The Rising Sea
http://therisingsea.org/
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Mon, 23 Jul 2018 00:11:01 0400

Welcome
http://therisingsea.org/post/intro/
Sun, 03 Jan 2016 05:16:08 0500
http://therisingsea.org/post/intro/
<p>My name is Daniel Murfet, I am a Lecturer (aka tenuretrack Assistant Professor) in the <a href="http://www.ms.unimelb.edu.au/">Mathematics Department</a> at the University of Melbourne. My CV is <a href="http://therisingsea.org/cv.pdf">here</a> and you can contact me by <a href="mailto:d.murfet@unimelb.edu.au">email</a>. My papers are <a href="https://arxiv.org/find/all/1/au:+murfet%5fdaniel/0/1/0/all/0/1">on the arXiv</a> with the exception of my PhD thesis which you can find <a href="http://therisingsea.org/thesis.pdf">here</a>. I run a <a href="http://therisingsea.org/post/seminarch/">seminar</a>, the videos from which may be found on <a href="https://www.youtube.com/channel/UCJTk6uSbSsclXN8v3b27_QQ/videos?flow=list&live_view=500&view=0&sort=dd">YouTube</a>.</p>
<p>My primary research interests are in algebraic geometry and mathematical logic. You can see some of the things I like to think about in my <a href="https://www.msri.org/attachments/media/news/emissary/EmissarySpring2013.pdf">Spring 2013 article in Emissary</a> on matrix factorisations and my <a href="http://arxiv.org/abs/1407.2650">survey of linear logic</a>. I have various code projects <a href="https://github.com/dmurfet/">on GitHub</a> including my joint project with Nils Carqueville on <a href="https://github.com/dmurfet/mf">computing with matrix factorisations</a>.</p>
<p>The title of this webpage refers to the following quote from Grothendieck, which I learned from Colin McLarty’s <a href="http://www.cwru.edu/artsci/phil/RisingSea.pdf">article</a> “The Rising Sea: Grothendieck on Simplicity and Generality”</p>
<blockquote>
<p>A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet it finally surrounds the resistant substance.</p>
</blockquote>

News
http://therisingsea.org/post/minicourse/
Sat, 26 Dec 2015 21:25:32 0500
http://therisingsea.org/post/minicourse/
<p>Recent papers and lectures:</p>
<ul>
<li>The <a href="http://therisingsea.org/post/mast30026/">notes and videos</a> for a course on Metric and Hilbert spaces (and <a href="https://gist.github.com/dmurfet/a5e1d06cfa558db85cecb7e655976d19">video HOWTO</a>).</li>
<li><a href="mailto:albert.y.zhang@gmail.com">Albert Zhang</a>’s masters thesis, October 2018, <a href="http://therisingsea.org/notes/MScThesisAlbertZhang.pdf">Vertex algebras, Hopf algebras and lattices</a> (and <a href="http://therisingsea.org/notes/albertthesistalk.pdf">talk slides</a>).</li>
<li><a href="https://arxiv.org/abs/1805.11813">Derivatives of Turing machines in Linear Logic</a> preprint with James Clift, May 2018.</li>
<li><a href="https://arxiv.org/abs/1805.10770">Encodings of Turing machines in Linear Logic</a> preprint with James Clift, May 2018.</li>
<li><a href="mailto:jamesedwardclift@gmail.com">James Clift</a>’s masters thesis, October 2017, <a href="http://therisingsea.org/notes/MScThesisJamesClift.pdf">Turing machines and differential linear logic</a>.</li>
<li><a href="mailto:p.cd.elliott@gmail.com">Patrick Elliott</a>’s masters thesis, October 2017, <a href="http://therisingsea.org/notes/MScThesisPatrickElliott.pdf">Ainfinity categories and matrix factorisations over hypersurface singularities</a>.</li>
</ul>
<p>Slides from my recent talks:</p>
<ul>
<li>Derivatives of Turing machines and inductive inference, Novermber 2018 at Peking University (<a href="http://therisingsea.org/notes/talkbeidatm2018.pdf">lecture notes</a>)</li>
<li>The computational content of LandauGinzburg models, November 2018 at BICMR, Beijing (<a href="http://therisingsea.org/notes/talkbeidalg2018.pdf">lecture notes</a>)</li>
<li>Derivatives of Turing machines in linear logic, May 2018 in the Melbourne pure math seminar (<a href="http://therisingsea.org/notes/talkturderiv.pdf">lecture notes</a>)</li>
<li>Algebra and Artificial Intelligence, May 2018 in the Melbourne logic seminar (<a href="http://therisingsea.org/notes/talkalgebraai.pdf">slides</a>  <a href="https://vimeo.com/268308026">video</a>)</li>
<li>Bar versus Koszul, April 2018 in the Melbourne topology seminar (<a href="http://therisingsea.org/notes/talkbarvskoszul.pdf">lecture notes</a>)</li>
<li>Minicourse on Ainfinity categories and matrix factorisations, September 2017 <a href="https://cgp.ibs.re.kr/conferences/String_Field_Theory/">at the IBS in Korea</a> (<a href="http://therisingsea.org/notes/talkibs20171.pdf">lecture 1</a>, <a href="http://therisingsea.org/notes/talkibs20172.pdf">lecture 2</a>, <a href="http://therisingsea.org/notes/talkibs20173.pdf">lecture 3</a>).</li>
<li>Turing machines and coalgebras, September 2017 at <a href="http://maths.anu.edu.au/events/60thbirthdayamnonneeman">Neeman’s 60th conference at ANU</a> (<a href="http://therisingsea.org/notes/talkloganu2017.pdf">lecture notes</a>).</li>
<li>Clifford algebras and 2D defect topological field theory, June 2017 at <a href="https://sites.google.com/view/tensorcategories2017/home">Tensor Categories and Field Theory</a> (<a href="http://therisingsea.org/notes/talkcliffordtft.pdf">lecture notes</a>).</li>
<li>A tour of wellgenerated triangulated categories, May 2017 at <a href="https://www.math.unibielefeld.de/birep/meetings/neeman2017/">Neeman’s 60th conference</a> (<a href="http://therisingsea.org/notes/talkneeman60th.pdf">lecture notes</a>).</li>
<li>Derivatives of proofs in linear logic (joint with James Clift), May 2017 in Melbourne (<a href="http://therisingsea.org/notes/logictalkdifflinearlogic.pdf">slides</a> and <a href="http://therisingsea.org/notes/logictalkdifflinearlogictranscript.pdf">transcript</a>).</li>
<li>The cobordism category, October 2016 in the <a href="http://therisingsea.org/post/seminartft/">TFT seminar</a> (<a href="http://therisingsea.org/notes/talk2cob.pdf">lecture notes</a>).</li>
<li>The CurryHoward principle, October 2016 in the <a href="http://therisingsea.org/post/seminarch/">CH seminar</a> (<a href="http://therisingsea.org/notes/talkch.pdf">lecture notes</a>).</li>
<li>The category of simplytyped lambda terms, September 2016 in the <a href="http://therisingsea.org/post/seminarch/">CH seminar</a> (<a href="http://therisingsea.org/notes/talkcatsimplytyped.pdf">lecture 1</a>, <a href="http://therisingsea.org/notes/talkcatsimplytyped2.pdf">lecture 2</a>).</li>
<li>Sheaves of Ainfinity algebras from matrix factorisations, September 2016 <a href="http://www.math.nagoyau.ac.jp/~ohta/conference/conference2016_1/">at the IPMU</a> (<a href="http://therisingsea.org/notes/talkipmuainfmf.pdf">lecture notes</a>).</li>
<li>Generalised orbifolding, September 2016 <a href="http://www.math.nagoyau.ac.jp/~ohta/conference/conference2016_1/">minicourse at the IPMU</a> (<a href="http://therisingsea.org/notes/talkipmugenorb1.pdf">lecture 1</a>, <a href="http://therisingsea.org/notes/talkipmugenorb2.pdf">lecture 2</a>, <a href="http://therisingsea.org/notes/talkipmugenorb3.pdf">lecture 3</a>).</li>
<li>Generalised orbifolding of simple singularities, August 2016 at <a href="https://mathspeople.anu.edu.au/~alperj/geometryattheanu.html">Geometry at the ANU</a> (<a href="http://therisingsea.org/notes/talkgenorb.pdf">lecture notes</a>).</li>
<li>Two odd things about computation, October 2015 in Vienna and August 2016 in Melbourne (<a href="http://therisingsea.org/notes/talktwothings.pdf">slides</a> and <a href="http://therisingsea.org/notes/talktwothingstranscript.pdf">transcript</a>).</li>
<li>Topological Quantum Field Theory in two dimensions, July 2016 in the <a href="http://therisingsea.org/post/seminartft/">TFT seminar</a> (<a href="http://therisingsea.org/notes/talk2dtqft.pdf">slides</a>).</li>
<li>Spectral sequences for vertex algebras, July 2016 in Melbourne (<a href="http://therisingsea.org/notes/talkspecseq.pdf">lecture notes</a>).</li>
<li>Linear logic and deep learning (joint with Huiyi Hu), June 2016 at the <a href="http://blogs.unimelb.edu.au/logic/aal2016/">AAL in Melbourne</a> (<a href="http://therisingsea.org/notes/talklldl.pdf">slides</a> and <a href="http://therisingsea.org/notes/talklldltranscript.pdf">transcript</a>).</li>
<li>Ainfinity algebras and matrix factorisations, June 2016 in <a href="http://www.birs.ca/events/2016/5dayworkshops/16w5040">Banff</a> (<a href="http://therisingsea.org/notes/talkainfmfbanff.pdf">lecture notes</a> and <a href="http://www.birs.ca/events/2016/5dayworkshops/16w5040/videos/watch/201606201531Murfet.html">video</a>).</li>
<li>The LandauGinzburg/Conformal Field Theory correspondence, May 2016 in Melbourne (<a href="http://therisingsea.org/notes/talklgcft.pdf">lecture notes</a>).</li>
<li>The superApolynomial and knot differentials, May 2016 in Melbourne (<a href="http://therisingsea.org/notes/talksupera.pdf">lecture notes</a>).</li>
<li>Ainfinity algebras and minimal models [Part 1], May 2016 in Melbourne (<a href="http://therisingsea.org/notes/talkainfminimal.pdf">lecture notes</a> and <a href="https://vimeo.com/165138188">screencast</a>).</li>
<li>Reading group on proofnets, April 2016 in the Melbourne logic seminar (<a href="http://therisingsea.org/notes/logicseminarproofnets.pdf">handout 1</a>, <a href="http://therisingsea.org/notes/logicseminarproofnets2.pdf">handout 2</a> and <a href="http://therisingsea.org/post/seminarproofnets/">notes</a>).</li>
<li>Stratifications and complexity in linear logic, March 2016 in the Melbourne logic seminar (<a href="http://therisingsea.org/notes/talkstratifications.pdf">slides</a> and <a href="https://vimeo.com/160036378">screencast</a>).</li>
<li>An introduction to Ainfinity algebras, November 2015 in Melbourne (<a href="http://therisingsea.org/notes/ainfintrotalk.pdf">lecture notes</a>).</li>
</ul>
<p>I gave a <a href="http://cgp.ibs.re.kr/conferences/MathematicalQuantumFieldTheory/">minicourse at the IBS in Korea</a> in January 2016 on topological field theory with defects, specifically the fusion of defects in topological LandauGinzburg models. This included several computer demos, the source for which can be found <a href="https://github.com/dmurfet/mf">on GitHub</a>.</p>
<ul>
<li>Lecture 1: 2D TFT with defects and matrix factorisations (<a href="http://therisingsea.org/notes/korealecture1.pdf">lecture notes</a> and <a href="https://vimeo.com/154577054">video</a>).</li>
<li>Lecture 2: The bicategory of LandauGinzburg models (<a href="http://therisingsea.org/notes/korealecture2.pdf">lecture notes</a> and <a href="https://vimeo.com/154711340">video</a>).</li>
<li>Lecture 3: The cut operation and computing fusions (<a href="http://therisingsea.org/notes/korealecture3.pdf">lecture notes</a> and <a href="https://vimeo.com/154843000">video</a>).</li>
</ul>
<p>These lectures were made with the excellent <a href="http://www.goodnotesapp.com/">GoodNotes</a> for the iPad Pro.</p>

Mass Surveillance
http://therisingsea.org/post/issues/
Sat, 26 Dec 2015 01:47:11 0500
http://therisingsea.org/post/issues/
<p>Mass surveillance is a serious threat to the ideal of free inquiry which makes mathematics (and other beautiful things) possible. Moreover, the technologies which enable it (e.g. machine learning) and the technologies which can be used to <a href="https://ssd.eff.org/en">protect oneself</a> from it (e.g. encryption) are themselves mathematical. Soon after the Snowden leaks my friend <a href="http://nils.carqueville.net/">Nils Carqueville</a> and I wrote <a href="http://nils.carqueville.net/MassSurveillanceEssay.pdf">an essay</a> on this subject, which you might find interesting.</p>

Lecture Notes
http://therisingsea.org/post/notes/
Sat, 26 Dec 2015 01:09:22 0500
http://therisingsea.org/post/notes/
<p>During my PhD under Amnon Neeman at the Australian National University, I took detailed notes. I am making some of these available in the hope that they may be useful. Many notes are heavily crossreferenced with other notes. I use an acronym system, where a reference of the form (MRS, Proposition 6) refers to my “Modules over Ringed Spaces” notes, for example. I have tried to document the relevant acronyms here, but there may be references to notes that I haven’t published online.</p>
<p>It goes without saying that most of the results in my notes are from one book or another. I’ve tried to list some of the main sources below, together with my shorthand for each book.</p>
<ul>
<li>Weibel: Weibel’s “An Introduction to Homological Algebra”.</li>
<li>H & S: Hilton & Stammbach’s “A Course in Homological Algebra”.</li>
<li>A & M: Atiyah & Macdonald’s “Introduction to Commutative Algebra”.</li>
<li>Z & S: Zariski and Samuel’s books on commutative algebra.</li>
<li>EFT: My Elementary Field Theory notes, based on Z & S’s chapter on field theory.</li>
<li>Mitchell: B. Mitchell’s “Category Theory”.</li>
<li>H or Hartshorne: Hartshorne’s “Algebraic Geometry”.</li>
</ul>
<h4 id="algebraicgeometry:db235fc914867ffea6b540c8732499d5">Algebraic Geometry</h4>
<ul>
<li><a href="http://therisingsea.org/notes/EGA1.pdf">Introduction to EGA I</a>: The motivating ideas of modern algebraic geometry, presented beautifully by Grothendieck (translated with the help of Tamah Murfet, way back in 2003).</li>
<li><a href="http://therisingsea.org/notes/SheavesOfSetsGroupsRings.pdf">Sheaves of Groups and Rings</a>: (SGR) Sheaves of sets (incomplete), sheaves of abelian groups, stalks, sheaf Hom, tensor products, inverse and direct image, extension by zero.</li>
<li><a href="http://therisingsea.org/notes/RingedSpaceModules.pdf">Modules over a Ringed Space</a>: (MRS) Inverse and direct image, tensor products, ideals, locally free sheaves, exponential tensor products, sheaf Hom, coinverse image and extension by zero. Sheaves of graded modules over sheaves of graded rings, quasistructures, modules over schemes, sheaves of algebras and sheaves of graded algebras (Quite rough in places, I’m in the process of typing written notes).</li>
<li><a href="http://therisingsea.org/notes/SheavesOfAlgebras.pdf">Sheaves of Algebras</a>: (SOA) Direct and inverse image for algebras, modules over sheaves of algebras, ideals, generating algebras, tensor products. Sheaves of graded algebras, their modules, and generating graded algebras. Sheaves of super algebras and tensor products.</li>
<li><a href="http://therisingsea.org/notes/SpecialSheavesOfAlgebras.pdf">Special Sheaves of Algebras</a>: (SSA) Sheaves of tensor algebras, symmetric algebras, exterior algebras, polynomial algebras and ideal products. Complete study of these constructions as adjoints.</li>
<li><a href="http://therisingsea.org/notes/ModulesOverAScheme.pdf">Modules over a Scheme</a>: (MOS) Ideals, special functors (extension by zero and coextension of scalars), locally free sheaves, sheaf Hom, extension of coherent sheaves.</li>
<li><a href="http://therisingsea.org/notes/TheProjConstruction.pdf">The Proj Construction</a>: (TPC) Functorial properties, products, linear morphisms, projective morphisms, dimensions of some schemes, points of projective space.</li>
<li><a href="http://therisingsea.org/notes/ModulesOverProjectiveSchemes.pdf">Modules over Projective Schemes</a>: (MPS) Properties of the functor associating a graded module with a quasicoherent sheaf on Proj.</li>
<li><a href="http://therisingsea.org/notes/AdjointModulesProj.pdf">An Adjunction for Modules over Projective Schemes</a>: (AAMPS) The adjoint for taking the associated sheaf of a graded module, the quasicoherent case.</li>
<li><a href="http://therisingsea.org/notes/EquivalenceModulesProj.pdf">An Equivalence for Modules over Projective Schemes</a>: (AEMPS) The projective version of some important theorems in the affine case.</li>
<li><a href="http://therisingsea.org/notes/RelativeAffineSchemes.pdf">Relative Affine Schemes</a>: (RAS) Affine morphisms, the Spec construction, the sheaf associated to a sheaf of quasicoherent modules over an algebra.</li>
<li><a href="http://therisingsea.org/notes/SegreEmbedding.pdf">The Segre Embedding</a>: (SEM) Pullback of Proj schemes, properties of projective morphisms.</li>
<li><a href="http://therisingsea.org/notes/ConcentratedSchemes.pdf">Concentrated Schemes</a>: (CON) Basic properties of quasicompact and quasiseparated schemes and morphisms. In our notes a quasicompact quasiseparated morphism (or scheme) is called concentrated.</li>
<li><a href="http://therisingsea.org/notes/Section2.6Divisors.pdf">Section 2.6  Divisors</a>: (DIV) Weil divisors, divisors on curves, cartier divisors, invertible sheaves.</li>
<li><a href="http://therisingsea.org/notes/Section2.7ProjectiveMorphisms.pdf">Section 2.7  Projective Morphisms</a>: (PM) Morphisms to Pn, the duple embedding, ample invertible sheaves, linear systems.</li>
<li><a href="http://therisingsea.org/notes/Section2.7.1BlowingUp.pdf">Section 2.7.1  Blowing Up</a>: (BU) Definition of the blowup, blowing up of varieties.</li>
<li><a href="http://therisingsea.org/notes/Section2.8Differentials.pdf">Section 2.8  Differentials</a>: (DIFF) Kahler differentials, sheaves of differentials, nonsingular varieties, rational maps, applications, some local algebra.</li>
<li><a href="http://therisingsea.org/notes/Section2.9FormalSchemes.pdf">Section 2.9  Formal Schemes</a>: (FS) Inverse limits, completion, adic rings (complete rings of fractions, local completion), affine formal schemes (this note is not yet complete).</li>
<li><a href="http://therisingsea.org/notes/Section3.2CohomologyOfSheaves.pdf">Section 3.2  Cohomology of Sheaves</a>: (COS) Definition of cohomology, the module structure and the presheaf of cohomology. A vanishing theorem of grothendieck, cohomology of noetherian schemes, Cech cohomology, the cohomology of projective space, Ext groups and sheaves.</li>
<li><a href="http://therisingsea.org/notes/Section3.8HigherDirectImageOfSheaves.pdf">Section 3.8  Higher Direct Images of Sheaves</a>: (HDIS) Definition of higher direct image functors, module structure and properties for schemes. Definition of the higher coinverse image functors, and their properties. Direct image and quasicoherent sheaves, uniqueness of cohomology.</li>
<li><a href="http://therisingsea.org/notes/Section3.7SerreDuality.pdf">Section 3.7  Serre Duality</a>: (SDT) Notes on Serre Duality and dualising sheaves as presented in Hartshorne.</li>
<li><a href="http://therisingsea.org/notes/TheRelativeProjConstruction.pdf">The Relative Proj Construction</a>: (TRPC) Associating a Proj with a sheaf of graded algebras. The sheaf associated to a sheaf of graded modules, the graded module associated to a quasicoherent sheaf, functorial properties, ideal sheaves and closed subschemes, the duple embedding, twisting with invertible sheaves.</li>
<li><a href="http://therisingsea.org/notes/AmpleFamilies.pdf">Ample Families</a>: (AMF) Ample sheaves and ample families of sheaves on arbitrary schemes, as described in EGA and later SGA.</li>
<li><a href="http://therisingsea.org/notes/SchemesFromLocalisation.pdf">Schemes via Noncommutative Localisation</a>: (SFL) Gabriel topologies and localisation with respect to them, the situation for commutative rings and how this relates to algebraic geometry. A lot of this is from Stenstrom’s book.</li>
<li><a href="http://therisingsea.org/notes/ZariskiTopology.pdf">The Zariski site</a>: (ZT) Definition of the Zariski site and the proof that schemes give sheaves on it. Probably directly from EGA, but I don’t recall.</li>
</ul>
<h4 id="commutativealgebra:db235fc914867ffea6b540c8732499d5">Commutative Algebra</h4>
<ul>
<li><a href="http://therisingsea.org/notes/Matsumura.pdf">Matsumura</a>: (MAT) General rings, flatness, depth, CohenMacaulay rings, normal and regular rings, koszul complexes, unique factorisation.</li>
<li><a href="http://therisingsea.org/notes/MatsumuraPart2.pdf">Matsumura Part II</a>: (MAT2) Extension of a ring by a module, derivations and differentials, separability.</li>
<li><a href="http://therisingsea.org/notes/NoetherNormalisation.pdf">Noether Normalisation</a>: First introduction to various versions of Noether normalisation.</li>
<li><a href="http://therisingsea.org/notes/MoreNoetherNormalisation.pdf">More Noether Normalisation</a>: A version of noether normalisation involving separability.</li>
<li><a href="http://therisingsea.org/notes/HenselsLemma.pdf">Hensel’s Lemma</a>: Hensel’s Lemma and a few small examples.</li>
<li><a href="http://therisingsea.org/notes/CohensTheorem.pdf">Cohen’s Theorem</a>.</li>
<li><a href="http://therisingsea.org/notes/GradedModules.pdf">Graded Rings and Modules</a>: (GRM) Definitions and basic properties, the category of graded modules, quasistructures, grading tensor products.</li>
<li><a href="http://therisingsea.org/notes/TensorExteriorSymmetric.pdf">Tensor, Exterior and Symmetric algebras</a>: (TES) The tensor algebra and properties, exterior algebra and properties, including: dimension theorems, the determinant formula (i.e. highest exterior powers), and duality properties, the symmetric algebra and properties.</li>
<li><a href="http://therisingsea.org/notes/AutoPowerSeries.pdf">Automorphisms of Power Series Rings</a>: (APSR) Constructing automorphisms of power series rings from an independent family of power series.</li>
<li><a href="http://therisingsea.org/notes/TopologicalRings.pdf">Topological Rings</a>: Topological groups and rings, fundamental systems of ideals and preparation for Gabriel topologies.</li>
</ul>
<h4 id="categorytheoryandnoncommutativealgebra:db235fc914867ffea6b540c8732499d5">Category Theory and Noncommutative Algebra</h4>
<ul>
<li><a href="http://therisingsea.org/notes/BasicSetTheory.pdf">Basic Set Theory</a>: (BST) Ordinal numbers, transfinite induction, cardinal numbers, cardinal operations, regular cardinals.</li>
<li><a href="http://therisingsea.org/notes/FoundationsForCategoryTheory.pdf">Foundations for Category Theory</a>: (FCT) Outline of the problem of foundations in category theory, first order theories, NBG and associated problems, review of ZFC and grothendieck universes. This forms the logical background for all my notes.</li>
<li><a href="http://therisingsea.org/notes/AbelianCategories.pdf">Abelian Categories</a>: (AC) Definition of categories, limits and colimits, functor categories, pointwise limits and colimits, adjoint functors, abelian categories, grothendieck abelian categories and reflective subcategories (mostly just to fix notation. These notes are not a complete reference on category theory).</li>
<li><a href="http://therisingsea.org/notes/DiagramChasingInAbelianCategories.pdf">Diagram Chasing in Abelian Categories</a>: (DCAC) Proving the Five Lemma in an abelian category, using an embedding to establish diagram chasing in arbitrary abelian categories (we use only the “first” embedding theorem into the category of abelian groups, since its proof is more accessible, and you probably have to do the same amount of work to avoid welldefinedness issues for the connecting morphism even with the better embeddings).</li>
<li><a href="http://therisingsea.org/notes/DerivedFunctors.pdf">Derived Functors</a>: (DF) (co)chain complexes in an abelian category, (co)homology, projective and injective resolutions, left and right derived functors of additive functors between abelian categories, long exact (co)homology sequences, long exact sequences of derived functors, dimension shifting and acyclic resolutions, change of base, homology and colimits, cohomology and limits, delta functors.</li>
<li><a href="http://therisingsea.org/notes/TriangulatedCategories.pdf">Triangulated Categories Part I</a>: (TRC) [Verdier quotients] Triangulated categories, triangulated functors, homotopy colimits, localising subcategories, right derived functors, left derived functors, portly considerations.</li>
<li><a href="http://therisingsea.org/notes/TriangulatedCategoriesPart2.pdf">Triangulated Categories Part II</a>: (TRC2) [Thomason localisation] Finer localising subcategories, perfect classes, small objects, compact objects, portly considerations, morphisms in the quotient.</li>
<li><a href="http://therisingsea.org/notes/TriangulatedCategoriesPart3.pdf">Triangulated Categories Part III</a>: (TRC3) [Brown representability] Representability theorems, compactly generated triangulated categories.</li>
<li><a href="http://therisingsea.org/notes/DerivedCategories.pdf">Derived Categories Part I</a>: (DTC) Homotopy categories, derived categories (extending functors, introduction to hearts, bounded derived categories), homotopy resolutions, homotopy direct limits, bousfield subcategories, existence of resolutions.</li>
<li><a href="http://therisingsea.org/notes/DerivedCategoriesPart2.pdf">Derived Categories Part II</a>: (DTC2) Derived functors, derived Hom, derived Tensor, brown representability.</li>
<li><a href="http://therisingsea.org/notes/DerivedCategoriesOfSheaves.pdf">Derived Categories Of Sheaves</a>: (DCOS) Representing cohomology, derived direct image, derived sheaf Hom, derived Tensor, derived inverse image.</li>
<li><a href="http://therisingsea.org/notes/DerivedCategoriesOfQuasicoherentSheaves.pdf">Derived Categories Of Quasicoherent Sheaves</a>: (DCOQS) Derived direct image, Derived inverse image, Sheaves with quasicoherent cohomology, local cohomology triangle, resolutions by Cech sheaves and the Cech triangles, Neeman’s unbounded Grothendieck duality, comparing the derived category of quasicoherent sheaves with the derived category of sheaves with quasicoherent cohomology, quasicoherent hypercohomology, perfect complexes and compactness, projection formula and friends.</li>
<li><a href="http://therisingsea.org/notes/Ext.pdf">Ext</a>: (EXT) Ext in general abelian categories, using injectives and projectives and balancing the two, Ext for linear categories, dimension shifting, Ext and coproducts, Ext for commutative rings, another characterisation of derived functors.</li>
<li><a href="http://therisingsea.org/notes/Tor.pdf">Tor</a>: (TOR) Tor on the left and right and balancing the two, dimension shifting, Tor and colimits, Tor for commutative rings and bimodules, criteria for flatness.</li>
<li><a href="http://therisingsea.org/notes/Dimensions.pdf">Dimensions</a>: (DIM) Projective dimension, injective dimension, global dimension, flat dimension and change of rings.</li>
<li><a href="http://therisingsea.org/notes/SpectralSequences.pdf">Spectral Sequences</a>: (SS) Definition, basic convergence properties, the spectral sequence of a complex filtration, the two spectral sequences of a bicomplex, the Grothendieck spectral sequence.</li>
<li><a href="http://therisingsea.org/notes/AlgebraInACategory.pdf">Algebra in a Category</a>: (ALCAT) Groups, rings and modules in an arbitrary category. Sheaves of groups, rings and modules and graded versions.</li>
<li><a href="http://therisingsea.org/notes/LinearisedCategories.pdf">Linearised Categories</a>: (LC) Generalise the group ring construction to the linearisation of any small category with respect to a sheaf of rings, the graded version of this construction. Includes proof that the category of graded sheaves of modules is grothendieck abelian.</li>
<li><a href="http://therisingsea.org/notes/Stenstrom.pdf">Rings of Quotients</a>: My notes from Stenstrom’s book “Rings and Modules of Quotients”, covering some basic material on modules, rings of fractions for noncommutative rings (Ore condition etc), Gabriel topologies, torsion theories, localisation with resect to a Gabriel topology, Giraud subcategories and their classification theorems.</li>
</ul>

Reading group on proofnets
http://therisingsea.org/post/seminarproofnets/
Mon, 26 Oct 2015 21:25:32 0500
http://therisingsea.org/post/seminarproofnets/
<p>These are the notes for the reading group on proofnets in linear logic, spanning April 8 and 15 of the <a href="http://blogs.unimelb.edu.au/logic/">logic seminar</a> at the University of Melbourne. The aim is to:</p>
<ul>
<li><strong>April 8</strong> (<a href="http://therisingsea.org/notes/logicseminarproofnets.pdf">handout</a>): understand the definition of proofnets and their cutelimination procedure, and see the statement of the two main theorems in the theory: the Sequentialisation Theorem (which identifies those proofnets coming from sequent calculus proofs) and the Strong Normalisation Theorem.</li>
<li><strong>April 15</strong> (<a href="http://therisingsea.org/notes/logicseminarproofnets2.pdf">handout</a>): work through details of the proof that in stratified linear logic, cutelimination is achieved in polynomial time (Theorem 16 of Baillot and Mazza’s “Linear logic by levels”) as stated in my <a href="http://therisingsea.org/notes/talkstratifications.pdf">previous talk</a> without details.</li>
</ul>
<p>The main references are:</p>
<ul>
<li>[G87] J.Y. Girard’s original paper “<a href="http://iml.univmrs.fr/~girard/linear.pdf">Linear logic</a>“</li>
<li>[G96] J.Y. Girard “<a href="http://iml.univmrs.fr/~girard/Proofnets.pdf.gz">Proofnets: the parallel syntax for prooftheory</a>“</li>
<li>[J91] J. Davoren “<a href="https://blogs.unimelb.edu.au/logic/files/2015/11/DavorenLLGLL2cedcbe.pdf">A Lazy Logician’s Guide to Linear Logic</a>“</li>
<li>[BM09] P. Baillot and D. Mazza “<a href="http://arxiv.org/abs/0801.1253">Linear Logic by Levels and Bounded Time Complexity</a>“</li>
<li>[PTF09] M. Pagani and L. Tortora de Falco “<a href="http://www.pps.univparisdiderot.fr/~pagani/snllTCS1.pdf">Strong Normalization Property for Second Order Linear Logic</a>“</li>
</ul>
<p>Here is a rough plan that makes sense to me, for the first seminar:</p>
<ol>
<li>General background on linear logic ([J91] Sections 0, 1, 3 and then [BM09] Section 1). Ideally we all would have skimmed this before Friday, to refresh our memories.</li>
<li>Definition of proofnets up to the definition of depth ([BM09] from p.8 to p.10).</li>
<li>Some examples of proofnets (Church numerals from [G87] Section 5.3.2 p. 86 and binary integers from p.26 of [BM09]).</li>
<li>Definition of cutelimination transformations and statement of Strong Normalisation Theorem ([BM09] p.12, p.13 and [PTF09]). This was proven in [G87] for a subsystem but only recently in [PTF09] for full linear logic.</li>
<li>Sequentialisation of sequent calculus proofs to proofnets ([BM09] p.11).</li>
<li>Examples of proofnets that are not sequentialisable, and proofnets that are the sequentialisation of multiple sequent calculus proofs; discussion of the advantages of proofnets vs sequent calculus ([J91] p.140, p.156, p.157).</li>
<li>More complicated examples, with duplication of nested boxes.</li>
<li>Definition of switchings and statement of the Sequentialisation Theorem (very brief statement in [BM09] Proposition 2, details from [G96], examples from [J91] Section 6).</li>
<li>If there is any time remaining, some details of the proof of the Sequentialisation Theorem from [G96].</li>
</ol>
<p>The canonical reference for proofnets and the Sequentialisation Theorem is Girard [G96], but in order to have notational consistency with the second seminar on light linear logic, and to see an overview free of complicating details, I think [BM09] is a better starting point for us. This means we would view [G96], [J91], [PTF09] as augmenting references for the real details (which are completely absent from [BM09]).</p>
<p>A rough plan for the second seminar, which will be taken almost entirely from [BM09].</p>
<ol>
<li>A brief recall of the relation between stratification and complexity from my earlier talk (<a href="http://therisingsea.org/notes/talkstratifications.pdf">slides</a> and <a href="https://vimeo.com/160036378">screencast</a>).</li>
<li>A brief recall of proofnets and their cutelimination steps from last time.</li>
<li>The definition of stratified proofnets (mL3 in BaillotMazza) from Section 2.1 of [BM09].</li>
<li>Then Section 3 of [BM09] in its entirety, which has three parts (A) weak normalisation for cutelimination in untyped stratified proofnets (Proposition 13) (B) the characterisation of elementary time by stratification (Theorem 16) © the characterisation of polynomial time by stratification (Theorem 23).</li>
</ol>

References on computation
http://therisingsea.org/post/comp/
Mon, 26 Oct 2015 21:25:32 0500
http://therisingsea.org/post/comp/
<p>My interest in the mathematical theory of computation is primarily due to the work of the brilliant logician <a href="https://en.wikipedia.org/wiki/JeanYves_Girard">JeanYves Girard</a> on <a href="http://plato.stanford.edu/entries/logiclinear/">linear logic</a>. Below is a (not very comprehensive) list of references for the parts of the theory of computation that I either know a little about, or am interested in.</p>
<p>General “big picture” references on computation and logic:</p>
<ul>
<li><a href="https://global.oup.com/academic/product/thenatureofcomputation9780199233212?cc=us&lang=en&">The Nature of Computation</a>, a textbook by Cristopher Moore and Stephan Mertens.</li>
<li><a href="https://jb55.com/linear/pdf/Towards%20a%20geometry%20of%20interaction.pdf">Towards a geometry of interaction</a> by JeanYves Girard.</li>
<li><a href="http://iml.univmrs.fr/~girard/coursang/coursang.html">The Blind Spot</a> also by Girard, quite eclectic and unpolished, but full of interesting ideas (especially the sections on Russell’s paradox and complexity).</li>
<li><a href="http://www.cs.tufts.edu/~nr/cs257/archive/permartinlof/constructivemath.pdf">Constructive mathematics and computer programming</a> by MartinLof.</li>
<li><a href="http://arxiv.org/abs/1403.4880">Two puzzles about computation</a> by Samson Abramsky.</li>
</ul>
<p>Some references on lambdacalculus, System F and proof theory:</p>
<ul>
<li><a href="http://www.paultaylor.eu/stable/prot.pdf">Proofs and types</a> by JeanYves Girard.</li>
</ul>
<p>Some references on computational complexity theory:</p>
<ul>
<li><a href="http://www.scottaaronson.com/papers/philos.pdf">Why philosophers should care about computational complexity</a> by Scott Aaronson.</li>
<li><a href="http://stationq.cnsi.ucsb.edu/~freedman/Publications/67.pdf">Topological views on computational complexity</a> by Michael Freedman (Fields medallist and now <a href="http://research.microsoft.com/enus/press/mfreedman.aspx">director of Station Q</a>).</li>
<li><a href="http://iml.univmrs.fr/~girard/LLL.pdf.gz">Light linear logic</a> by JeanYves Girard.</li>
<li><a href="http://www.kurims.kyotou.ac.jp/~terui/phd.pdf">Light logic and polynomial time computation</a> by Kazushige Terui.</li>
</ul>
<p>Relations between computation and category theory:</p>
<ul>
<li><a href="https://www.elsevier.com/books/lecturesonthecurryhowardisomorphism/srensen/9780444520777">Lectures on the CurryHoward isomorphism</a> by Morten Heine Sorensen and Pawel Urzyczyn.</li>
<li><a href="http://math.ucr.edu/home/baez/rosetta.pdf">Physics, Topology, Logic and Computation:
A Rosetta Stone</a> by John Baez and Mike Stay.</li>
<li><a href="http://www.pps.univparisdiderot.fr/~mellies/mpri/mpriens/biblio/categoricalsemanticsoflinearlogic.pdf">Categorical semantics of linear logic</a> by PaulAndre Mellies.</li>
<li><a href="https://core.ac.uk/download/files/145/21173011.pdf">Notions of computation and monads</a> by Eugenio Moggi.</li>
<li><a href="http://homotopytypetheory.org/">Homotopy type theory</a> is a semantics of MartinLof type theory defined using homotopy theory.</li>
</ul>
<p>Relations between computation and physics:</p>
<ul>
<li><a href="http://www.amazon.com/FeynmanLecturesOnComputationRichard/dp/0738202967">The Feynman lectures on computation</a>, seminal work in the field of quantum computing, also has a good discussion of basics including Bennett’s work on irreversibility (find a PDF on bookzz).</li>
<li><a href="http://www.cs.cornell.edu/selman/papers/pdf/99.nature.phase.pdf">Complexity and phase transitions</a>, Nature paper from 1999.</li>
<li>There is a deep and interesting connection between Maxwell’s demon and topics in computation, for which see <a href="https://www.weizmann.ac.il/complex/tlusty/courses/InfoInBio/Papers/Szilard1929.pdf">Szilard</a>, <a href="http://adsabs.harvard.edu/abs/1961AmJPh..29..318B">Brillouin</a>, <a href="http://worrydream.com/refs/Landauer%20%20Irreversibility%20and%20Heat%20Generation%20in%20the%20Computing%20Process.pdf">Landauer</a> and the papers of Bennett below.</li>
<li><a href="http://www.pitt.edu/~jdnorton/lectures/Rotman_Summer_School_2013/thermo_computing_docs/Bennett_1982.pdf">The thermodynamics of computation</a> by Bennett 1982.</li>
<li><a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.364.3669&rep=rep1&type=pdf">Timespace tradeoffs for reversible computation</a> by Bennett.</li>
</ul>
<p>Some interesting videos:</p>
<ul>
<li><a href="https://www.youtube.com/watch?v=7cPtCpyBPNI">Lambda calculus, then and now</a> by Dana Scott.</li>
<li><a href="https://www.youtube.com/watch?v=IOiZatlZtGU">Propositions as types</a> by Philip Wadler.</li>
<li><a href="https://www.youtube.com/watch?v=8pTEmbeENF4">The future of programming</a> by Bret Victor.</li>
</ul>