@online{abeProofVolumeConjecture2021, title = {Proof of {{Volume Conjecture}} for Twist Knots}, author = {Abe, Sukuse}, date = {2021-01-11}, eprint = {2012.05441}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/2012.05441}, urldate = {2021-02-01}, abstract = {We prove the volume conjecture for any hyperbolic knots by using Twisted Alexander polynomials and Vassiliev invariants.}, archiveprefix = {arXiv}, langid = {english}, keywords = {Mathematics - Algebraic Topology,Mathematics - Geometric Topology}, file = {/home/zack/Dropbox/Zotero/storage/GT8GUCWJ/Abe - 2021 - Proof of Volume Conjecture for twist knots.pdf} } @online{abouzaidArnoldConjectureMorava2021, title = {Arnold {{Conjecture}} and {{Morava K}}-Theory}, author = {Abouzaid, Mohammed and Blumberg, Andrew J.}, date = {2021-03-02}, eprint = {2103.01507}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/2103.01507}, urldate = {2021-03-05}, abstract = {We prove that the rank of the cohomology of a closed symplectic manifold with coefficients in a field of characteristic \$p\$ is smaller than the number of periodic orbits of any non-degenerate Hamiltonian flow. Following Floer, the proof relies on constructing a homology group associated to each such flow, and comparing it with the homology of the ambient symplectic manifold. The proof does not proceed by constructing a version of Floer's complex with characteristic \$p\$ coefficients, but uses instead the canonical (stable) complex orientations of moduli spaces of Floer trajectories to construct a version of Floer homology with coefficients in Morava's \$K\$-theories, and can thus be seen as an implementation of Cohen, Jones, and Segal's vision for a Floer homotopy theory. The key feature of Morava K-theory that allows the construction to be carried out is the fact that the corresponding homology and cohomology groups of classifying spaces of finite groups satisfy Poincar\textbackslash 'e duality.}, archiveprefix = {arXiv}, keywords = {homotopy,symplectic}, file = {/home/zack/Dropbox/Zotero/storage/JJ2KVQWW/Abouzaid and Blumberg - 2021 - Arnold Conjecture and Morava K-theory.pdf;/home/zack/Dropbox/Zotero/storage/38QZKKST/2103.html} } @article{adamsIfTHEORYHOPFINVARIANT, title = {If-{{THEORY AND THE HOPF INVARIANT}}}, author = {Adams, J F and Atiyah, M F}, pages = {8}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/F9JHJX2W/Adams and Atiyah - if-THEORY AND THE HOPF INVARIANT.pdf} } @online{ahrensUnivalencePrinciple2021, title = {The {{Univalence Principle}}}, author = {Ahrens, Benedikt and North, Paige Randall and Shulman, Michael and Tsementzis, Dimitris}, date = {2021-02-11}, eprint = {2102.06275}, eprinttype = {arxiv}, primaryclass = {cs, math}, url = {http://arxiv.org/abs/2102.06275}, urldate = {2021-02-15}, abstract = {The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk's completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is expressed in Voevodsky's Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.}, archiveprefix = {arXiv}, keywords = {18N99; 03B38; 03G30; 55U35,Computer Science - Logic in Computer Science,Mathematics - Category Theory,Mathematics - Logic}, file = {/home/zack/Downloads/Zotero_Source/arXiv2102.06275 [cs, math]/2021/Ahrens et al. - 2021 - The Univalence Principle.pdf;/home/zack/Dropbox/Zotero/storage/7DZJPZQI/2102.html} } @online{alexeevCompactificationsModuliElliptic2020, title = {Compactifications of Moduli of Elliptic {{K3}} Surfaces: Stable Pair and Toroidal}, shorttitle = {Compactifications of Moduli of Elliptic {{K3}} Surfaces}, author = {Alexeev, Valery and Brunyate, Adrian and Engel, Philip}, date = {2020-03-18}, eprint = {2002.07127}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/2002.07127}, urldate = {2021-03-03}, abstract = {We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space.}, archiveprefix = {arXiv}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/QNWWHF7U/Alexeev et al. - 2020 - Compactifications of moduli of elliptic K3 surface.pdf} } @incollection{avramovLookingGlassDictionary1986, title = {Through the Looking Glass: {{A}} Dictionary between Rational Homotopy Theory and Local Algebra}, shorttitle = {Through the Looking Glass}, booktitle = {Algebra, {{Algebraic Topology}} and Their {{Interactions}}}, author = {Avramov, Luchezar and Halperin, Stephen}, editor = {Roos, Jan-Erik}, date = {1986}, series = {Lecture {{Notes}} in {{Mathematics}}}, volume = {1183}, pages = {1--27}, publisher = {{Springer Berlin Heidelberg}}, location = {{Berlin, Heidelberg}}, doi = {10.1007/BFb0075446}, url = {http://link.springer.com/10.1007/BFb0075446}, urldate = {2021-02-15}, isbn = {978-3-540-16453-1 978-3-540-39790-8}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/AHCCYK4K/Avramov and Halperin - 1986 - Through the looking glass A dictionary between ra.pdf} } @online{barthelChromaticStructuresStable2019, title = {Chromatic Structures in Stable Homotopy Theory}, author = {Barthel, Tobias and Beaudry, Agnès}, date = {2019-04-29}, eprint = {1901.09004}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/1901.09004}, urldate = {2021-01-28}, abstract = {In this survey, we review how the global structure of the stable homotopy category gives rise to the chromatic filtration. We then discuss computational tools used in the study of local chromatic homotopy theory, leading up to recent developments in the field. Along the way, we illustrate the key methods and results with explicit examples.}, archiveprefix = {arXiv}, keywords = {Mathematics - Algebraic Topology}, file = {/home/zack/Dropbox/Zotero/storage/49462PWI/Barthel and Beaudry - 2019 - Chromatic structures in stable homotopy theory.pdf;/home/zack/Dropbox/Zotero/storage/U8TCXW4N/1901.html} } @online{behrensTopologicalModularAutomorphic2019, title = {Topological Modular and Automorphic Forms}, author = {Behrens, Mark}, date = {2019-02-05}, eprint = {1901.07990}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/1901.07990}, urldate = {2021-03-03}, abstract = {This article is a brief survey of the theory of topological modular forms (TMF) and the theory of topological automorphic forms (TAF). It will be a chapter in forthcoming "Handbook of Homotopy Theory" edited by Haynes Miller.}, archiveprefix = {arXiv}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/UEZDG98S/Behrens - 2019 - Topological modular and automorphic forms.pdf} } @online{bhattRevisitingRhamWittComplex2020, title = {Revisiting the de {{Rham}}-{{Witt}} Complex}, author = {Bhatt, Bhargav and Lurie, Jacob and Mathew, Akhil}, date = {2020-02-18}, eprint = {1805.05501}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/1805.05501}, urldate = {2021-03-03}, abstract = {The goal of this paper is to offer a new construction of the de Rham–Witt complex of a smooth variety over a perfect field of characteristic p {$>$} 0. We introduce a category of cochain complexes which are equipped with an endomorphism F of underlying graded abelian groups satisfying dF = pF d, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham–Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot–Ogus operator Lηp on the p-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison for the AΩ-cohomology theory introduced in [BMS18].}, archiveprefix = {arXiv}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/B8ZQIKTF/Schemes 233bnotesIurie.pdf;/home/zack/Dropbox/Zotero/storage/C5AUWXVL/Bhatt et al. - 2020 - Revisiting the de Rham-Witt complex.pdf} } @article{brunerAdamsSpectralSequence, title = {An {{Adams Spectral Sequence Primer}}}, author = {Bruner, R R}, pages = {78}, abstract = {My aim with these notes is to quickly get the student started with the Adams spectral sequence. To see its power requires that some concrete calculations be done. However, the algebra required can quickly become overwhelming if one starts with the generalized Adams spectral sequence. The classical Adams spectral sequence, in contrast, can be quickly set up and used to do some calculations which would be quite difficult by any other technique. Further, the classical Adams spectral sequence is still a useful calculational and theoretical tool, and is an excellent introduction to the general case.}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/C7JT78K8/Bruner - An Adams Spectral Sequence Primer.pdf} } @online{camarenaWhirlwindTourWorld2013, title = {A {{Whirlwind Tour}} of the {{World}} of \$(\textbackslash infty,1)\$-Categories}, author = {Camarena, Omar Antolín}, date = {2013-09-05}, eprint = {1303.4669}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/1303.4669}, urldate = {2021-01-30}, abstract = {This introduction to higher category theory is intended to a give the reader an intuition for what \$(\textbackslash infty,1)\$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.}, archiveprefix = {arXiv}, keywords = {Mathematics - Category Theory}, file = {/home/zack/Dropbox/Zotero/storage/ULWGCTS6/Camarena - 2013 - A Whirlwind Tour of the World of $(infty,1)$-cate.pdf;/home/zack/Dropbox/Zotero/storage/GE3QQY32/1303.html} } @online{casalaina-martinIntroductionModuliStacks2017, title = {An Introduction to Moduli Stacks, with a View towards {{Higgs}} Bundles on Algebraic Curves}, author = {Casalaina-Martin, Sebastian and Wise, Jonathan}, date = {2017-08-27}, eprint = {1708.08124}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/1708.08124}, urldate = {2021-02-15}, abstract = {This article is based in part on lecture notes prepared for the summer school "The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a "bootstrap" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a projective moduli space of semi-stable Higgs bundles, and we describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin's criterion.}, archiveprefix = {arXiv}, keywords = {14D20; 14D23; 14H60,Mathematics - Algebraic Geometry}, file = {/home/zack/Dropbox/Zotero/storage/I8J6AIRC/Casalaina-Martin and Wise - 2017 - An introduction to moduli stacks, with a view towa.pdf;/home/zack/Dropbox/Zotero/storage/Z95ST5QG/1708.html} } @article{chernPARTIIIALGEBRAIC, title = {{{PART III}}. {{ALGEBRAIC SHEAF THEORY}}}, author = {Chern, Shiing-Shen}, journaltitle = {AMERICAN MATHEMATICAL SOCIETY}, pages = {25}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/4DMN26F6/Chern - PART III. ALGEBRAIC SHEAF THEORY.pdf} } @article{ciriciSHORTCOURSEINTERACTIONS, title = {A {{SHORT COURSE ON THE INTERACTIONS OF RATIONAL HOMOTOPY THEORY AND COMPLEX}} ({{ALGEBRAIC}}) {{GEOMETRY}}}, author = {Cirici, Joana}, pages = {14}, abstract = {These notes were written for the Summer School on “Rational Homotopy Theory and its Interactions” (July 2016, Rabat, Morocco). The objective of this course is to do several computations in rational homotopy theory for topological spaces that carry extra structure coming from complex geometry. We review results on the rational homotopy theory of complex manifolds, compact Ka¨hler manifolds and (singular) complex projective varieties.}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/CZLS783G/Cirici - A SHORT COURSE ON THE INTERACTIONS OF RATIONAL HOM.pdf} } @article{cohenTopologyFiberBundles, title = {The {{Topology}} of {{Fiber Bundles Lecture Notes}}}, author = {Cohen, Ralph L}, pages = {178}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/XPZGI7AG/Cohen - The Topology of Fiber Bundles Lecture Notes.pdf} } @article{duggerGEOMETRICINTRODUCTIONKTHEORY, title = {A {{GEOMETRIC INTRODUCTION TO K}}-{{THEORY}}}, author = {Dugger, Daniel}, pages = {293}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/JJEIBWSC/Dugger - A GEOMETRIC INTRODUCTION TO K-THEORY.pdf} } @article{eiflerRealVariablesNamed, title = {Real {{Variables Named Theorems}}}, author = {Eifler, Kari}, pages = {8}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/3NTS62YF/Eifler - Real Variables Named Theorems.pdf} } @article{eiflerRealVariablesNameda, title = {Real {{Variables Named Theorems}}}, author = {Eifler, Kari}, pages = {8}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/P2V4G5FX/Eifler - Real Variables Named Theorems.pdf} } @online{farbRepresentationStability2014, title = {Representation {{Stability}}}, author = {Farb, Benson}, date = {2014-04-15}, eprint = {1404.4065}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/1404.4065}, urldate = {2021-02-02}, abstract = {Representation stability is a phenomenon whereby the structure of certain sequences \$X\_n\$ of spaces can be seen to stabilize when viewed through the lens of representation theory. In this paper I describe this phenomenon and sketch a framework, the theory of FI-modules, that explains the mechanism behind it.}, archiveprefix = {arXiv}, keywords = {Mathematics - Algebraic Topology,Mathematics - Geometric Topology,Mathematics - Representation Theory}, file = {/home/zack/Dropbox/Zotero/storage/WE994W2Z/Farb - 2014 - Representation Stability.pdf;/home/zack/Dropbox/Zotero/storage/J6Y7RV9X/1404.html} } @book{FoundationsStableHomotopy, title = {Foundations of {{Stable Homotopy}}}, file = {/home/zack/Dropbox/Library/Barnes, Roitzheim/Foundations of Stable Homotopy Theory (779)/Foundations of Stable Homotopy Theory - Barnes, Roitzheim.pdf} } @article{galatiusDerivedGaloisDeformation2018, title = {Derived {{Galois}} Deformation Rings}, author = {Galatius, Soren and Venkatesh, Akshay}, date = {2018-03}, journaltitle = {Advances in Mathematics}, volume = {327}, eprint = {1608.07236}, eprinttype = {arxiv}, pages = {470--623}, issn = {00018708}, doi = {10.1016/j.aim.2017.08.016}, url = {http://arxiv.org/abs/1608.07236}, urldate = {2021-03-03}, abstract = {We define a derived version of Mazur’s Galois deformation ring. It is a pro-simplicial ring R classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group π0R recovers Mazur’s deformation ring.}, archiveprefix = {arXiv}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/9HDLU2TZ/Galatius and Venkatesh - 2018 - Derived Galois deformation rings.pdf} } @online{GaloisDescentPdf, title = {Galois Descent.Pdf}, url = {https://users.math.msu.edu/users/ruiterj2/math/Documents/Notes%20and%20talks/Galois%20descent.pdf}, urldate = {2021-01-17}, file = {/home/zack/Dropbox/Zotero/storage/95RQCYAV/Galois descent.pdf} } @online{gomezAlgebraicStacks1999, title = {Algebraic Stacks}, author = {Gomez, T.}, date = {1999-11-25}, eprint = {math/9911199}, eprinttype = {arxiv}, url = {http://arxiv.org/abs/math/9911199}, urldate = {2021-01-31}, abstract = {This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector budles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.}, archiveprefix = {arXiv}, keywords = {Mathematics - Algebraic Geometry}, file = {/home/zack/Dropbox/Zotero/storage/G8DBP2CA/Gomez - 1999 - Algebraic stacks.pdf;/home/zack/Dropbox/Zotero/storage/UXZVWLKG/9911199.html} } @article{granvilleMultiplicativeNumberTheory, title = {Multiplicative Number Theory: {{The}} Pretentious Approach}, author = {Granville, Andrew and Soundararajan, K}, pages = {28}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/PJX4CKNI/Granville and Soundararajan - Multiplicative number theory The pretentious appr.pdf} } @online{greuelDeltaParameterizedCurve2021, title = {On {{Delta}} for Parameterized {{Curve Singularities}}}, author = {Greuel, Gert-Martin and Pfister, Gerhard}, date = {2021-01-05}, eprint = {2101.01784}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/2101.01784}, urldate = {2021-01-07}, abstract = {We consider families of parameterizations of reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each curve singularity in the family is the image of a parameterization and not the fiber of a morphism. The problem came up in connection with the right-left classification of parameterizations of curve singularities defined over a field of positive characteristic. We prove a bound for right-left determinacy of a parameterization in terms of delta and the semicontinuity theorem provides a simultaneous bound for the determinacy in a family. The fact that the base space can be an arbitrary Noetherian scheme causes some difficulties but is (not only) of interest for computational purposes.}, archiveprefix = {arXiv}, keywords = {Mathematics - Algebraic Geometry,Mathematics - Commutative Algebra}, file = {/home/zack/Dropbox/Zotero/storage/UBS8R3JK/Greuel and Pfister - 2021 - On Delta for parameterized Curve Singularities.pdf;/home/zack/Dropbox/Zotero/storage/75AX6UF9/2101.html} } @online{guidolinComputingSerreSpectral2019, title = {Computing {{Serre}} Spectral Systems of Towers of Fibrations}, author = {Guidolin, Andrea and Romero, Ana}, date = {2019-12-10}, eprint = {1912.04848}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/1912.04848}, urldate = {2021-01-17}, abstract = {The Serre spectral system associated with a tower of fibrations represents a generalization of the classical Serre spectral sequence of a fibration. In this work, we present algorithms to compute the Serre spectral system leveraging the effective homology technique, which allows to perform computations involving infinitely generated chain complexes associated with interesting objects in algebraic topology. In order to develop the programs, implemented as a new module for the Computer Algebra system Kenzo, we translated the original construction of the Serre spectral system in a simplicial framework and studied some of its fundamental properties.}, archiveprefix = {arXiv}, keywords = {Mathematics - Algebraic Topology}, file = {/home/zack/Dropbox/Zotero/storage/T9KW4R9P/Guidolin and Romero - 2019 - Computing Serre spectral systems of towers of fibr.pdf;/home/zack/Dropbox/Zotero/storage/MTEFRQRJ/1912.html} } @article{haineDifferentialCohomologyTheories, title = {Differential Cohomology Theories as Sheaves on Manifolds}, author = {Haine, Peter J}, pages = {48}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/PEGDCC45/Haine - Differential cohomology theories as sheaves on man.pdf} } @online{heinlothUniformizationMathcalGbundles2009, title = {Uniformization of \textbackslash mathcal\{\vphantom\}{{G}}\vphantom\{\}-Bundles}, author = {Heinloth, Jochen}, date = {2009-10-28}, eprint = {0711.4450}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/0711.4450}, urldate = {2021-03-03}, abstract = {We show some of the conjectures of Pappas and Rapoport concerning the moduli stack BunG of G-torsors on a curve C, where G is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of BunG in case G is simply connected.}, archiveprefix = {arXiv}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/2BNTNNDK/Heinloth - 2009 - Uniformization of mathcal G -bundles.pdf} } @article{hillHHRKervaireProblem, title = {{{HHR Kervaire Problem}} ({{Equivariant Stable Homotopy Theory}})}, author = {Hill, Michael A and Hopkins, Michael J and Ravenel, Douglas C}, pages = {923}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/N59SRS64/Hill et al. - WORK IN PROGRESS This book in being written now an.pdf} } @article{hopkinsMathematicalWorkDouglas2008, title = {The Mathematical Work of {{Douglas C}}. {{Ravenel}}}, author = {Hopkins, Michael J.}, date = {2008}, journaltitle = {Homology, Homotopy and Applications}, volume = {10}, number = {3}, pages = {1--13}, issn = {15320073, 15320081}, doi = {10.4310/HHA.2008.v10.n3.a1}, url = {http://www.intlpress.com/site/pub/pages/journals/items/hha/content/vols/0010/0003/a001/}, urldate = {2021-03-03}, abstract = {This is a transcription of my talk on the work of Doug Ravenel. While it was intended as a one-time performance, the organizers felt it would be a good idea to have some representation appear as part of the conference proceedings. Don Davis went to the great trouble of transcribing the lecture from a video, and he sent it to me. After reading it I agreed to have it included in these proceedings. Even though the text is ungrammatical, repetitious and meandering (apparently that’s how I talk), it feels to me as if it conveys something about how the world of algebraic topology has felt all these years with Doug in it. So with heartfelt thanks to Don Davis, here it is.}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/68ZNK5BQ/Hopkins - 2008 - The mathematical work of Douglas C. Ravenel.pdf} } @online{hoveySpectraSymmetricSpectra2000, title = {Spectra and Symmetric Spectra in General Model Categories}, author = {Hovey, Mark}, date = {2000-06-09}, eprint = {math/0004051}, eprinttype = {arxiv}, url = {http://arxiv.org/abs/math/0004051}, urldate = {2021-04-03}, abstract = {(This is an updated version; following an idea of Voevodsky, we have strengthened our results so all of them apply to one form of motivic homotopy theory). We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as motivic homotopy theory of schemes. One is based on the standard notion of spectra originated by Boardman. Its input is a well-behaved model category C and an endofunctor G, generalizing the suspension. Its output is a model category on which G is a Quillen equivalence. Under strong hypotheses the weak equivalences in this model structure are the appropriate analogue of stable homotopy isomorphisms. The second construction is based on symmetric spectra, and is of value only when C has some monoidal structure that G preserves. In this case, ordinary spectra generally will not have monoidal structure, but symmetric spectra will. Our abstract approach makes constructing the stable model category of symmetric spectra straightforward. We study properties of these stabilizations; most importantly, we show that the two different stabilizations are Quillen equivalent under some hypotheses (that also hold in the motivic example).}, archiveprefix = {arXiv}, keywords = {homotopy,motivic,spectra}, file = {/home/zack/Dropbox/Zotero/storage/EJPF2HWR/Hovey - 2000 - Spectra and symmetric spectra in general model cat.pdf;/home/zack/Dropbox/Zotero/storage/JS55WK5W/0004051.html} } @online{hoyoisHilbertSchemeInfinite2020, title = {The {{Hilbert}} Scheme of Infinite Affine Space and Algebraic {{K}}-Theory}, author = {Hoyois, Marc and Jelisiejew, Joachim and Nardin, Denis and Totaro, Burt and Yakerson, Maria}, date = {2020-09-15}, eprint = {2002.11439}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/2002.11439}, urldate = {2021-02-01}, abstract = {We study the Hilbert scheme Hilbd(A∞) from an A1-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme Hilbd(A∞) is A1-equivalent to the Grassmannian of (d − 1)-planes in A∞. We then describe the A1-homotopy type of Hilbd(An) in a range, for n large compared to d. For example, we compute the integral cohomology of Hilbd(An)(C) in a range. We also deduce that the forgetful map FFlat → Vect from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an A1-equivalence after group completion. This implies that the moduli stack FFlat, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum kgl representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the kgl-homology of smooth proper schemes over a perfect field.}, archiveprefix = {arXiv}, langid = {english}, keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology,Mathematics - K-Theory and Homology}, file = {/home/zack/Dropbox/Zotero/storage/2EVBRXIQ/Hoyois et al. - 2020 - The Hilbert scheme of infinite affine space and al.pdf} } @article{hutchingsIntroductionHigherHomotopy, title = {Introduction to Higher Homotopy Groups and Obstruction Theory}, author = {Hutchings, Michael}, pages = {38}, abstract = {These are some notes to accompany the beginning of a secondsemester algebraic topology course. The goal is to introduce homotopy groups and their uses, and at the same time to prepare a bit for the study of characteristic classes which will come next. These notes are not intended to be a comprehensive reference (most of this material is covered in much greater depth and generality in a number of standard texts), but rather to give an elementary introduction to selected basic ideas.}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/NKMC4WHZ/Hutchings - Introduction to higher homotopy groups and obstruc.pdf} } @misc{InfinityCategoryLearning, title = {Infinity {{Category Learning Seminar Resources}}}, url = {http://homepages.math.uic.edu/~jlv/seminars/infinitycats/} } @online{InfinityCategoryTheory, title = {Infinity Category Theory from Scratch, {{Emily Riehl}}'s Lecture 1}, url = {https://tube.switch.ch/videos/8d2e27f1}, urldate = {2021-01-31}, file = {/home/zack/Dropbox/Zotero/storage/M2NF8LQ3/8d2e27f1.html} } @online{IntroductionHigherCategories, title = {Introduction to Higher Categories - {{Mathematical Institute}} of the {{University}} of {{Bonn}}}, url = {http://www.math.uni-bonn.de/people/fhebestr/Higher%20Cats/}, urldate = {2021-03-13}, file = {/home/zack/Dropbox/Zotero/storage/HZ9P97I3/Higher Cats.html} } @online{IntroHigherCategory, title = {Intro to {{Higher Category Theory}}}, url = {https://www.math.uni-hamburg.de/home/dyckerhoff/higher/notes.pdf}, urldate = {2021-01-30}, file = {/home/zack/Dropbox/Zotero/storage/NA37EMPC/Intro to Higher Category Theory.pdf} } @online{KevinBuzzardMSRI, title = {Kevin {{Buzzard}}'s {{MSRI Summer School}} on Automorphic Forms.}, url = {http://wwwf.imperial.ac.uk/~buzzard/MSRI/}, urldate = {2021-02-22}, file = {/home/zack/Dropbox/Zotero/storage/M3ZB89AQ/MSRI.html} } @article{ladymanPrimerHomotopyType2014, title = {A {{Primer}} on {{Homotopy Type Theory Part}} 1: {{The Formal Type Theory}}}, author = {Ladyman, James and Presnell, Stuart}, date = {2014}, pages = {85}, abstract = {This Primer is an introduction to Homotopy Type Theory (HoTT). The original source for the ideas presented here is the “HoTT Book” – Homotopy Type Theory: Univalent Foundations of Mathematics published by The Univalent Foundations Program, Institute for Advanced Study, Princeton. In what follows we freely borrow and adapt definitions, arguments and proofs from the HoTT Book throughout without always giving a specific citation.1 However, whereas that book provides an introduction to the subject that rapidly involves the reader in advanced technical material, the exposition in this Primer is more gently paced for the beginner. We also do more to motivate, justify, and explain some aspects of the theory in greater detail, and we address foundational and philosophical issues that the HoTT Book does not.}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/IPVSQJ4R/Ladyman and Presnell - 2014 - A Primer on Homotopy Type Theory Part 1 The Forma.pdf} } @article{lenstraGaloisTheorySchemes, title = {Galois Theory for Schemes}, author = {Lenstra, H W}, pages = {113}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/C3FZ485J/Lenstra - Galois theory for schemes.pdf} } @online{lentnerHochschildCohomologyModular2020, title = {Hochschild {{Cohomology}}, {{Modular Tensor Categories}}, and {{Mapping Class Groups}}}, author = {Lentner, Simon and Mierach, Svea Nora and Schweigert, Christoph and Sommerhaeuser, Yorck}, date = {2020-03-13}, eprint = {2003.06527}, eprinttype = {arxiv}, primaryclass = {math-ph}, url = {http://arxiv.org/abs/2003.06527}, urldate = {2021-04-08}, abstract = {Given a finite modular tensor category, we associate with each compact surface with boundary a cochain complex in such a way that the mapping class group of the surface acts projectively on its cohomology groups. In degree zero, this action coincides with the known projective action of the mapping class group on the space of chiral conformal blocks. In the case that the surface is a torus and the category is the representation category of a factorizable ribbon Hopf algebra, we recover our previous result on the projective action of the modular group on the Hochschild cohomology groups of the Hopf algebra.}, archiveprefix = {arXiv}, file = {/home/zack/Downloads/Zotero_Source/arXiv2003.06527 [math-ph]/2020/Lentner et al. - 2020 - Hochschild Cohomology, Modular Tensor Categories, .pdf;/home/zack/Dropbox/Zotero/storage/TDHDKJSJ/2003.html} } @article{loefflerTCCHomologicalAlgebra, title = {{{TCC Homological Algebra}}: {{Assignment}} \#3 ({{Solutions}})}, author = {Loeffler, David}, pages = {6}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/UMYKKV5G/Loeffler - TCC Homological Algebra Assignment #3 (Solutions).pdf} } @book{lurieHigherToposTheory2009, title = {Higher Topos Theory}, author = {Lurie, Jacob}, date = {2009}, series = {Annals of Mathematics Studies}, number = {no. 170}, publisher = {{Princeton University Press}}, location = {{Princeton, N.J}}, isbn = {978-0-691-14048-3 978-0-691-14049-0}, langid = {english}, pagetotal = {925}, keywords = {Categories (Mathematics),Toposes}, annotation = {OCLC: ocn244702012}, file = {/home/zack/Dropbox/Zotero/storage/9R3LRC8Y/Lurie - 2009 - Higher topos theory.pdf} } @article{mackeyHARMONICANALYSISEXPLOITATION, title = {{{HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY}}-{{A HISTORICAL SURVEY}}}, author = {Mackey, George W}, pages = {156}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/DXFTBZYJ/Mackey - HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY-.pdf} } @article{mathewPicardGroupTopological2016, title = {The {{Picard}} Group of Topological Modular Forms via Descent Theory}, author = {Mathew, Akhil and Stojanoska, Vesna}, date = {2016-12-21}, journaltitle = {Geom. Topol.}, volume = {20}, number = {6}, pages = {3133--3217}, issn = {1364-0380, 1465-3060}, doi = {10.2140/gt.2016.20.3133}, url = {http://msp.org/gt/2016/20-6/p03.xhtml}, urldate = {2021-02-19}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/SPECQVF9/Mathew and Stojanoska - 2016 - The Picard group of topological modular forms via .pdf} } @article{muellerCalculatingCobordismRings, title = {Calculating {{Cobordism Rings}}}, author = {Mueller, Michael}, pages = {23}, abstract = {The notion of (unoriented, oriented, ...) cobordism yields an equivalence relation on closed manifolds, and can be used to construct generalized (co)homology theories. In 1954, Thom [15] determined the structure of the unoriented cobordism ring by reducing the problem to a question in stable homotopy theory. We provide an exposition of this approach, introducing basic concepts in cobordism theory and covering the computations of the unoriented and complex cobordism rings.}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/4RAEIGEW/Mueller - Calculating Cobordism Rings.pdf} } @online{NilesJohnsonHomological, title = {Niles {{Johnson}} : {{Homological}} Algebra}, url = {https://nilesjohnson.net/homological-algebra-topics.html}, urldate = {2021-02-25}, file = {/home/zack/Dropbox/Zotero/storage/4Q4EAI89/homological-algebra-topics.html} } @article{noohiQUICKINTRODUCTIONFIBERED, title = {A {{QUICK INTRODUCTION TO FIBERED CATEGORIES AND TOPOLOGICAL STACKS}}}, author = {Noohi, Behrang}, pages = {9}, abstract = {This is a quick introduction to fibered categories and topological stacks.}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/PE5NU5F7/Noohi - A QUICK INTRODUCTION TO FIBERED CATEGORIES AND TOP.pdf} } @online{programHomotopyTypeTheory2013, title = {Homotopy {{Type Theory}}: {{Univalent Foundations}} of {{Mathematics}}}, shorttitle = {Homotopy {{Type Theory}}}, author = {Program, The Univalent Foundations}, date = {2013-08-03}, eprint = {1308.0729}, eprinttype = {arxiv}, primaryclass = {cs, math}, url = {http://arxiv.org/abs/1308.0729}, urldate = {2021-01-21}, abstract = {Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.}, archiveprefix = {arXiv}, keywords = {Computer Science - Programming Languages,Mathematics - Algebraic Topology,Mathematics - Category Theory,Mathematics - Logic}, file = {/home/zack/Dropbox/Zotero/storage/EUSNKHA2/Program - 2013 - Homotopy Type Theory Univalent Foundations of Mat.pdf;/home/zack/Dropbox/Zotero/storage/CEXGG9S5/1308.html} } @article{putmanClassifyingSpacesBrown, title = {Classifying Spaces and {{Brown}} Representability}, author = {Putman, Andrew}, pages = {3}, abstract = {We sketch the proof of the Brown representability theorem and give a few applications of it, the most important being the construction of the classifying space for principal G-bundles.}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/9H8PXPA6/Putman - Classifying spaces and Brown representability.pdf} } @article{riehlCategoricalHomotopyTheory, title = {Categorical Homotopy Theory}, author = {Riehl, Emily}, pages = {292}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/YTSAZ4LP/Riehl - Categorical homotopy theory.pdf} } @online{robaloNoncommutativeMotivesUniversal2013, title = {Noncommutative {{Motives I}}: {{A Universal Characterization}} of the {{Motivic Stable Homotopy Theory}} of {{Schemes}}}, shorttitle = {Noncommutative {{Motives I}}}, author = {Robalo, Marco}, date = {2013-06-17}, eprint = {1206.3645}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/1206.3645}, urldate = {2021-04-03}, abstract = {Let \$\textbackslash V\$ be a symmetric monoidal model category and let \$X\$ be an object in \$\textbackslash V\$. From this we can construct a new symmetric monoidal model category \$Sp\^\{\textbackslash Sigma\}(\textbackslash V,X)\$ of symmetric spectra objects in \$\textbackslash V\$ with respect to \$X\$, together with a left Quillen monoidal map \$\textbackslash V\textbackslash to Sp\^\{\textbackslash Sigma\}(\textbackslash V,X)\$ sending \$X\$ to an invertible object. In this paper we use the recent developments in the subject of Higher Algebra to understand the nature of this construction. Every symmetric monoidal model category has an underlying symmetric monoidal \$(\textbackslash infty,1)\$-category and the first notion should be understood as a mere "presentation" of the second. Our main result is the characterization of the underlying symmetric monoidal \$\textbackslash infty\$-category of \$Sp\^\{\textbackslash Sigma\}(\textbackslash V,X)\$, by means of a universal property inside the world of symmetric monoidal \$(\textbackslash infty,1)\$-categories. In the process we also describe the link between the construction of ordinary spectra and the one of symmetric spectra. As a corollary, we obtain a precise universal characterization for the motivic stable homotopy theory of schemes with its symmetric monoidal structure. This characterization trivializes the problem of finding motivic monoidal realizations and opens the way to compare the motivic theory of schemes with other motivic theories. As an application we provide a new approach to the theory of noncommutative motives by constructing a stable motivic homotopy theory for the noncommutative spaces of Kontsevich. For that we introduce an analogue for the Nisnevich topology in the noncommutative setting. Our universal property for the classical theory for schemes provides a canonical monoidal map towards these new noncommutative motives and allows us to compare the two theories.}, archiveprefix = {arXiv}, keywords = {homotopy,motivic,spectra}, file = {/home/zack/Dropbox/Zotero/storage/KDCWW7L4/Robalo - 2013 - Noncommutative Motives I A Universal Characteriza.pdf;/home/zack/Dropbox/Zotero/storage/MCJ8Y6NN/1206.html} } @article{romanovLanglandsCorespondenceBezrukavnikov, title = {Langlands {{Corespondence}} and {{Bezrukavnikov}}’s {{Equivalence}}}, author = {Romanov, Anna}, pages = {86}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/PLC585KX/Romanov - Langlands Corespondence and Bezrukavnikov’s Equiva.pdf} } @inbook{serreSingularHomologyFiber2012, title = {Singular Homology of Fiber Spaces}, booktitle = {Series on {{Knots}} and {{Everything}}}, author = {Serre, J.-P.}, date = {2012-09}, volume = {50}, pages = {1--104}, publisher = {{WORLD SCIENTIFIC}}, doi = {10.1142/9789814401319_0001}, url = {http://www.worldscientific.com/doi/abs/10.1142/9789814401319_0001}, urldate = {2021-03-03}, bookauthor = {Novikov, S P and Taimanov, I A and Golubyatnikov, V P}, isbn = {978-981-4401-30-2 978-981-4401-31-9}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/G4CDPBQ5/Serre - 2012 - Singular homology of fiber spaces.pdf} } @inproceedings{todaCohomologyClassifyingSpaces, title = {Cohomology of {{Classifying Spaces}}}, author = {Toda, Hiroshi}, pages = {75--108}, location = {{Kyoto University, Kyoto, Japan}}, doi = {10.2969/aspm/00910075}, url = {https://projecteuclid.org/euclid.aspm/1525310151}, urldate = {2021-03-03}, eventtitle = {Symposium on {{Homotopy Theory}} and {{Related Topics}}}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/JW3WNV28/Toda - Cohomology of Classifying Spaces.pdf} } @article{vakilModuliSpaceCurves2003, title = {The {{Moduli Space}} of {{Curves}} and {{Its Tautological Ring}}}, author = {Vakil, Ravi}, date = {2003}, volume = {50}, number = {6}, pages = {12}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/E287Y2GG/Vakil - 2003 - The Moduli Space of Curves and Its Tautological Ri.pdf} } @article{webbFiniteGroupRepresentations, title = {Finite {{Group Representations}} for the {{Pure Mathematician}}}, author = {Webb, Peter}, pages = {105}, langid = {english}, file = {/home/zack/Dropbox/Zotero/storage/5IQ3KL52/Webb - Finite Group Representations for the Pure Mathemat.pdf} } @online{zanardiniStabilityPencilsPlane2021, title = {Stability of Pencils of Plane Curves, Log Canonical Thresholds and Multiplicities}, author = {Zanardini, Aline}, date = {2021-01-05}, eprint = {2101.01756}, eprinttype = {arxiv}, primaryclass = {math}, url = {http://arxiv.org/abs/2101.01756}, urldate = {2021-01-21}, abstract = {In this paper we study the problem of classifying pencils of curves of degree \$d\$ in \$\textbackslash mathbb\{P\}\^2\$ using geometric invariant theory. We consider the action of \$SL(3)\$ and we relate the stability of a pencil to the stability of its generators, to the log canonical threshold of its members, and to the multiplicities of its base points, thus obtaining explicit stability criteria.}, archiveprefix = {arXiv}, keywords = {14L24 (Primary); 14E99 (Secondary),Mathematics - Algebraic Geometry}, file = {/home/zack/Dropbox/Zotero/storage/CSGFNDIQ/Zanardini - 2021 - Stability of pencils of plane curves, log canonica.pdf;/home/zack/Dropbox/Zotero/storage/FBWBF8EJ/2101.html} }