez replies (Jun. 17, 2013): In particular, if anything you do is able to 'clog the system', the system needs to be fixed. We should count on the existence of thousands of people who are less careful than you.
[24] Comment on T-Duality for Langlands Dual Groups (May 28, 2013): Show discussion.

Some relationships between topological T-duality and Langlands dual groups for those not of type B or C appear at the end of this (is a slight improvement on a previous paper arXiv:1211.0763 by others, which only covered ADE groups). A nice application of gerbes here.

http://arxiv.org/abs/1305.6050

Title: T-Duality via Gerby Geometry and Reductions
Authors: Ulrich Bunke, Thomas Nikolaus

Abstract: We consider topological T-duality of torus bundles equipped with S^{1}-gerbes. We show how a geometry on the gerbe determines a reduction of its band to the subsheaf of S^{1}-valued functions which are constant along the torus fibres. We observe that such a reduction is exactly the additional datum needed for the construction of a T-dual pair. We illustrate the theory by working out the example of the canonical lifting gerbe on a compact Lie group which is a torus bundles over the associated flag manifold. It was a recent observation of Daenzer and van Erp (arXiv1211.0763) that for certain compact Lie groups and a particular choice of the gerbe, the T-dual torus bundle is given by the Langlands dual group.

arxiv:1305.6050 #spnetwork
[25] Comment on The homology of $\mathrm{tmf}$ (May 28, 2013): Show discussion.

A calculation of the mod 2 cohomology of the connective spectrum tmf. This is very nice. If you've seen diagrams of tmf homotopy or cohomology groups in terms of the Steenrod algebra before, they are utterly intimidating. Here Akhil warms up with the cohomology of the spectrum bo, which is related to K-theory, and one gets little baby diagrams that look like dumbbells and question marks.

http://arxiv.org/abs/1305.6100

Title: The homology of $\mathrm{tmf}$
Author: Akhil Mathew

Abstract: We compute the mod 2 cohomology of the spectrum $\tmf$ of topological modular forms by proving a 2-local equivalence $\tmf \wedge DA(1) \simeq BP\left \langle 2\right\rangle$, where DA(1) is an eight cell complex whose cohomology is isomorphic to the subalgebra of the mod 2 Steenrod algebra generated by $\mathrm{Sq}^2$ and $\mathrm{Sq}^4$. To do so, we give, with use of the language of stacks, a modular description of the elliptic homology of DA(1) via level three structures. We briefly discuss analogs at odd primes and recover the stack-theoretic description of the Adams-Novikov spectral sequence for $\tmf$.

arXiv:1305.6100 #spnetwork
[26] Comment on A Whirlwind Tour of the World of $(\infty,1)$-categories RE: topics: #arXiv (May 14, 2013): Show discussion.

If you want to know about (∞,1)-categories, but are afraid to ask, then this is the paper for you. What I especially good are the applications it lists.

http://arxiv.org/abs/1303.4669

Title: A Whirlwind Tour of the World of (\infty,1)-categories
Author: Omar Antolín Camarena

Abstract: This introduction to higher category theory is intended to a give the reader an intuition for what (\infty,1)-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.

#arXiv #spnetwork arXiv:1303.4669
John Baez replies (May 14, 2013): The correct syntax is #spnetwork arXiv:1303.4669 ... if you want #spnetwork to pick up on this. You can see if it did or not - maybe the link will suffice. The tag #arXiv does not help it, though you might want it for your own purposes.

Thanks for pointing out this paper!

David Roberts replies (May 14, 2013): No, that was my mistake. I've fixed it.

Adeel Ahmad Khan replies (May 14, 2013): Nice! I've added the reference to the nLab.

David Roberts replies (May 14, 2013): Thanks, +Adeel Ahmad Khan. For everyone else: don't be afraid to add stuff to the nLab!
[27] Comment on Are elite journals declining? (Apr. 26, 2013): Show discussion.

If you are a scientist lured by the glow of a glossy journal, have a read of this:

http://arxiv.org/abs/1304.6460

(I've made some points for mathematicians further down)

To quote from the abstract:

"Since the late 1980s and early 1990s elite journals have been publishing a decreasing proportion of these top cited papers. This also applies to the two journals that are typically considered as the top venues and often used as bibliometric indicators of "excellence", Science and Nature. On the other hand, several new and established journals are publishing an increasing proportion of most cited papers."

Some quotes from the paper itself:

"The digital age has transformed the manner in which scientists find papers. We used to search for papers on the bookshelves of the library but now we do so via the internet, often at our library’s website. Furthermore, papers are now directly accessible independently and one does not have to even look at the corresponding issue or volume of the journal. Hence, whether papers get cited or ignored is increasingly independent of the journal in which they appear."

"Papers that are freely accessible have a greater probability of being cited (Gagouri et al., 2010). Researchers now access papers from a greater variety of journals, not just the so-called premier journals in a given field, or elite journals in general. Technically, a paper could now be in any journal, and it would still be found by internet searches, downloaded if available, and cited if deemed relevant."

This is especially true of a field like mathematics, where papers appear on the arXiv long before publication (my goodness, if we had to wait until publication before we found a paper, things would be so incredibly slow. And for people in Australia, like me, who cannot pop over to the next state (US) or country (EU) for a conference, it would be very frustrating). And better, if people are blogging about their research (post-hoc, or even during, as at he n-category cafe), then the research results can be spread even sooner - it should get more citations regardless of the journal it appears in.

So the correlation between historical prestige and actual proportion of attention garnered seems to be going down. Unfortunately in mathematics, new general mid-to-high generalist journals analogous to the PLoS suite and PeerJ are thin on the ground, until the episciences project gets a bit more underway, and the Forum of Mathematics gets out some articles.

From the Times Higher Education article "Fields medallists’ open-access track" by Paul Jump [1]:

"The handful of epijournals (Professor Demailly estimated between five and 10) that will launch at the same time as the platform itself..."

From the Episciences website [2]:

"The Episciences.org platform is hosted and developed by the CCSD and will be launched in the first half of 2013 with Episciences-Maths."

Do we have any idea when that might be? +Timothy Gowers ?

(Update: +Terence Tao said on 17 January that the launch would be in April [3], but this has clearly slipped us by. Any further news on this front?)

And since _Forum of Mathematics has been open for submissions since October, any projection on when the first papers might appear? Will the publisher/editors wait until there is enough material for an issue, or will they release papers as they are accepted? If they wait for a full issue's worth of papers, will this be a habit, or will they settle down to a steady stream of releases?

---
[1] http://www.timeshighereducation.co.uk/news/fields-medallists-open-access-track/2002944.article

[2] http://episciences.org/

[3] https://plus.google.com/114134834346472219368/posts/Mph8iJ3S8XT

#spnetwork arxiv:1304.6460
Timothy Gowers replies (Apr. 26, 2013): I've heard that there has indeed been some slippage. I think September now looks more likely as a start date. (Well, I suppose it's trivially more likely, but I also mean reasonably likely.)

David Roberts replies (Apr. 26, 2013): Thanks, +Timothy Gowers .
[28] Comment on The classification of subfactors of index at most 5 (Apr. 24, 2013): Show discussion.

http://arxiv.org/abs/1304.6141

Via +Scott Morrison , who writes

"Our survey paper (joint with Vaughan Jones and +Noah Snyder) on small index subfactors has just appeared on the arXiv!"

#spnetwork arXiv:1304.6141

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