@book{hatcherAlgebraicTopologya,
title = {Algebraic {{Topology}}},
language = {en},
author = {Hatcher, Allen},
file = {/home/zack/Dropbox/Library/Allen Hatcher/Algebraic Topology (609)/Algebraic Topology - Allen Hatcher.pdf}
}
@book{dummitAbstractAlgebra,
title = {Abstract {{Algebra}}},
language = {en},
author = {Dummit, David Steven and Foote, Richard M},
file = {/home/zack/Dropbox/Library/David Steven Dummit/Abstract Algebra (656)/Abstract Algebra - David Steven Dummit.pdf}
}
@book{rudinRealComplexAnalysis1987,
address = {New York},
edition = {3rd ed},
title = {Real and Complex Analysis},
isbn = {978-0-07-054234-1},
lccn = {QA300 .R82 1987},
language = {en},
publisher = {{McGraw-Hill}},
author = {Rudin, Walter},
year = {1987},
keywords = {Mathematical analysis},
file = {/home/zack/Dropbox/Library/Walter Rudin/Real and Complex Analysis (439)/Real and Complex Analysis - Walter Rudin.pdf}
}
@book{rudinPrinciplesMathematicalAnalysis1976,
address = {New York},
edition = {3d ed},
series = {International Series in Pure and Applied Mathematics},
title = {Principles of Mathematical Analysis},
isbn = {978-0-07-054235-8},
lccn = {QA300 .R8 1976},
language = {en},
publisher = {{McGraw-Hill}},
author = {Rudin, Walter},
year = {1976},
keywords = {Mathematical analysis},
file = {/home/zack/Dropbox/Library/Walter Rudin/Principles of Mathematical Analysis (431)/Rudin - 1976 - Principles of mathematical analysis.pdf}
}
@book{griffithsRationalHomotopyTheory2013,
address = {New York, NY},
series = {Progress in {{Mathematics}}},
title = {Rational {{Homotopy Theory}} and {{Differential Forms}}},
volume = {16},
isbn = {978-1-4614-8467-7 978-1-4614-8468-4},
language = {en},
publisher = {{Springer New York}},
author = {Griffiths, Phillip and Morgan, John},
year = {2013},
file = {/home/zack/Dropbox/Library/Philip Griffiths, John Morgan/Rational Homotopy Theory and Differential Forms (436)/Griffiths and Morgan - 2013 - Rational Homotopy Theory and Differential Forms.pdf},
doi = {10.1007/978-1-4614-8468-4}
}
@book{milnorCharacteristicClasses,
title = {Characteristic {{Classes}}},
language = {en},
author = {Milnor, John and Stasheff, James D},
file = {/home/zack/Dropbox/Library/John Milnor/Characteristic Classes (654)/Characteristic Classes - John Milnor.pdf}
}
@book{bottDifferentialFormsAlgebraic1982,
address = {New York, NY},
series = {Graduate {{Texts}} in {{Mathematics}}},
title = {Differential {{Forms}} in {{Algebraic Topology}}},
volume = {82},
isbn = {978-1-4419-2815-3 978-1-4757-3951-0},
language = {en},
publisher = {{Springer New York}},
author = {Bott, Raoul and Tu, Loring W.},
year = {1982},
file = {/home/zack/Dropbox/Library/Raoul Bott/Differential Forms in Algebraic Topology (Graduate Texts in Mathematics) (655)/Differential Forms in Algebraic Topology ( - Raoul Bott.pdf},
doi = {10.1007/978-1-4757-3951-0}
}
@article{angelini-knollSegalConjectureTopological2017,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1705.03343},
primaryClass = {math},
title = {The {{Segal Conjecture}} for Topological {{Hochschild}} Homology of the {{Ravenel}} Spectra \${{X}}(n)\$ and \${{T}}(n)\$},
abstract = {In [27], Ravenel introduced sequences of spectra X(n) and T (n) which played an important role in the proof of the Nilpotence Theorem of Devinatz-Hopkins-Smith [11]. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic group of prime order. We prove the same theorem for T (n) under the assumption that T (n) is an E2-ring spectrum. This is also a first step towards computing algebraic K-theory of X(n) and T (n) using trace methods.},
language = {en},
journal = {arXiv:1705.03343 [math]},
author = {{Angelini-Knoll}, Gabe and Quigley, J. D.},
month = may,
year = {2017},
keywords = {Mathematics - Algebraic Topology,Mathematics - K-Theory and Homology,SSP42; 18D55,THH},
file = {/home/zack/Dropbox/Zotero/storage/STRA8APJ/Angelini-Knoll and Quigley - 2017 - The Segal Conjecture for topological Hochschild ho.pdf}
}
@article{rietschNewtonOkounkovBodiesCluster2017,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1712.00447},
primaryClass = {math},
title = {Newton-{{Okounkov}} Bodies, Cluster Duality, and Mirror Symmetry for {{Grassmannians}}},
abstract = {We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian \$X=Gr\_\{n-k\}(\textbackslash{}mathbb C\^n)\$, as well as the mirror dual Landau-Ginzburg model \$(\textbackslash{}check\{X\}\^\textbackslash{}circ, W\_q:\textbackslash{}check\{X\}\^\textbackslash{}circ \textbackslash{}to \textbackslash{}mathbb C)\$, where \$\textbackslash{}check\{X\}\^\textbackslash{}circ\$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian \$\textbackslash{}check\{X\} = Gr\_k((\textbackslash{}mathbb C\^n)\^*)\$, and the superpotential W\_q has a simple expression in terms of Pl\textbackslash{}"ucker coordinates. Grassmannians simultaneously have the structure of an \$\textbackslash{}mathcal\{A\}\$-cluster variety and an \$\textbackslash{}mathcal\{X\}\$-cluster variety. Given a cluster seed G, we consider two associated coordinate systems: a \$\textbackslash{}mathcal X\$-cluster chart \$\textbackslash{}Phi\_G:(\textbackslash{}mathbb C\^*)\^\{k(n-k)\}\textbackslash{}to X\^\{\textbackslash{}circ\}\$ and a \$\textbackslash{}mathcal A\$-cluster chart \$\textbackslash{}Phi\_G\^\{\textbackslash{}vee\}:(\textbackslash{}mathbb C\^*)\^\{k(n-k)\}\textbackslash{}to \textbackslash{}check\{X\}\^\textbackslash{}circ\$. To each \$\textbackslash{}mathcal X\$-cluster chart \$\textbackslash{}Phi\_G\$ and ample `boundary divisor' \$D\$ in \$X\textbackslash{}setminus X\^\{\textbackslash{}circ\}\$, we associate a Newton-Okounkov body \$\textbackslash{}Delta\_G(D)\$ in \$\textbackslash{}mathbb R\^\{k(n-k)\}\$, which is defined as the convex hull of rational points. On the other hand using the \$\textbackslash{}mathcal A\$-cluster chart \$\textbackslash{}Phi\_G\^\{\textbackslash{}vee\}\$ on the mirror side, we obtain a set of rational polytopes, described by inequalities, by writing the superpotential \$W\_q\$ in the \$\textbackslash{}mathcal A\$-cluster coordinates, and then "tropicalising". Our main result is that the Newton-Okounkov bodies \$\textbackslash{}Delta\_G(D)\$ and the polytopes obtained by tropicalisation coincide. As an application, we construct degenerations of the Grassmannian to toric varieties corresponding to these Newton-Okounkov bodies. Additionally, when \$G\$ corresponds to a plabic graph, we give a formula for the lattice points of the Newton-Okounkov bodies, which has an interpretation in terms of quantum Schubert calculus.},
language = {en},
journal = {arXiv:1712.00447 [math]},
author = {Rietsch, Konstanze and Williams, Lauren},
month = nov,
year = {2017},
keywords = {Mathematics - Algebraic Geometry,Mathematics - Combinatorics,Mirror Symmetry},
file = {/home/zack/Dropbox/Zotero/storage/9JG23WXA/Rietsch and Williams - 2017 - Newton-Okounkov bodies, cluster duality, and mirro.pdf}
}
@article{bartholdiHomotopyGroupsSpheres2018,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1805.10894},
primaryClass = {math},
title = {Homotopy Groups of Spheres and Dimension Quotients},
abstract = {We construct for every prime p a finitely presented group G in which the dimension quotient (G {$\cap$} (1 + ̟n))/{$\gamma$}n(G) has p-torsion for some n. The construction uses Serre's element of order p in the homotopy group {$\pi$}2p(S2), and derives a result analogous to the main claim in the context of Lie algebras.},
language = {en},
journal = {arXiv:1805.10894 [math]},
author = {Bartholdi, Laurent and Mikhailov, Roman},
month = may,
year = {2018},
keywords = {Mathematics - Algebraic Topology,Mathematics - Group Theory},
file = {/home/zack/Dropbox/Zotero/storage/IK9B7CMD/Bartholdi and Mikhailov - 2018 - Homotopy groups of spheres and dimension quotients.pdf}
}
@article{littArithmeticRepresentationsFundamental2018,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1809.03524},
primaryClass = {math},
title = {Arithmetic Representations of Fundamental Groups {{II}}: Finiteness},
shorttitle = {Arithmetic Representations of Fundamental Groups {{II}}},
abstract = {Let X be a smooth curve over a finitely generated field k, and let {$\mathscr{l}$} be a prime different from the characteristic of k. We analyze the dynamics of the Galois action on the deformation rings of mod {$\mathscr{l}$} representations of the geometric fundamental group of X. Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero.},
language = {en},
journal = {arXiv:1809.03524 [math]},
author = {Litt, Daniel},
month = sep,
year = {2018},
keywords = {Mathematics - Algebraic Geometry,Mathematics - Number Theory},
file = {/home/zack/Dropbox/Zotero/storage/RPDSKAJF/Litt - 2018 - Arithmetic representations of fundamental groups I.pdf}
}
@article{jaffeTheoreticalMathematicsCultural1993,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {math/9307227},
title = {``{{Theoretical}} Mathematics'': {{Toward}} a Cultural Synthesis of Mathematics and Theoretical Physics},
shorttitle = {``{{Theoretical}} Mathematics''},
abstract = {Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.},
language = {en},
journal = {arXiv:math/9307227},
author = {Jaffe, Arthur and Quinn, Frank},
month = jun,
year = {1993},
keywords = {Mathematics - History and Overview},
file = {/home/zack/Dropbox/Zotero/storage/YPMQ6BFA/Jaffe and Quinn - 1993 - ``Theoretical mathematics'' Toward a cultural syn.pdf}
}
@article{thurstonProofProgressMathematics1994,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {math/9404236},
title = {On Proof and Progress in Mathematics},
abstract = {In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.},
language = {en},
journal = {arXiv:math/9404236},
author = {Thurston, William P.},
month = mar,
year = {1994},
keywords = {Mathematics - History and Overview},
file = {/home/zack/Dropbox/Zotero/storage/WE6JNAIL/Thurston - 1994 - On proof and progress in mathematics.pdf}
}
@article{farbResolventDegreeHilbert,
title = {Resolvent Degree, {{Hilbert}}'s 13th {{Problem}} and Geometry},
abstract = {We develop the theory of resolvent degree, introduced by Brauer [Bra2] in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.},
language = {en},
author = {Farb, Benson and Wolfson, Jesse},
pages = {66},
file = {/home/zack/Dropbox/Zotero/storage/STJ6D6NM/Farb and Wolfson - Resolvent degree, Hilbert’s 13th Problem and geome.pdf}
}
@article{losevInductiveConstructionProcesi2019,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1901.05862},
primaryClass = {math},
title = {On Inductive Construction of {{Procesi}} Bundles},
abstract = {We give a new proof of Haiman's n! theorem from 2001. This theorem states that the so called isospectral Hilbert scheme of n points in C2 is Cohen-Macaulay and Gorenstein. Unlike Haiman's original proof, ours is purely geometric/representation-theoretic and does not use complicated combinatorial arguments. In the proof we show that Procesi bundles on Hilbn(C2) for various n, as constructed by Bezrukavnikov and Kaledin, admit an inductive description via the so called nested Hilbert schemes. Our proof of the latter fact is based on studying quantizations of the nested Hilbert schemes in zero and positive characteristic.},
language = {en},
journal = {arXiv:1901.05862 [math]},
author = {Losev, Ivan},
month = jan,
year = {2019},
keywords = {Mathematics - Algebraic Geometry,14E16; 53D55; 16G99,Mathematics - Representation Theory},
file = {/home/zack/Dropbox/Zotero/storage/9IS6DNWR/Losev - 2019 - On inductive construction of Procesi bundles.pdf}
}
@article{hutchingsIntroductionSpectralSequences,
title = {Introduction to Spectral Sequences},
abstract = {The words ``spectral sequence'' strike fear into the hearts of many hardened mathematicians. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful.},
language = {en},
author = {Hutchings, Michael},
keywords = {Spectral Sequences},
pages = {13},
file = {/home/zack/Dropbox/Zotero/storage/Z5Z4B779/Hutchings - Introduction to spectral sequences.pdf}
}
@article{hallSubgroupsFreeProducts1953,
title = {Subgroups of Free Products},
volume = {3},
issn = {0030-8730, 0030-8730},
doi = {10.2140/pjm.1953.3.115},
language = {en},
number = {1},
journal = {Pacific Journal of Mathematics},
author = {Hall, Marshall},
month = mar,
year = {1953},
pages = {115-120},
file = {/home/zack/Dropbox/Zotero/storage/LIBUMTJV/Hall - 1953 - Subgroups of free products.pdf}
}
@incollection{dwyerHomotopyTheoriesModel1995,
title = {Homotopy {{Theories}} and {{Model Categories}}},
isbn = {978-0-444-81779-2},
language = {en},
booktitle = {Handbook of {{Algebraic Topology}}},
publisher = {{Elsevier}},
author = {Dwyer, W.G. and Spalinski, J.},
year = {1995},
keywords = {Model Categories},
pages = {73-126},
file = {/home/zack/Dropbox/Zotero/storage/Y7XNUSMG/Dwyer and Spalinski - 1995 - Homotopy Theories and Model Categories.pdf},
doi = {10.1016/B978-044481779-2/50003-1}
}
@book{hirschhornModelCategoriesTheir2009,
address = {Providence, Rhode Island},
series = {Mathematical {{Surveys}} and {{Monographs}}},
title = {Model {{Categories}} and {{Their Localizations}}},
volume = {99},
isbn = {978-0-8218-4917-0 978-1-4704-1326-2},
abstract = {We begin with the basic definitions and ideas of model categories, giving complete arguments in order to make this accessible to the novice. We go on to develop all of the homotopy theory that we need for our localization results.},
language = {en},
publisher = {{American Mathematical Society}},
author = {Hirschhorn, Philip},
month = aug,
year = {2009},
keywords = {Model Categories,Localization},
file = {/home/zack/Dropbox/Zotero/storage/CMATLHEK/Hirschhorn - 2009 - Model Categories and Their Localizations.pdf},
doi = {10.1090/surv/099}
}
@misc{LocalizationHomotopyTheory,
title = {Localization in {{Homotopy Theory}}.Pdf},
keywords = {Localization},
file = {/home/zack/Dropbox/Zotero/storage/UTCTBMBN/Localization in Homotopy Theory.pdf}
}
@misc{RationalHomotopyPdf,
title = {Rational {{Homotopy}}.Pdf},
file = {/home/zack/Dropbox/Zotero/storage/A9RLL6PY/Rational Homotopy.pdf}
}
@misc{ClassificationBundlesRyan,
title = {Classification of {{Bundles}} - {{Ryan Grady}}.Pdf},
keywords = {Fiber Bundles},
file = {/home/zack/Dropbox/Zotero/storage/PQGFCQ2T/Classification of Bundles - Ryan Grady.pdf}
}
@article{chanPrimerHomologicalAlgebra,
title = {A {{Primer}} on {{Homological Algebra}}},
language = {en},
author = {Chan, Henry Y},
pages = {7},
file = {/home/zack/Dropbox/Zotero/storage/WXPNXLDP/Chan - A Primer on Homological Algebra.pdf}
}
@article{blasiakCOHOMOLOGYCOMPLEXGRASSMANNIAN,
title = {{{COHOMOLOGY OF THE COMPLEX GRASSMANNIAN}}},
abstract = {The Grassmannian is a generalization of projective spaces\textendash{}instead of looking at the set of lines of some vector space, we look at the set of all n-planes. It can be given a manifold structure, and we study the cohomology ring of the Grassmannian manifold in the case that the vector space is complex. The multiplicative structure of the ring is rather complicated and can be computed using the fact that for smooth oriented manifolds, cup product is Poincar\textasciiacute{}e dual to intersection. There is some nice combinatorial machinery for describing the intersection numbers. This includes the symmetric Schur polynomials, Young tableaux, and the Littlewood-Richardson rule. Sections 1, 2, and 3 introduce notation and the necessary topological tools. Section 4 uses linear algebra to prove Pieri's formula, which describes the cup product of the cohomology ring in a special case. Section 5 describes the combinatorics and algebra that allow us to deduce all the multiplicative structure of the cohomology ring from Pieri's formula.},
language = {en},
author = {Blasiak, Jonah},
pages = {16},
file = {/home/zack/Dropbox/Zotero/storage/4DMEPZYU/Blasiak - COHOMOLOGY OF THE COMPLEX GRASSMANNIAN.pdf}
}
@article{hirschTubularNeighbourhoodsManifolds1966,
title = {On Tubular Neighbourhoods of Manifolds. {{I}}},
volume = {62},
issn = {0305-0041, 1469-8064},
doi = {10.1017/S0305004100039712},
language = {en},
number = {02},
journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
author = {Hirsch, Morris W.},
month = apr,
year = {1966},
pages = {177},
file = {/home/zack/Dropbox/Zotero/storage/THMGNWWV/Hirsch - 1966 - On tubular neighbourhoods of manifolds. I.pdf}
}
@misc{TubularNeighborhoodsPdf,
title = {Tubular {{Neighborhoods}}.Pdf},
file = {/home/zack/Dropbox/Zotero/storage/EMV8RS84/Tubular Neighborhoods.pdf}
}
@article{hatcherNotesBasic3Manifold,
title = {Notes on {{Basic}} 3-{{Manifold Topology}}},
language = {en},
author = {Hatcher, Allen},
keywords = {Low Dimensional,Manifolds},
pages = {61},
file = {/home/zack/Dropbox/Zotero/storage/DNE34B7Z/Hatcher - Notes on Basic 3-Manifold Topology.pdf}
}
@article{aurouxBeginnerIntroductionFukaya2013,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1301.7056},
primaryClass = {math},
title = {A Beginner's Introduction to {{Fukaya}} Categories},
abstract = {The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic topology, mirror symmetry and low-dimensional topology. This text is based on a series of lectures given at a Summer School on Contact and Symplectic Topology at Universit\textbackslash{}'e de Nantes in June 2011.},
language = {en},
journal = {arXiv:1301.7056 [math]},
author = {Auroux, Denis},
month = jan,
year = {2013},
keywords = {Mirror Symmetry,Mathematics - Symplectic Geometry},
file = {/home/zack/Dropbox/Zotero/storage/QAC7MVU3/Auroux - 2013 - A beginner's introduction to Fukaya categories.pdf}
}
@article{rasmussenFloerHomologyLow,
title = {Floer {{Homology}} and {{Low Dimensional Topology}}},
language = {en},
author = {Rasmussen, Jacob and Watson, Liam},
keywords = {Low Dimensional,Floer},
pages = {64},
file = {/home/zack/Dropbox/Zotero/storage/3ZS2DAGH/Rasmussen and Watson - Floer Homology and Low Dimensional Topology.pdf}
}
@book{axlerLinearAlgebraDone1997,
address = {New York},
edition = {2nd ed},
series = {Undergraduate Texts in Mathematics},
title = {Linear Algebra Done Right},
isbn = {978-0-387-98259-5 978-0-387-98258-8},
lccn = {QA184 .A96 1997},
language = {en},
publisher = {{Springer}},
author = {Axler, Sheldon Jay},
year = {1997},
keywords = {Algebras; Linear},
file = {/home/zack/Dropbox/Library/Sheldon Axler/Linear Algebra Done Right (Undergraduate Texts in Mathematics) (471)/Linear Algebra Done Right (Undergraduate T - Sheldon Axler.pdf}
}
@article{garrettHilbertSchmidtOperatorsNuclear2014,
title = {Hilbert-{{Schmidt}} Operators, Nuclear Spaces, Kernel Theorem {{I}}},
language = {en},
author = {Garrett, Paul},
year = {2014},
pages = {11},
file = {/home/zack/Dropbox/Zotero/storage/9PLQV6Q6/Garrett - 2014 - Hilbert-Schmidt operators, nuclear spaces, kernel .pdf}
}
@misc{HomologicalAlgebraIntroductory,
title = {Homological Algebra - Introductory Book on Spectral Sequences},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/22188/introductory-book-on-spectral-sequences},
file = {/home/zack/Dropbox/Zotero/storage/PFNPVU9C/introductory-book-on-spectral-sequences.html}
}
@misc{HomologicalAlgebraIntroductorya,
title = {Homological Algebra - Introductory Book on Spectral Sequences},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/22188/introductory-book-on-spectral-sequences},
file = {/home/zack/Dropbox/Zotero/storage/NIDJG35Q/introductory-book-on-spectral-sequences.html}
}
@article{vakilSPECTRALSEQUENCESFRIEND,
title = {{{SPECTRAL SEQUENCES}}: {{FRIEND OR FOE}}?},
language = {en},
author = {Vakil, Ravi},
pages = {12},
file = {/home/zack/Dropbox/Zotero/storage/2S7X32CY/Vakil - SPECTRAL SEQUENCES FRIEND OR FOE.pdf}
}
@article{chowYouCouldHave2006,
title = {You {{Could Have Invented Spectral Sequences}}},
volume = {53},
language = {en},
number = {1},
author = {Chow, Timothy Y},
year = {2006},
pages = {5},
file = {/home/zack/Dropbox/Zotero/storage/UKR4GPUY/Chow - 2006 - You Could Have Invented Spectral Sequences.pdf}
}
@misc{rachaelSpectralSequences2016,
title = {Spectral Sequences},
abstract = {Last post Anna talked about representing a large tensor by a collection of smaller ones, which gave me the inspiration to write this post. A lot of the time in topology (or at least in the p\ldots{}},
language = {en},
journal = {Picture this maths},
author = {{Rachael}},
month = feb,
year = {2016},
keywords = {examples},
file = {/home/zack/Dropbox/Zotero/storage/DEH7VCHQ/spectral-sequences.html}
}
@misc{HomologicalAlgebraAre,
title = {Homological Algebra - {{Are}} Long Exact Sequences in Homology a Special Case of Spectral Sequences?},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/2204494/are-long-exact-sequences-in-homology-a-special-case-of-spectral-sequences?rq=1},
file = {/home/zack/Dropbox/Zotero/storage/FVLRBIB8/are-long-exact-sequences-in-homology-a-special-case-of-spectral-sequences.html}
}
@misc{CohomologyWhatCo,
title = {Cohomology - {{What}} Is (Co)Homology, and How Does a Beginner Gain Intuition about It?},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it},
file = {/home/zack/Dropbox/Zotero/storage/FWSSLBSI/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it.html}
}
@misc{IntuitionGroupCohomology,
title = {Intuition for {{Group Cohomology}}},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/10879/intuition-for-group-cohomology},
file = {/home/zack/Dropbox/Zotero/storage/QF8NM5AJ/intuition-for-group-cohomology.html}
}
@misc{SerreSpectralSequence,
title = {Serre Spectral Sequence - {{Wikipedia}}},
howpublished = {https://en.wikipedia.org/wiki/Serre\_spectral\_sequence},
file = {/home/zack/Dropbox/Zotero/storage/LKPTRNXX/Serre_spectral_sequence.html}
}
@article{ZigzagLemma2018,
title = {Zig-Zag Lemma},
copyright = {Creative Commons Attribution-ShareAlike License},
abstract = {In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category.},
language = {en},
journal = {Wikipedia},
month = nov,
year = {2018},
file = {/home/zack/Dropbox/Zotero/storage/WME57AV5/index.html},
note = {Page Version ID: 871048974}
}
@article{SnakeLemma2018,
title = {Snake Lemma},
copyright = {Creative Commons Attribution-ShareAlike License},
abstract = {The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called connecting homomorphisms.},
language = {en},
journal = {Wikipedia},
month = jun,
year = {2018},
file = {/home/zack/Dropbox/Zotero/storage/P9SQABL6/index.html},
note = {Page Version ID: 848161219}
}
@book{jamesHandbookAlgebraicTopology1995,
address = {Amsterdam ; New York},
title = {Handbook of Algebraic Topology},
isbn = {978-0-444-81779-2},
lccn = {QA612 .H36 1995},
language = {en},
publisher = {{Elsevier Science B.V}},
editor = {James, I. M.},
year = {1995},
keywords = {Algebraic topology},
file = {/home/zack/Dropbox/Zotero/storage/JZTHCN5V/James - 1995 - Handbook of algebraic topology.pdf}
}
@misc{WhatModernAlgebraic,
title = {What Is Modern Algebraic Topology(Homotopy Theory) About?},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/228388/what-is-modern-algebraic-topologyhomotopy-theory-about},
file = {/home/zack/Dropbox/Zotero/storage/QRKK54MQ/what-is-modern-algebraic-topologyhomotopy-theory-about.html}
}
@misc{OpenProblemsAlgebraic,
title = {Open Problems in Algebraic Topology},
howpublished = {http://www-users.math.umn.edu/\textasciitilde{}tlawson/hovey/},
file = {/home/zack/Dropbox/Zotero/storage/JPI6HD9X/hovey.html}
}
@misc{SimpleExamplesUse,
title = {Simple Examples for the Use of Spectral Sequences},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/23297/simple-examples-for-the-use-of-spectral-sequences},
file = {/home/zack/Dropbox/Zotero/storage/EZALFUP5/simple-examples-for-the-use-of-spectral-sequences.html}
}
@misc{IntuitionWhatComes,
title = {Intuition - {{What}} Comes after Diagram Chasing?},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/1761124/what-comes-after-diagram-chasing?rq=1},
file = {/home/zack/Dropbox/Zotero/storage/4MJ5SGTQ/what-comes-after-diagram-chasing.html}
}
@misc{CategoryTheoryBigger,
title = {Category Theory - {{The}} Bigger Picture the {{Five Lemma}} Fits Into},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/1764913/the-bigger-picture-the-five-lemma-fits-into?rq=1},
file = {/home/zack/Dropbox/Zotero/storage/593SRXL5/the-bigger-picture-the-five-lemma-fits-into.html}
}
@misc{AlgebraicTopologyWhy,
title = {At.Algebraic Topology - {{Why}} Is the Definition of the Higher Homotopy Groups the "Right One"?},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/284041/why-is-the-definition-of-the-higher-homotopy-groups-the-right-one},
keywords = {intuition,motivation,obstruction},
file = {/home/zack/Dropbox/Zotero/storage/ANIZFVVL/why-is-the-definition-of-the-higher-homotopy-groups-the-right-one.html}
}
@misc{HomologicalAlgebraSpectral,
title = {Homological Algebra - {{Spectral}} Sequences: Opening the Black Box Slowly with an Example - {{MathOverflow}}},
howpublished = {https://mathoverflow.net/questions/45036/spectral-sequences-opening-the-black-box-slowly-with-an-example?rq=1},
keywords = {examples},
file = {/home/zack/Dropbox/Zotero/storage/DRQ85XN3/spectral-sequences-opening-the-black-box-slowly-with-an-example.html}
}
@misc{HomologicalAlgebraIntroductoryb,
title = {Homological Algebra - Introductory Book on Spectral Sequences},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/22188/introductory-book-on-spectral-sequences/22234},
keywords = {references},
file = {/home/zack/Dropbox/Zotero/storage/8F6D6MAJ/22234.html}
}
@misc{AlgebraicTopologyTwo,
title = {Algebraic Topology - {{Two CW}} Complexes with Isomorphic Homotopy Groups and Homology, yet Not Homotopy Equivalent},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/88943/two-cw-complexes-with-isomorphic-homotopy-groups-and-homology-yet-not-homotopy},
keywords = {counterexamples},
file = {/home/zack/Dropbox/Zotero/storage/LBFQLCTE/two-cw-complexes-with-isomorphic-homotopy-groups-and-homology-yet-not-homotopy.html}
}
@misc{AlgebraicTopologySpaces,
title = {Algebraic Topology - {{Spaces}} with Equal Homotopy Groups but Different Homology Groups?},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/99302/spaces-with-equal-homotopy-groups-but-different-homology-groups},
keywords = {counterexamples},
file = {/home/zack/Dropbox/Zotero/storage/A9ZH7VNP/spaces-with-equal-homotopy-groups-but-different-homology-groups.html}
}
@misc{limsupExactSequencesGrothendieck2015,
title = {Exact {{Sequences}} and the {{Grothendieck Group}}},
abstract = {As before, all rings are not commutative in general. Definition. An~exact sequence of R-modules is a collection of R-modules \$latex M\_i\$ and a sequence of R-module homomorphisms: \$latex \textbackslash{}ldots \textbackslash{}sta\ldots{}},
language = {en},
journal = {Mathematics and Such},
author = {{limsup}},
month = jan,
year = {2015},
file = {/home/zack/Dropbox/Zotero/storage/8A6ZH5J6/exact-sequences-and-the-grothendieck-group.html}
}
@misc{AlgebraicTopologyWhat,
title = {At.Algebraic Topology - {{What}} Is the Best Way to Study {{Rational Homotopy Theory}}},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/56809/what-is-the-best-way-to-study-rational-homotopy-theory/70792},
keywords = {references},
file = {/home/zack/Dropbox/Zotero/storage/89VVE99N/70792.html}
}
@misc{martinsInjectivityProjectivitySemisimplicity2013,
title = {Injectivity, {{Projectivity}} and {{Semisimplicity}}},
abstract = {In this text, I'd like to try to put three different key concepts in module theory under the same light. Let \$latex R\$ be a ring. Recall that an exact sequence of left \$latex R\$-modules \$late\ldots{}},
language = {en},
journal = {The Math of Khan},
author = {Martins, Gabriel},
month = jun,
year = {2013},
keywords = {definitions},
file = {/home/zack/Dropbox/Zotero/storage/FMUQCUJE/injectivity-projectivity-and-semisimplicity.html}
}
@misc{AlgebraicTopologyHomotopy,
title = {Algebraic Topology - {{Homotopy}} Groups of \${{S}}\^2\$},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/50377/homotopy-groups-of-s2},
file = {/home/zack/Dropbox/Zotero/storage/R8CVDP28/homotopy-groups-of-s2.html}
}
@misc{ReferenceRequestHomotopy,
title = {Reference Request - {{Homotopy}} Groups of \${{S}}\^2\$},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/64131/homotopy-groups-of-s2},
keywords = {references},
file = {/home/zack/Dropbox/Zotero/storage/9RKJBVSC/homotopy-groups-of-s2.html}
}
@misc{AlgebraicTopologyExtension,
title = {Algebraic Topology - Extension Problem of a Spectral Sequence},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/805180/extension-problem-of-a-spectral-sequence},
file = {/home/zack/Dropbox/Zotero/storage/IW5HP2H6/extension-problem-of-a-spectral-sequence.html}
}
@article{ivanovNontrivialityHomotopyGroups2015,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1506.00952},
primaryClass = {math},
title = {On Nontriviality of Homotopy Groups of Spheres},
abstract = {For \$n\textbackslash{}geq 2\$, the homotopy groups \$\textbackslash{}pi\_n(S\^2)\$ are non-zero.},
journal = {arXiv:1506.00952 [math]},
author = {Ivanov, Sergei O. and Mikhailov, Roman and Wu, Jie},
month = jun,
year = {2015},
keywords = {Mathematics - Algebraic Topology,papers},
file = {/home/zack/Dropbox/Zotero/arXiv1506.00952 [math]/2015/Ivanov et al_2015_On nontriviality of homotopy groups of spheres.pdf;/home/zack/Dropbox/Zotero/storage/K2Y96X7A/1506.html}
}
@misc{SoftQuestionDerived,
title = {Soft Question - {{Derived}} Algebraic Geometry: How to Reach Research Level Math?},
shorttitle = {Soft Question - {{Derived}} Algebraic Geometry},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/217792/derived-algebraic-geometry-how-to-reach-research-level-math/219361},
file = {/home/zack/Dropbox/Zotero/storage/ZHYDFWSE/219361.html}
}
@misc{AlgebraicTopologyComputational,
title = {At.Algebraic Topology - {{Computational}} Complexity of Computing Homotopy Groups of Spheres},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/31004/computational-complexity-of-computing-homotopy-groups-of-spheres},
file = {/home/zack/Dropbox/Zotero/storage/ZPSFUVM3/computational-complexity-of-computing-homotopy-groups-of-spheres.html}
}
@misc{NCategoryCafe,
title = {The N-{{Category Caf\'e}}},
language = {en},
howpublished = {https://golem.ph.utexas.edu/category/2013/08/what\_is\_a\_spectral\_sequence.html},
file = {/home/zack/Dropbox/Zotero/storage/WIXM9E3K/what_is_a_spectral_sequence.html}
}
@article{guozhenSurveyComputationsHomotopy,
title = {A {{Survey}} of {{Computations}} of {{Homotopy Groups}} of {{Spheres}} and {{Cobordisms}}},
language = {en},
author = {Guozhen, Wang and Zhouli, Xu},
pages = {115},
file = {/home/zack/Dropbox/Zotero/storage/BDGPMAXD/Guozhen and Zhouli - A Survey of Computations of Homotopy Groups of Sph.pdf}
}
@misc{ModernResearchAlgebraic,
title = {Modern {{Research}} in {{Algebraic Topology}}},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/918508/modern-research-in-algebraic-topology},
keywords = {research problems},
file = {/home/zack/Dropbox/Zotero/storage/AP2GBCQX/modern-research-in-algebraic-topology.html}
}
@article{hessFINITENESSSTABLEHOMOTOPY,
title = {{{FINITENESS OF THE STABLE HOMOTOPY GROUPS OF SPHERES}}},
language = {en},
author = {Hess, Daniel},
pages = {31},
file = {/home/zack/Dropbox/Zotero/storage/QK4W37TE/Hess - FINITENESS OF THE STABLE HOMOTOPY GROUPS OF SPHERE.pdf}
}
@misc{OrthogonalGroupNLab,
title = {Orthogonal Group in {{nLab}}},
howpublished = {https://ncatlab.org/nlab/show/orthogonal+group},
file = {/home/zack/Dropbox/Zotero/storage/8LEBXG53/orthogonal+group.html}
}
@book{gelfandMethodsHomologicalAlgebra,
title = {Methods of {{Homological Algebra}}},
author = {Gelfand, Sergei},
file = {/home/zack/Dropbox/Library/Sergei I. Gelfand/Methods of Homological Algebra (662)/Methods of Homological Algebra - Sergei I. Gelfand.pdf}
}
@misc{AlgebraicTopologyList,
title = {At.Algebraic Topology - {{List}} of {{Classifying Spaces}} and {{Covers}}},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/56363/list-of-classifying-spaces-and-covers},
file = {/home/zack/Dropbox/Zotero/storage/RC8RG55N/list-of-classifying-spaces-and-covers.html}
}
@misc{ClassifyingSpacesMade,
title = {Classifying {{Spaces Made Easy}}},
howpublished = {http://math.ucr.edu/home/baez/calgary/BG.html},
file = {/home/zack/Dropbox/Zotero/storage/Q4QPB8DF/BG.html}
}
@misc{TextbookRecommendationAlgebraic,
title = {Textbook Recommendation - {{Algebraic Topology Beyond}} the {{Basics}}:{{Any Texts Bridging The Gap}}?},
shorttitle = {Textbook Recommendation - {{Algebraic Topology Beyond}} the {{Basics}}},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/18041/algebraic-topology-beyond-the-basicsany-texts-bridging-the-gap},
keywords = {recommendations,textbook reference},
file = {/home/zack/Dropbox/Zotero/storage/9DFZ2284/algebraic-topology-beyond-the-basicsany-texts-bridging-the-gap.html}
}
@misc{AlgebraicTopologySplitting,
title = {At.Algebraic Topology - {{Splitting}} of the {{Universal Coefficients}} Sequence},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/27985/splitting-of-the-universal-coefficients-sequence},
file = {/home/zack/Dropbox/Zotero/storage/FSG4ZJQD/splitting-of-the-universal-coefficients-sequence.html}
}
@misc{HomotopyTheoryNotes,
title = {Homotopy {{Theory Notes}}},
howpublished = {https://www.math.upenn.edu/\textasciitilde{}siegelch/Notes/at1.pdf},
file = {/home/zack/Dropbox/Zotero/storage/CGE9QB9D/at1.pdf}
}
@misc{SpectralSequencesApplication,
title = {Spectral {{Sequences}} and an {{Application}}},
howpublished = {https://people.math.osu.edu/flicker.1/orlich.pdf},
file = {/home/zack/Dropbox/Zotero/storage/KWZASC2G/orlich.pdf}
}
@misc{MotivationWhyAre,
title = {Motivation - {{Why}} Are Spectral Sequences so Ubiquitous?},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/8052/why-are-spectral-sequences-so-ubiquitous},
keywords = {motivation,expository},
file = {/home/zack/Dropbox/Zotero/storage/7RM8QXJF/why-are-spectral-sequences-so-ubiquitous.html}
}
@misc{AlgebraicTopologyWhya,
title = {At.Algebraic Topology - {{Why}} Is It Difficult to Obtain the next Differential in a Spectral Sequence?},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/280900/why-is-it-difficult-to-obtain-the-next-differential-in-a-spectral-sequence/280933},
keywords = {motivation},
file = {/home/zack/Dropbox/Zotero/storage/HXALFLBM/280933.html}
}
@article{romeroSpectralSequencesComputing,
title = {Spectral Sequences for Computing Persistent Homology},
language = {en},
journal = {d r},
author = {Romero, Ana},
pages = {146},
file = {/home/zack/Dropbox/Zotero/storage/VHE7TNR3/Romero - Spectral sequences for computing persistent homolo.pdf}
}
@article{EilenbergMacLaneSpace2019,
title = {Eilenberg\textendash{{MacLane}} Space},
copyright = {Creative Commons Attribution-ShareAlike License},
abstract = {In mathematics, and algebraic topology in particular, an Eilenberg\textendash{}MacLane space is a topological space with a single nontrivial homotopy group. As such, an Eilenberg\textendash{}MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.
Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg\textendash{}MacLane space of type K(G, n), if it has n-th homotopy group {$\pi$}n(X) isomorphic to G and all other homotopy groups trivial. If n {$>$} 1 then G must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence. By abuse of language, any such space is often called just K(G, n).},
language = {en},
journal = {Wikipedia},
month = jan,
year = {2019},
file = {/home/zack/Dropbox/Zotero/storage/9MC767DW/index.html},
note = {Page Version ID: 876701063}
}
@misc{TopologyOpenProblem,
title = {Topology | {{Open Problem Garden}}},
howpublished = {http://www.openproblemgarden.org/topology?sort=asc\&order=Imp.\%C2\%B9},
keywords = {research problems},
file = {/home/zack/Dropbox/Zotero/storage/3TZLETDV/topology.html}
}
@misc{DifferentialTopologyVisualization,
title = {Differential Topology - {{Visualization}} of {{Lens Spaces}}},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/1186778/visualization-of-lens-spaces},
file = {/home/zack/Dropbox/Zotero/storage/PF32CV2G/visualization-of-lens-spaces.html}
}
@misc{AlgebraicTopologyCo,
title = {At.Algebraic Topology - ({{Co}})Homology of the {{Eilenberg}}-{{MacLane}} Spaces {{K}}({{G}},N)},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/24754/cohomology-of-the-eilenberg-maclane-spaces-kg-n},
file = {/home/zack/Dropbox/Zotero/storage/GXM7C55N/cohomology-of-the-eilenberg-maclane-spaces-kg-n.html}
}
@misc{AlgebraicTopologyGood,
title = {At.Algebraic Topology - {{Good}} Reference for Homology of \${{K}}(\textbackslash{}mathbb\{\vphantom\}{{Z}}\vphantom\{\}, 2n)\$?},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/27367/good-reference-for-homology-of-k-mathbbz-2n},
file = {/home/zack/Dropbox/Zotero/storage/YNPEF6LY/good-reference-for-homology-of-k-mathbbz-2n.html}
}
@article{VectorBundle2019,
title = {Vector Bundle},
copyright = {Creative Commons Attribution-ShareAlike License},
abstract = {In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x) = V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X \texttimes{} V over X. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles: for example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if and only if its tangent bundle is trivial.
Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.},
language = {en},
journal = {Wikipedia},
month = jan,
year = {2019},
file = {/home/zack/Dropbox/Zotero/storage/YET4FI2Y/index.html},
note = {Page Version ID: 876701284}
}
@misc{TopicsGeometricTopology,
title = {Topics in {{Geometric Topology}} (18.937)},
howpublished = {http://www.math.harvard.edu/\textasciitilde{}lurie/937.html},
keywords = {advanced},
file = {/home/zack/Dropbox/Zotero/storage/9J225JVS/937.html}
}
@misc{AlgebraicTopologyWhata,
title = {Algebraic Topology - {{What}} Are the Best Known Results for the Stable Homotopy Groups of Spheres? - {{Mathematics Stack Exchange}}},
howpublished = {https://math.stackexchange.com/questions/85365/what-are-the-best-known-results-for-the-stable-homotopy-groups-of-spheres},
file = {/home/zack/Dropbox/Zotero/storage/ATJ6XBNM/what-are-the-best-known-results-for-the-stable-homotopy-groups-of-spheres.html}
}
@article{HomotopyCategoryChain2018,
title = {Homotopy Category of Chain Complexes},
copyright = {Creative Commons Attribution-ShareAlike License},
abstract = {In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the derived category D(A) of A when A is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) makes isomorphisms of any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, K(A) is more understandable than D(A).},
language = {en},
journal = {Wikipedia},
month = dec,
year = {2018},
file = {/home/zack/Dropbox/Zotero/storage/54SV3P82/index.html},
note = {Page Version ID: 875445293}
}
@misc{llcListAlgebraicTopology,
title = {"{{List}} of Algebraic Topology Topics" on {{Revolvy}}.Com},
abstract = {List of algebraic topology topics This is a list of algebraic topology topics, by Wikipedia page. See also: Glossary of algebraic topology topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Homology (mathematics) Simplex Simplicial complex Polytope Triangulation Barycentric subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial category Chain (algebraic topolo},
language = {en},
howpublished = {https://www.revolvy.com/page/List-of-algebraic-topology-topics?smv=2827599},
author = {LLC, Revolvy},
file = {/home/zack/Dropbox/Zotero/storage/SNGX8C6R/List-of-algebraic-topology-topics.html}
}
@misc{BigListHonest,
title = {Big List - {{Honest}} Application of Category Theory},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/1208375/honest-application-of-category-theory},
file = {/home/zack/Dropbox/Zotero/storage/APFMGQ3Z/honest-application-of-category-theory.html}
}
@misc{AlgebraicTopologyRational,
title = {Algebraic Topology - {{Rational}} Homotopy Groups of Spheres},
journal = {Mathematics Stack Exchange},
howpublished = {https://math.stackexchange.com/questions/2086300/rational-homotopy-groups-of-spheres},
file = {/home/zack/Dropbox/Zotero/storage/CHZ26FRQ/rational-homotopy-groups-of-spheres.html}
}
@book{selickIntroductionHomotopyTheory,
title = {Introduction to {{Homotopy Theory}}},
language = {en},
author = {Selick, Paul},
file = {/home/zack/Dropbox/Library/Paul Selick/Introduction to Homotopy Theory (663)/Introduction to Homotopy Theory - Paul Selick.pdf}
}
@article{allOhioStateUniversity,
title = {Ohio {{State University Department}} of {{Mathematics Algebra Qualifier Exam Solutions}}},
language = {en},
author = {All, Timothy and Belfanti, Michael},
keywords = {quals,solutions},
pages = {90},
file = {/home/zack/Dropbox/Zotero/storage/JUSQ72LA/All and Belfanti - Ohio State University Department of Mathematics Al.pdf}
}
@misc{IndexIacoleyHw,
title = {Index of /\textasciitilde{}iacoley/Hw/Alghwfall},
howpublished = {https://www.math.ucla.edu/\textasciitilde{}iacoley/hw/alghwfall/},
keywords = {solutions},
file = {/home/zack/Dropbox/Zotero/storage/VRFLRB45/alghwfall.html}
}
@misc{IndexIacoleyHwa,
title = {Index of /\textasciitilde{}iacoley/Hw/Alghwwinter},
howpublished = {https://www.math.ucla.edu/\textasciitilde{}iacoley/hw/alghwwinter/},
file = {/home/zack/Dropbox/Zotero/storage/Z9MXESV3/alghwwinter.html}
}
@article{coleyUCLAAlgebraQualifying,
title = {{{UCLA Algebra Qualifying Exam Solutions}}},
language = {en},
author = {Coley, Ian},
keywords = {solutions},
pages = {50},
file = {/home/zack/Dropbox/Zotero/storage/72B27P2C/Coley - UCLA Algebra Qualifying Exam Solutions.pdf}
}
@misc{IndexIacoleyHwb,
title = {Index of /\textasciitilde{}iacoley/Hw/Diffhwfall},
howpublished = {https://www.math.ucla.edu/\textasciitilde{}iacoley/hw/diffhwfall/},
keywords = {solutions},
file = {/home/zack/Dropbox/Zotero/storage/PZTXET3T/diffhwfall.html}
}
@misc{IndexIacoleyHwc,
title = {Index of /\textasciitilde{}iacoley/Hw/Diffhwwinter},
howpublished = {https://www.math.ucla.edu/\textasciitilde{}iacoley/hw/diffhwwinter/},
keywords = {solutions},
file = {/home/zack/Dropbox/Zotero/storage/MPIWP72V/diffhwwinter.html}
}
@article{coleyUCLAGeometryTopology,
title = {{{UCLA Geometry}}/{{Topology Qualifying Exam Solutions}}},
language = {en},
author = {Coley, Ian},
keywords = {solutions},
pages = {77},
file = {/home/zack/Dropbox/Zotero/storage/FBSKBZHY/Coley - UCLA GeometryTopology Qualifying Exam Solutions.pdf}
}
@misc{HatcherAlgebraicTopology,
title = {Hatcher's {{Algebraic Topology Solutions}} | Riemannian Hunger},
howpublished = {https://riemannianhunger.wordpress.com/solutions-to-algebraic-topology-by-allen-hatcher/},
keywords = {solutions},
file = {/home/zack/Dropbox/Zotero/storage/HBCAY98J/solutions-to-algebraic-topology-by-allen-hatcher.html}
}
@misc{HomotopicalAlgebraLecture,
title = {Homotopical {{Algebra Lecture Notes}}},
howpublished = {http://pi.math.cornell.edu/\textasciitilde{}apatotski/7400-notes-2015.pdf},
file = {/home/zack/Dropbox/Zotero/storage/GANXXDZ6/7400-notes-2015.pdf}
}
@misc{SpectralSequences,
title = {Spectral {{Sequences}}},
howpublished = {https://neil-strickland.staff.shef.ac.uk/courses/bestiary/ss.pdf},
file = {/home/zack/Dropbox/Zotero/storage/AWJ2ZFXH/ss.pdf}
}
@article{filakovskyComputingSimplicialRepresentatives2017,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1706.00380},
primaryClass = {cs, math},
title = {Computing Simplicial Representatives of Homotopy Group Elements},
abstract = {A central problem of algebraic topology is to understand the homotopy groups \$\textbackslash{}pi\_d(X)\$ of a topological space \$X\$. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group \$\textbackslash{}pi\_1(X)\$ of a given finite simplicial complex \$X\$ is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex \$X\$ that is simply connected (i.e., with \$\textbackslash{}pi\_1(X)\$ trivial), compute the higher homotopy group \$\textbackslash{}pi\_d(X)\$ for any given \$d\textbackslash{}geq 2\$. \%The first such algorithm was given by Brown, and more recently, \textbackslash{}v\{C\}adek et al. However, these algorithms come with a caveat: They compute the isomorphism type of \$\textbackslash{}pi\_d(X)\$, \$d\textbackslash{}geq 2\$ as an \textbackslash{}emph\{abstract\} finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of \$\textbackslash{}pi\_d(X)\$. Converting elements of this abstract group into explicit geometric maps from the \$d\$-dimensional sphere \$S\^d\$ to \$X\$ has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a\textasciitilde{}simply connected space \$X\$, computes \$\textbackslash{}pi\_d(X)\$ and represents its elements as simplicial maps from a suitable triangulation of the \$d\$-sphere \$S\^d\$ to \$X\$. For fixed \$d\$, the algorithm runs in time exponential in \$size(X)\$, the number of simplices of \$X\$. Moreover, we prove that this is optimal: For every fixed \$d\textbackslash{}geq 2\$, we construct a family of simply connected spaces \$X\$ such that for any simplicial map representing a generator of \$\textbackslash{}pi\_d(X)\$, the size of the triangulation of \$S\^d\$ on which the map is defined, is exponential in \$size(X)\$.},
journal = {arXiv:1706.00380 [cs, math]},
author = {Filakovsky, Marek and Franek, Peter and Wagner, Uli and Zhechev, Stephan},
month = jun,
year = {2017},
keywords = {Mathematics - Algebraic Topology,Computer Science - Computational Geometry},
file = {/home/zack/Dropbox/Zotero/arXiv1706.00380 [cs, math]/2017/Filakovsky et al_2017_Computing simplicial representatives of homotopy group elements.pdf;/home/zack/Dropbox/Zotero/storage/Y5CHMHWP/1706.html}
}
@misc{ModestAdviceGraduate,
title = {Some {{Modest Advice}} for {{Graduate Students}} | {{Stearns Lab}}},
howpublished = {https://stearnslab.yale.edu/some-modest-advice-graduate-students},
keywords = {advice,graduate school},
file = {/home/zack/Dropbox/Zotero/storage/JV6QSHHZ/some-modest-advice-graduate-students.html}
}
@article{ericksonONEDIMENSIONALFORMALGROUPS,
title = {{{ONE}}-{{DIMENSIONAL FORMAL GROUPS}}},
language = {en},
author = {Erickson, Carl},
pages = {8},
file = {/home/zack/Dropbox/Zotero/storage/7BPK8D5X/Erickson - ONE-DIMENSIONAL FORMAL GROUPS.pdf}
}
@misc{ChromaticHomotopyTheory,
title = {Chromatic {{Homotopy Theory}} (252x)},
howpublished = {http://www.math.harvard.edu/\textasciitilde{}lurie/252x.html},
file = {/home/zack/Dropbox/Zotero/storage/CNMLQZ3M/252x.html}
}
@article{beardsleyUserGuideRelative2017,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1709.07543},
primaryClass = {math},
title = {A {{User}}'s {{Guide}}: {{Relative Thom Spectra}} via {{Operadic Kan Extensions}}},
shorttitle = {A {{User}}'s {{Guide}}},
abstract = {This is an expository paper about the paper Relative Thom Spectra via Operadic Kan Extensions.},
journal = {arXiv:1709.07543 [math]},
author = {Beardsley, Jonathan},
month = sep,
year = {2017},
keywords = {Mathematics - Algebraic Topology,55-02; 55P43; 00A35},
file = {/home/zack/Dropbox/Zotero/arXiv1709.07543 [math]/2017/Beardsley_2017_A User's Guide.pdf;/home/zack/Dropbox/Zotero/storage/RY6U76SY/1709.html}
}
@misc{AlgebraicTopologyWhatb,
title = {At.Algebraic Topology - {{What}} Is 'formal' ?},
journal = {MathOverflow},
howpublished = {https://mathoverflow.net/questions/13005/what-is-formal/13063},
keywords = {formal,rational homotopy},
file = {/home/zack/Dropbox/Zotero/storage/2E4FF6SK/13063.html}
}
@article{wilsonHOPFRINGSALGEBRAIC,
title = {{{HOPF RINGS IN ALGEBRAIC TOPOLOGY}}},
abstract = {These are colloquium style lecture notes about Hopf rings in algebraic topology. They were designed for use by non-topologists and graduate students but have been found helpful for those who want to start learning about Hopf rings. They are not ``up to date,'' nor are they intended to be, but instead they are intended to be introductory in nature. Although these are ``old'' notes, Hopf rings are thriving and these notes give a relatively painless introduction which should prepare the reader to approach the current literature.},
language = {en},
author = {Wilson, W Stephen},
pages = {20},
file = {/home/zack/Dropbox/Zotero/storage/5P7Z8D7P/Wilson - HOPF RINGS IN ALGEBRAIC TOPOLOGY.pdf}
}
@article{kottkeBUNDLESCLASSIFYINGSPACES,
title = {{{BUNDLES}}, {{CLASSIFYING SPACES AND CHARACTERISTIC CLASSES}}},
language = {en},
author = {Kottke, Chris},
pages = {21},
file = {/home/zack/Dropbox/Zotero/storage/T3MM2TSR/Kottke - BUNDLES, CLASSIFYING SPACES AND CHARACTERISTIC CLA.pdf}
}
@article{hopkinsLecturesLubinTateSpaces,
title = {Lectures on {{Lubin}}-{{Tate}} Spaces {{Arizona Winter School March}} 2019 ({{Incomplete}} Draft)},
language = {en},
author = {Hopkins, M J},
pages = {46},
file = {/home/zack/Dropbox/Zotero/storage/PD7BJDTM/Hopkins - Lectures on Lubin-Tate spaces Arizona Winter Schoo.pdf}
}
@article{levineAlgebraicTheoryWelschinger2018,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1808.02238},
primaryClass = {math},
title = {Toward an Algebraic Theory of {{Welschinger}} Invariants},
abstract = {Let \$S\$ be a smooth del Pezzo surface over a field \$k\$ of characteristic \$\textbackslash{}neq 2, 3\$. We define an invariant in the Grothendieck-Witt ring \$GW(k)\$ for "counting" rational curves in a curve class \$D\$ of fixed positive degree (with respect to the anti-canonical bundle \$-K\_S\$) and containing a collection of distinct closed points \$\textbackslash{}mathfrak\{p\}=\textbackslash{}sum\_ip\_i\$ of total degree \$r:=-D\textbackslash{}cdot K\_S-1\$ on \$S\$. This recovers Welschinger's invariant in case \$k=\textbackslash{}mathbb\{R\}\$ by applying the signature map. The main result is that this quadratic invariant depends only on the \$\textbackslash{}mathbb\{A\}\^1\$-connected component containing \$\textbackslash{}mathfrak\{p\}\$ in \$Sym\^r(S)\^0(k)\$, where \$Sym\^r(S)\^0\$ is the open subscheme of \$Sym\^r(S)\$ parametrizing geometrically reduced 0-cycles.},
journal = {arXiv:1808.02238 [math]},
author = {Levine, Marc},
month = aug,
year = {2018},
keywords = {Mathematics - Algebraic Geometry,14F42; 55N20; 55N35,gromov,invariants,witten},
file = {/home/zack/Dropbox/Zotero/arXiv1808.02238 [math]/2018/Levine_2018_Toward an algebraic theory of Welschinger invariants.pdf;/home/zack/Dropbox/Zotero/storage/9HS4A9C2/1808.html}
}
@article{brownCOHOMOLOGYTHEORIES,
title = {{{COHOMOLOGY THEORIES}}},
language = {en},
author = {Brown, H},
pages = {20},
file = {/home/zack/Dropbox/Zotero/storage/KU9YP8YP/Brown - COHOMOLOGY THEORIES.pdf}
}
@article{milnorSteenrodAlgebraIts,
title = {The {{Steenrod Algebra}} and {{Its Dual}}},
language = {en},
author = {Milnor, John},
pages = {23},
file = {/home/zack/Dropbox/Zotero/storage/QMH2LFSW/Milnor - The Steenrod Algebra and Its Dual.pdf}
}
@article{milnorStructureHopfAlgebras1965,
title = {On the {{Structure}} of {{Hopf Algebras}}},
volume = {81},
issn = {0003486X},
doi = {10.2307/1970615},
language = {en},
number = {2},
journal = {The Annals of Mathematics},
author = {Milnor, John W. and Moore, John C.},
month = mar,
year = {1965},
pages = {211},
file = {/home/zack/Dropbox/Zotero/storage/I3G5ZJVP/Milnor and Moore - 1965 - On the Structure of Hopf Algebras.pdf}
}
@article{sternPoorManAttempt,
title = {A {{Poor Man}}'s {{Attempt}} to {{Learn Stable Homotopy Theory}}},
language = {en},
author = {Stern, Reuben},
pages = {12},
file = {/home/zack/Dropbox/Zotero/storage/H5Q5B7IA/Stern - A Poor Man’s Attempt to Learn Stable Homotopy Theo.pdf}
}
@article{holmberg-perouxSTABLEHOMOTOPYTHEORY,
title = {{{TO STABLE HOMOTOPY THEORY}}},
language = {en},
author = {{Holmberg-P\'eroux}, Maximilien and {Hess-Bellwald}, Kathryn},
pages = {93},
file = {/home/zack/Dropbox/Zotero/storage/P27ZPJD5/Holmberg-Péroux and Hess-Bellwald - TO STABLE HOMOTOPY THEORY.pdf}
}
@article{kupersAdvancedAlgebraicTopology,
title = {Advanced Algebraic Topology, Version {{April}} 28, 2018},
language = {en},
author = {Kupers, Alexander},
keywords = {reading list,textbook},
pages = {314},
file = {/home/zack/Dropbox/Zotero/storage/SNIWCT5J/Kupers - Advanced algebraic topology, version April 28, 201.pdf}
}
@article{kupersLecturesDiffeomorphismGroups,
title = {Lectures on Diffeomorphism Groups of Manifolds, Version {{February}} 22, 2019},
language = {en},
author = {Kupers, Alexander},
keywords = {to read,low dimensional},
pages = {329},
file = {/home/zack/Dropbox/Zotero/storage/MWDWU3QT/Kupers - Lectures on diffeomorphism groups of manifolds, ve.pdf}
}
@misc{8803StableHomotopy,
title = {8803 {{Stable Homotopy Theory}}},
howpublished = {http://people.math.gatech.edu/\textasciitilde{}kwickelgren3/8803\_Stable/},
keywords = {stable homotopy},
file = {/home/zack/Dropbox/Zotero/storage/SPFMZSY6/8803_Stable.html}
}
@article{quillenFORMALGROUPLAWS,
title = {{{ON THE FORMAL GROUP LAWS OF UNORIENTED AND COMPLEX COBORDISM THEORY}}},
language = {en},
author = {Quillen, Daniel},
pages = {6},
file = {/home/zack/Dropbox/Zotero/storage/SYUV8XAC/Quillen - ON THE FORMAL GROUP LAWS OF UNORIENTED AND COMPLEX.pdf}
}
@article{milnorCobordismRingComplex1960,
title = {On the {{Cobordism Ring $\Omega$}}* and a {{Complex Analogue}}, {{Part I}}},
volume = {82},
issn = {0002-9327},
doi = {10.2307/2372970},
number = {3},
journal = {American Journal of Mathematics},
author = {Milnor, J.},
year = {1960},
keywords = {\#nosource},
pages = {505-521}
}
@article{milnorManifoldsHomeomorphic7Sphere1956,
title = {On {{Manifolds Homeomorphic}} to the 7-{{Sphere}}},
volume = {64},
issn = {0003486X},
doi = {10.2307/1969983},
language = {en},
number = {2},
journal = {The Annals of Mathematics},
author = {Milnor, John},
month = sep,
year = {1956},
pages = {399},
file = {/home/zack/Dropbox/Zotero/storage/G8LPJIFD/Milnor - 1956 - On Manifolds Homeomorphic to the 7-Sphere.pdf}
}
@incollection{quillenHigherAlgebraicKtheory1973,
address = {Berlin, Heidelberg},
title = {Higher Algebraic {{K}}-Theory: {{I}}},
volume = {341},
isbn = {978-3-540-06434-3 978-3-540-37767-2},
shorttitle = {Higher Algebraic {{K}}-Theory},
language = {en},
booktitle = {Higher {{K}}-{{Theories}}},
publisher = {{Springer Berlin Heidelberg}},
author = {Quillen, Daniel},
editor = {Bass, H.},
year = {1973},
pages = {85-147},
file = {/home/zack/Dropbox/Zotero/storage/LKVYIXRL/Quillen - 1973 - Higher algebraic K-theory I.pdf},
doi = {10.1007/BFb0067053}
}
@article{milnorGROUPSHOMOTOPYSPHERES,
title = {{{GROUPS OF HOMOTOPY SPHERES}}: {{I}}},
language = {en},
author = {Milnor, Michela Kervaireand Johnw},
pages = {35},
file = {/home/zack/Dropbox/Zotero/storage/MQY8NG6T/Milnor - GROUPS OF HOMOTOPY SPHERES I.pdf}
}
@article{milnorGeometricRealizationSemiSimplicial1957,
title = {The {{Geometric Realization}} of a {{Semi}}-{{Simplicial Complex}}},
volume = {65},
issn = {0003486X},
doi = {10.2307/1969967},
language = {en},
number = {2},
journal = {The Annals of Mathematics},
author = {Milnor, John},
month = mar,
year = {1957},
pages = {357},
file = {/home/zack/Dropbox/Zotero/storage/X5E36DWH/Milnor - 1957 - The Geometric Realization of a Semi-Simplicial Com.pdf}
}
@article{doldHomologySymmetricProducts1958,
title = {Homology of {{Symmetric Products}} and {{Other Functors}} of {{Complexes}}},
volume = {68},
issn = {0003486X},
doi = {10.2307/1970043},
language = {en},
number = {1},
journal = {The Annals of Mathematics},
author = {Dold, Albrecht},
month = jul,
year = {1958},
pages = {54},
file = {/home/zack/Dropbox/Zotero/storage/F5LALJEG/Dold - 1958 - Homology of Symmetric Products and Other Functors .pdf}
}
@article{segalEquivariantKtheory,
title = {Equivariant {{K}}-Theory},
language = {en},
author = {Segal, Graeme},
pages = {24},
file = {/home/zack/Dropbox/Zotero/storage/H8ZZBNXJ/Segal - Equivariant K-theory.pdf}
}
@misc{TopicsGeometricTopologya,
title = {Topics in {{Geometric Topology}} (18.937)},
howpublished = {http://www.math.harvard.edu/\textasciitilde{}lurie/937.html},
file = {/home/zack/Dropbox/Zotero/storage/HWCBQCBX/937.html}
}
@book{zotero-3956,
keywords = {\#nosource}
}
@article{duggerMULTIPLICATIVESTRUCTURESHOMOTOPY,
title = {{{MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II}}},
language = {en},
author = {Dugger, Daniel},
pages = {19},
file = {/home/zack/Dropbox/Zotero/storage/BTAR87VH/Dugger - MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQ.pdf}
}
@article{conradSPLITTINGSHORTEXACT,
title = {{{SPLITTING OF SHORT EXACT SEQUENCES FOR GROUPS}}},
language = {en},
author = {Conrad, Keith},
pages = {9},
file = {/home/zack/Dropbox/Zotero/storage/PEL8I59T/Conrad - SPLITTING OF SHORT EXACT SEQUENCES FOR GROUPS.pdf}
}
@article{voSpectralSequence,
title = {{{SpectralSequence}}},
language = {en},
author = {Vo, Huan},
pages = {16},
file = {/home/zack/Dropbox/Zotero/storage/CV4ZUKXW/Vo - SpectralSequence.pdf}
}
@article{claderSpectralSequences,
title = {{{SpectralSequences}}},
language = {en},
author = {Clader, Emily},
pages = {7},
file = {/home/zack/Dropbox/Zotero/storage/DZ9KQNII/Clader - SpectralSequences.pdf}
}
@article{robertsThoughtsDoingPhD,
title = {Some Thoughts on Doing a {{PhD}} in Topology/Geometry},
language = {en},
author = {Roberts, Justin},
pages = {4},
file = {/home/zack/Dropbox/Zotero/storage/PFGPN8TW/Roberts - Some thoughts on doing a PhD in topologygeometry.pdf}
}
@article{brownSerreSpectralSequences,
title = {Serre {{Spectral Sequences}} and {{Applications}}},
abstract = {Spectral sequences are a key theoretical and computational tool in algebraic topology. This paper is a tutorial essay, intended for students and researchers as a quick introduction, providing a few nice examples, applications, and a short list of references for further inquiry.},
language = {en},
author = {Brown, David},
pages = {18},
file = {/home/zack/Dropbox/Zotero/storage/XE9HM2SP/Brown - Serre Spectral Sequences and Applications.pdf}
}
@article{duggerMULTIPLICATIVESTRUCTURESHOMOTOPYa,
title = {{{MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES}}, {{PART I}}},
language = {en},
author = {Dugger, Daniel},
pages = {30},
file = {/home/zack/Dropbox/Zotero/storage/YRABTJC9/Dugger - MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQ.pdf}
}
@article{duggerNOTESMILNORCONJECTURES,
title = {{{NOTES ON THE MILNOR CONJECTURES}}},
language = {en},
author = {Dugger, Daniel},
pages = {30},
file = {/home/zack/Dropbox/Zotero/storage/26TSBAL7/Dugger - NOTES ON THE MILNOR CONJECTURES.pdf}
}
@article{hillNonexistenceElementsKervaire2009,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {0908.3724},
primaryClass = {math},
title = {On the Non-Existence of Elements of {{Kervaire}} Invariant One},
abstract = {We show that the Kervaire invariant one elements \texttheta{}j {$\in$} {$\pi$}2j+1-2S0 exist only for j {$\leq$} 6. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.},
language = {en},
journal = {arXiv:0908.3724 [math]},
author = {Hill, Michael A. and Hopkins, Michael J. and Ravenel, Douglas C.},
month = aug,
year = {2009},
keywords = {Mathematics - Algebraic Topology,55Q45; 55Q91; 55P91; 57R55; 57R60; 57R77; 57R85,Mathematics - Geometric Topology},
file = {/home/zack/Dropbox/Zotero/storage/U5PBDZH8/Hill et al. - 2009 - On the non-existence of elements of Kervaire invar.pdf}
}
@article{boothIntroductionModelCategories,
title = {An {{Introduction}} to {{Model Categories}}},
language = {en},
author = {Booth, Matt},
pages = {14},
file = {/home/zack/Dropbox/Zotero/storage/JFVLQFUM/Booth - An Introduction to Model Categories.pdf}
}
@article{banerjeeHopfInvariantOne,
title = {The {{Hopf}} Invariant One Problem},
abstract = {This paper will discuss the Adams-Atiyah solution to the Hopf invariant problem. We will first define and prove some identities concerning the Adams operations. Then we will look at the proof of the ordinary Hopf invariant one problem. Finally we will look at some results concerning the pth cup power mod p and the mod p version of the Hopf invariant problem.},
language = {en},
author = {Banerjee, Ishan},
pages = {12},
file = {/home/zack/Dropbox/Zotero/storage/CI4ILYKL/Banerjee - The Hopf invariant one problem.pdf}
}
@article{hatcherListRecommendedBooks,
title = {A {{List}} of {{Recommended Books}} in {{Topology}}},
language = {en},
author = {Hatcher, Allen},
pages = {9},
file = {/home/zack/Dropbox/Zotero/storage/575R6QUR/Hatcher - A List of Recommended Books in Topology.pdf}
}
@article{murrayLineBundlesHonours,
title = {Line {{Bundles}}. {{Honours}} 1996},
language = {en},
author = {Murray, Michael},
pages = {13},
file = {/home/zack/Dropbox/Zotero/storage/IGULWNHS/Murray - Line Bundles. Honours 1996.pdf}
}
@article{EilenbergMacLaneSpaces,
title = {Eilenberg {{MacLane Spaces}}},
abstract = {This paper discusses basic properties of Eilenberg-MacLane spaces K(G, n), their cohomology groups and some classic applications. We construct K(G, n) using both CW complexes and classifying spaces. Two proofs are given to the basic fact that cohomology of a CW complex X has 1-1 correspondence with homotopy classes of maps from X into Eilenberg-MacLane spaces: a barehanded proof (original) and a categorical proof using loop spaces. We use it to demonstrate the connection between cohomology operations and cohomology groups of K(G, n)'s. Finally we use the technique of spectral sequence to compute the cohomology of some classes of Eilenberg-MacLane spaces, and apply it to the calculation {$\pi$}5(S3).},
language = {en},
pages = {16},
file = {/home/zack/Dropbox/Zotero/storage/YUQ8P3XL/Eilenberg MacLane Spaces.pdf}
}
@article{duggerZARISKINISNEVICHDESCENT,
title = {{{THE ZARISKI AND NISNEVICH DESCENT THEOREMS}}},
language = {en},
author = {Dugger, Daniel},
pages = {7},
file = {/home/zack/Dropbox/Zotero/storage/ZN9LFUXZ/Dugger - THE ZARISKI AND NISNEVICH DESCENT THEOREMS.pdf}
}
@article{hutchingsCupProductIntersections,
title = {Cup Product and Intersections},
abstract = {This is a handout for an algebraic topology course. The goal is to explain a geometric interpretation of the cup product. Namely, if X is a closed oriented smooth manifold, if A and B are oriented submanifolds of X, and if A and B intersect transversely, then the Poincar\textasciiacute{}e dual of A {$\cap$} B is the cup product of the Poincar\textasciiacute{}e duals of A and B. As an application, we prove the Lefschetz fixed point formula on a manifold. As a byproduct of the proof, we explain why the Euler class of a smooth oriented vector bundle is Poincar\textasciiacute{}e dual to the zero set of a generic section.},
language = {en},
author = {Hutchings, Michael},
pages = {13},
file = {/home/zack/Dropbox/Zotero/storage/RD3E4IIZ/Hutchings - Cup product and intersections.pdf}
}
@article{mitchellNotesPrincipalBundles,
title = {Notes on Principal Bundles and Classifying Spaces},
language = {en},
author = {Mitchell, Stephen A},
pages = {25},
file = {/home/zack/Dropbox/Zotero/storage/BJWWS5H7/Mitchell - Notes on principal bundles and classifying spaces.pdf}
}
@article{kottkeBUNDLESCLASSIFYINGSPACESa,
title = {{{BUNDLES}}, {{CLASSIFYING SPACES AND CHARACTERISTIC CLASSES}}},
language = {en},
author = {Kottke, Chris},
pages = {21},
file = {/home/zack/Dropbox/Zotero/storage/VJA4N4UR/Kottke - BUNDLES, CLASSIFYING SPACES AND CHARACTERISTIC CLA.pdf}
}
@article{banerjeeHopfInvariantOnea,
title = {The {{Hopf}} Invariant One Problem},
abstract = {This paper will discuss the Adams-Atiyah solution to the Hopf invariant problem. We will first define and prove some identities concerning the Adams operations. Then we will look at the proof of the ordinary Hopf invariant one problem. Finally we will look at some results concerning the pth cup power mod p and the mod p version of the Hopf invariant problem.},
language = {en},
author = {Banerjee, Ishan},
pages = {12},
file = {/home/zack/Dropbox/Zotero/storage/HSW6M82X/Banerjee - The Hopf invariant one problem.pdf}
}
@article{wallFinitenessConditionsCWComplexes1965,
title = {Finiteness {{Conditions}} for {{CW}}-{{Complexes}}},
volume = {81},
issn = {0003486X},
doi = {10.2307/1970382},
language = {en},
number = {1},
journal = {The Annals of Mathematics},
author = {Wall, C. T. C.},
month = jan,
year = {1965},
pages = {56},
file = {/home/zack/Dropbox/Zotero/storage/ZTI7S82Z/Wall - 1965 - Finiteness Conditions for CW-Complexes.pdf}
}
@article{hatcherShortExpositionMadsenWeiss2011,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1103.5223},
primaryClass = {math},
title = {A Short Exposition of the {{Madsen}}-{{Weiss}} Theorem},
abstract = {This is an exposition of a proof of the Madsen-Weiss Theorem, which asserts that the homology of mapping class groups of surfaces, in a stable dimension range, is isomorphic to the homology of a certain infinite loopspace that arises naturally when one applies the "scanning method". The proof given here utilizes simplifications introduced by Galatius and Randal-Williams.},
language = {en},
journal = {arXiv:1103.5223 [math]},
author = {Hatcher, Allen},
month = mar,
year = {2011},
keywords = {Mathematics - Algebraic Topology,Mathematics - Geometric Topology},
file = {/home/zack/Dropbox/Zotero/storage/KWLS95B7/Hatcher - 2011 - A short exposition of the Madsen-Weiss theorem.pdf}
}
@article{abouzaidSheaftheoreticModelSL2017,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1708.00289},
primaryClass = {math},
title = {A Sheaf-Theoretic Model for {{SL}}(2,{{C}}) {{Floer}} Homology},
abstract = {Given a Heegaard splitting of a three-manifold Y , we consider the SL(2, C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an intersection a perverse sheaf of vanishing cycles. We prove that in our setting, the perverse sheaf is an invariant of Y , i.e., it is independent of the Heegaard splitting. The hypercohomology of the perverse sheaf can be viewed as a model for (the dual of) SL(2, C) instanton Floer homology. We also present a framed version of this construction, which takes into account reducible representations. We give explicit computations for lens spaces and Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology.},
language = {en},
journal = {arXiv:1708.00289 [math]},
author = {Abouzaid, Mohammed and Manolescu, Ciprian},
month = aug,
year = {2017},
keywords = {Mathematics - Algebraic Geometry,Mathematics - Symplectic Geometry,Mathematics - Geometric Topology,57M27 (Primary); 53D40; 57R58 (Secondary)},
file = {/home/zack/Dropbox/Zotero/storage/ZJ92DCVE/Abouzaid and Manolescu - 2017 - A sheaf-theoretic model for SL(2,C) Floer homology.pdf}
}
@article{garrettHilbertSchmidtOperatorsNuclear2014a,
title = {Hilbert-{{Schmidt}} Operators, Nuclear Spaces, Kernel Theorem {{I}}},
language = {en},
author = {Garrett, Paul},
year = {2014},
pages = {11},
file = {/home/zack/Dropbox/Zotero/storage/PTHSYNM4/Garrett - 2014 - Hilbert-Schmidt operators, nuclear spaces, kernel .pdf}
}
@book{rudyakThomSpectraOrientability1998,
address = {Berlin ; New York},
series = {Springer Monographs in Mathematics},
title = {On {{Thom}} Spectra, Orientability, and Cobordism},
isbn = {978-3-540-62043-3},
lccn = {QA613.66 .R83 1998},
language = {en},
publisher = {{Springer}},
author = {Rudyak, Yuli B.},
year = {1998},
keywords = {Cobordism theory,Homology theory,Topological manifolds},
file = {/home/zack/Dropbox/Library/Budyak/On Thom Spectra, Orientability, and Cobordism (694)/Rudyak - 1998 - On Thom spectra, orientability, and cobordism.pdf}
}
@misc{ModuliPdf,
title = {Moduli.Pdf},
file = {/home/zack/Dropbox/Zotero/storage/PB3EKC4K/moduli.pdf}
}
@article{mayConciseCourseAlgebraic,
title = {A {{Concise Course}} in {{Algebraic Topology}}},
language = {en},
author = {May, J Peter},
pages = {251},
file = {/home/zack/Dropbox/Library/J. Peter May/A Concise Course in Algebraic Topology (628)/May - A Concise Course in Algebraic Topology.pdf}
}
@article{MoreConciseAlgebraic,
title = {More {{Concise Algebraic Topology}}: {{Localization}}, {{Completion}}, and {{Model Categories}}},
language = {en},
pages = {404},
file = {/home/zack/Dropbox/Library/J. P. May/More Concise Algebraic Topology_ Localization, Completion, and Model Categories (461)/More Concise Algebraic Topology Localization, Com.pdf}
}
@misc{19LawsonthefuturePdf,
title = {19-{{Lawson}}-Thefuture.Pdf},
file = {/home/zack/Dropbox/Zotero/storage/R8SXT85E/19-Lawson-thefuture.pdf}
}
@misc{ComplexAlgebraicGeometry,
title = {Complex {{Algebraic Geometry}}.Pdf},
file = {/home/zack/Dropbox/Zotero/storage/EVEESZTC/Complex Algebraic Geometry.pdf}
}
@misc{FRGLecturePdf,
title = {{{FRGLecture}}.Pdf},
howpublished = {http://math.bu.edu/people/jsweinst/FRGLecture.pdf},
file = {/home/zack/Dropbox/Zotero/storage/3V33B9PB/FRGLecture.pdf}
}
@misc{HomotopyProjectPdf,
title = {{{HomotopyProject}}.Pdf},
howpublished = {https://www.maths.gla.ac.uk/\textasciitilde{}nnabijou/HomotopyProject.pdf},
file = {/home/zack/Dropbox/Zotero/storage/73ZCWFN4/HomotopyProject.pdf}
}
@article{levineAlgebraicTheoryWelschinger2018a,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1808.02238},
primaryClass = {math},
title = {Toward an Algebraic Theory of {{Welschinger}} Invariants},
abstract = {Let \$S\$ be a smooth del Pezzo surface over a field \$k\$ of characteristic \$\textbackslash{}neq 2, 3\$. We define an invariant in the Grothendieck-Witt ring \$GW(k)\$ for "counting" rational curves in a curve class \$D\$ of fixed positive degree (with respect to the anti-canonical bundle \$-K\_S\$) and containing a collection of distinct closed points \$\textbackslash{}mathfrak\{p\}=\textbackslash{}sum\_ip\_i\$ of total degree \$r:=-D\textbackslash{}cdot K\_S-1\$ on \$S\$. This recovers Welschinger's invariant in case \$k=\textbackslash{}mathbb\{R\}\$ by applying the signature map. The main result is that this quadratic invariant depends only on the \$\textbackslash{}mathbb\{A\}\^1\$-connected component containing \$\textbackslash{}mathfrak\{p\}\$ in \$Sym\^r(S)\^0(k)\$, where \$Sym\^r(S)\^0\$ is the open subscheme of \$Sym\^r(S)\$ parametrizing geometrically reduced 0-cycles.},
journal = {arXiv:1808.02238 [math]},
author = {Levine, Marc},
month = aug,
year = {2018},
keywords = {Mathematics - Algebraic Geometry,14F42; 55N20; 55N35}
}
@article{NielsenThurstonClassification2017,
title = {Nielsen\textendash{{Thurston}} Classification},
copyright = {Creative Commons Attribution-ShareAlike License},
abstract = {In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by Jakob Nielsen (1944).
Given a homeomorphism f : S {$\rightarrow$} S, there is a map g isotopic to f such that at least one of the following holds:
g is periodic, i.e. some power of g is the identity;
g preserves some finite union of disjoint simple closed curves on S (in this case, g is called reducible); or
g is pseudo-Anosov.The case where S is a torus (i.e., a surface whose genus is one) is handled separately (see torus bundle) and was known before Thurston's work. If the genus of S is two or greater, then S is naturally hyperbolic, and the tools of Teichm\"uller theory become useful. In what follows, we assume S has genus at least two, as this is the case Thurston considered. (Note, however, that the cases where S has boundary or is not orientable are definitely still of interest.)
The three types in this classification are not mutually exclusive, though a pseudo-Anosov homeomorphism is never periodic or reducible. A reducible homeomorphism g can be further analyzed by cutting the surface along the preserved union of simple closed curves {$\Gamma$}. Each of the resulting compact surfaces with boundary is acted upon by some power (i.e. iterated composition) of g, and the classification can again be applied to this homeomorphism.},
language = {en},
journal = {Wikipedia},
month = sep,
year = {2017},
file = {/home/zack/Dropbox/Zotero/storage/8NTM2AEQ/2017 - Nielsen–Thurston classification.html},
note = {Page Version ID: 799659651}
}
@misc{msriMathematicalSciencesResearch,
title = {Mathematical {{Sciences Research Institute}}},
abstract = {The Mathematical Sciences Research Institute (MSRI), founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions. The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to Grizzly Peak, on the hills overlooking Berkeley.},
howpublished = {http://www.msri.org},
author = {MSRI}
}