Glossary

Letter: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Other

Mac LaneWrite comment View comments

See Selected Papers

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MahantaWrite comment View comments

Webpage

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MaltsiniotisWrite comment View comments

Web page

MathSciNet

arXiv

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ManifoldWrite comment View comments

For the various types of manifolds, see Dieudonne: Panorama. The definition of manifold includes the existence of charts and transition functions. The definition of PL manifold, differentiable mfd, real analytic and complex analytic mfd involves conditions on the transition functions.

Given a (differentiable???) mfd M, have a notion of G-structure on M. Specific choices of G leads to the notions of Riemannian structure (orthogonal gp), pseudo-Riemannian str (Lorentz group), symplectic str (symplectic grp), and almost complex str (complex general linear group). Any complex analytic mfd comes with an almost complex str, but the converse is true only if the almost complex str is "integrable" (I think).

http://www.ncatlab.org/nlab/show/G-structure

http://nlab.mathforge.org/nlab/show/manifold

There are various things by Thurston available on giga.

http://mathoverflow.net/questions/116814/torsion-in-cohomology-of-smooth-manifolds

Manifolds: Many references in the Manifolds folder.

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ManinWrite comment View comments

MathSciNet

arXiv

Wikipedia entry

Check Algebra, Arithmetic and Geometry (in honor of Manin), 2008.

Selected papers, see scan

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Manin conjectureWrite comment View comments

arXiv:1009.2364 Manin's Conjecture for a Singular Sextic del Pezzo Surface from arXiv Front: math.NT by Daniel Loughran We prove Manin's conjecture for a del Pezzo surface of degree six which has one singularity of type $\mathbf{A}_2$. Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

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MarcolliWrite comment View comments

Web page

MathSciNet

arXiv

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MathematiciansWrite comment View comments

Some of my favourite mathematicians. "S" means that I have downloaded their interesting articles and listed the missing ones in scratchbook. "I" means that I have indexed his/her work in the DB, i.e. skimmed and taken brief notes.

Add the following: Bourbaki people, and automorphic people: Frenkel, Gelbart, Arthur, Shahidi, Piatetiskii-Shapiro, Rapoport, Zink, Yekutieli, Wells.

See the Fields medallists volume for Atiyah, Novikov, Mumford, Voevoedsky, Mori, Connes, Witten, Kontsevich.

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MathJaxWrite comment View comments

http://www.mathjax.org/

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Maurer-Cartan equationWrite comment View comments

http://ncatlab.org/nlab/show/Maurer-Cartan+equation

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MayWrite comment View comments

Papers of May

Web page of May

MathSciNet

Some publications:

  • Parametrized homotopy theory, by J. P. May and J. Sigurdsson
  • The geometry of iterated loop spaces.
  • Equivariant homotopy and cohomology theory.
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Mayer-VietorisWrite comment View comments

http://mathoverflow.net/questions/23175/mathematically-mature-way-to-think-about-mayervietoris

http://mathoverflow.net/questions/97621/mayer-vietoris-implies-excision

http://mathoverflow.net/questions/108819/is-there-a-mayer-vietoris-spectral-sequence-of-motivic-cohomology-for-closed-cove

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MazurWrite comment View comments

Web page

MathSciNet

arXiv

Some publications:

  • Artin and Mazur: Etale homotopy. LNM 100, 1969.
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MessingWrite comment View comments
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Meta-mathematicsWrite comment View comments

http://mathoverflow.net/questions/38639/thinking-and-explaining

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Metric geometryWrite comment View comments

Metric geometry: See Metric stuff folder. Among other things, book by Gromov on Metric structures

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MilneWrite comment View comments

Web page

MathSciNet

arXiv

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MilnorWrite comment View comments

Collected papers, at least 3 vols

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Milnor ConjectureWrite comment View comments

One of the main outcomes of Voevodsky's work on motivic cohomology. Here is his original paper, and here are some notes by Kahn. Here are Voevodsky's Seattle Lectures. This might be an updated version of the original article.

Another paper by Orlov, Vishik and Voevodsky

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Mirror symmetryWrite comment View comments

Many things in folder AG/Mirror symmetry

HMS for toric varieties: Abouzaid

http://mathoverflow.net/questions/40062/roadmap-for-mirror-symmetry

Meet HMS: http://arxiv.org/abs/0801.2014

HMS: See all by Seidel, books and arXiv

Names: Katzarkov, Sheridan, Orlov, Kontsevich. Orlov has a couple of surveys on arxiv.

MR2336692 (2008h:53151) Neeman, Amnon An infinite version of homological mirror symmetry. Real and complex singularities, 290--298, World Sci. Publ., Hackensack, NJ, 2007

Seidel book: Fukaya categories and Picard-Lefschetz theory

http://mathoverflow.net/questions/2905/is-the-fukaya-category-defined

arXiv:0908.1256 String modular motives of mirrors of rigid Calabi-Yau varieties from arXiv Front: math.AG by Savan Kharel, Monika Lynker, Rolf Schimmrigk The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the L-function of their $\Omega-$motives. It is shown that the emerging modular forms are string theoretic in origin, derived from the characters of the underlying rational conformal field theory. The definition of the class of Fano varieties of special type is motivated by the goal to find candidates for a geometric realization of the mirrors of rigid Calabi-Yau varieties. We consider explicitly the cubic sevenfold and the quartic fivefold, and show that their motivic L-functions agree with the L-functions of their rigid mirror Calabi-Yau varieties. We also show that the cubic fourfold is string theoretic, with a modular form that is determined by that of an exactly solvable K3 surface.

[arXiv:0907.3903] Homological mirror symmetry for curves of higher genus from arXiv Front: math.AG by Alexander I. Efimov Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. Seidel \cite{Se} has proved a version of this conjecture in the simplest case of the genus two curve. Basing on the paper of Seidel, we prove the conjecture (in the same version) for curves of genus $g\geq 3,$ relating the Fukaya category of a genus $g$ curve to the category of Landau-Ginzburg branes on a certain singular surface. We also prove a kind of reconstruction theorem for hypersurface singularities. Namely, formal type of hypersurface singularity (i.e. a formal power series up to a formal change of variables) can be reconstructed, with some technical assumptions, from its D$(\Z/2)$-G category of Landau-Ginzburg branes. The precise statement is Theorem 1.2.

[arXiv:0910.2014] Homological mirror symmetry of Fermat polynomials from arXiv Front: math.AG by So Okada We discuss homological mirror symmetry of Fermat polynomials in terms of derived Morita equivalence between derived categories of coherent sheaves and Fukaya-Seidel categories (a.k.a. perfect derived categories of directed Fukaya categories), and some related aspects such as stability conditions, (kinds of) modular forms, and Hochschild homologies.

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MitchellWrite comment View comments

Mitchell

MathSciNet

arXiv

Selected publications:

  • On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint (1994)
  • Hypercohomology spectra and Thomason's descent theorem (1997)
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Mixed Hodge modulesWrite comment View comments

Elementary introduction by Saito in Asterisque 179-180

ICM talk 1990, by Saito I guess.

Schuermann on characteristic classes of mixed Hodge modules

arXiv:0907.0584 Characteristic classes of mixed Hodge modules. from arXiv Front: math.AG by Joerg Schuermann. This paper gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito's deep theory of mixed Hodge modules as a "black box", thinking about them as "constructible or perverse sheaves of Hodge structures", having the same functorial calculus of Grothendieck functors. For the "constant Hodge sheaf", one gets the "motivic characteristic classes" of Brasselet-Schuermann-Yokura, whereas the classes of the "intersection homology Hodge sheaf" were studied by Cappell-Maxim-Shaneson.

There are two versions of these characteristic classes. The K-theoretical classes capture information about the graded pieces of the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. Application of a suitable Todd class transformation then gives classes in homology. These classes are functorial for proper pushdown and exterior products, together with some other properties one would expect for a "good" theory of characteristic classes for singular spaces. For "admissible variation of mixed Hodge structures" they have an explicit classical description in terms of "logarithmic de Rham complexes". On a point space they correspond to a specialization of the Hodge polynomial of a mixed Hodge structure, which one gets by forgetting the weight filtration.

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Mixed Hodge structuresWrite comment View comments

Some articles of Wojtkowiak. A more fundamental paper is probably Deligne-Beilinson in Motives (Interpretation motivique...).

Some paper by Kashiwara.

Gelfand-Manin: Algebra V

http://mathoverflow.net/questions/73924/book-on-mixed-hodge-structures

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Model 2-categoryWrite comment View comments

n-Lab

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Model categoryWrite comment View comments

nLab Quillen equivalence

http://ncatlab.org/nlab/show/simplicial+model+category

http://ncatlab.org/nlab/show/cofibrantly+generated+model+category

http://mathoverflow.net/questions/16183/infty-1-categories-and-model-categories

Toen Essen talk: Any model category is naturally enriched over the homotopy cat of simplicial sets. This exposition is by the way a very concise intro to some key concepts in model cats, including localization and infinity-cat thinking, and homotopy limits.

http://ncatlab.org/nlab/show/global+model+structure+on+functors

http://mathoverflow.net/questions/78400/do-we-still-need-model-categories

References: Dwyer and Spalinski, Hovey's book, Goerss and Schemmerhorn.

Hirschhorn: Model cats and their localizations

Hirschhorn et al: Homotopy Limit Functors on Model Categories and Homotopical Categories (AMS)

Dwyer-Spalinski in the homotopy theory folder: Model categories, Homotopy limits brief intro, localization wrt a homology theory: very brief intro on p. 54.

Definition: A model category is a category Mathematical formula with all small limits and colimits, together with a model structure on Mathematical formula. A model structure on a category consists of three subcategories, called cofibrations (cofibs), fibrations (fibs), and weak equivalences (WEs), and two functorial factorizations Mathematical formula and Mathematical formula, satisfying:

  1. (2-out-of-3) If two of Mathematical formula, Mathematical formula, Mathematical formula are WEs, then so is the third.
  2. (Retracts) The three classes of morphisms are closed under retracts.
  3. (Lifting) Trivial cofibs have the LLP wrt fibs, and cofibs have the LLP wrt trivial fibs.
  4. (Factorization) For any morphism Mathematical formula, Mathematical formula is a cofib, Mathematical formula is a trivial fib, Mathematical formula is a trivial cofib, and Mathematical formula is a fib.

This is the definition given in Hovey, which differs slightly from earlier definitions, for example Quillen's original definition. Some people (ref?) have suggested that the first axiom should be replaced by a (4-out-of-6) axiom, to obtain a more general setting for homotopy theory in some cases.

Hovey suggests (p. 21) that the 2-category of model categories might behave like a model category.

Examples of model cats:

  • Sset: Simplicial sets
  • Top: Topological spaces
  • Chain complexes of R-modules
  • Modules over a Frobenius ring
  • Cochain complexes of comodules over a Hopf algebra

See also nLab entry on homotopy theory

Christensen, Dwyer, Isaksen: Obstruction theory in model cats

It seems like all cofibrantly generated model cats are combinatorial: http://arxiv.org/abs/0905.0595

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Model category axiomsWrite comment View comments

Variants on model structure axioms (taken from Sevilla lectures): Baues, Brown, Cisinski, Thomason. See also Waldhausen category.

http://nlab.mathforge.org/nlab/show/Thomason+model+structure

http://mathoverflow.net/questions/29635/what-determines-a-model-structure

arXiv:1102.2512 Partial model categories and their simplicial nerves from arXiv Front: math.CT by C. Barwick, D. M. Kan In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More precisely, a partial model category will be a relative category that has the two out of six property and admits a 3-arrow calculus.

We then show that Charles Rezk's result that the simplicial space obtained from a simplicial model category by taking a Reedy fibrant replacement of its simplicial nerve is a complete Segal space also holds for these partial model categories.

We also note that conversely every complete Segal space is