The Rising Sea
Sun, Jan 3, 2016
My name is Daniel Murfet, I am a Lecturer (aka tenure-track Assistant Professor) in the
at the University of Melbourne and part of the . My CV is
and you can contact me by
or find me . My papers are
with the exception of my PhD thesis which you can find . My primary research interests are in algebraic geometry, category theory and mathematical logic. You can see some of the things I like to think about in my
on matrix factorisations and my . I currently run two seminars: one on the
and another on . I have various code projects
including my joint project with Nils Carqueville on , and my joint project with Huiyi Hu on .
The title of this webpage refers to the following quote from Grothendieck, which I learned from Colin McLarty&s
&The Rising Sea: Grothendieck on Simplicity and Generality&
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.
Sat, Dec 26, 2015
Slides from my recent talks:
The cobordism category, October 2016 in the
The Curry-Howard principle, October 2016 in the
The category of simply-typed lambda terms, September 2016 in the
Sheaves of A-infinity algebras from matrix factorisations, September 2016
Generalised orbifolding, September 2016
Generalised orbifolding of simple singularities, August 2016 at
Two odd things about computation, October 2015 in Vienna and August 2016 in Melbourne ( and ).
Topological Quantum Field Theory in two dimensions, July 2016 in the
Spectral sequences for vertex algebras, July 2016 in Melbourne ().
Linear logic and deep learning (joint with Huiyi Hu), June 2016 at the
A-infinity algebras and matrix factorisations, June 2016 in
The Landau-Ginzburg/Conformal Field Theory correspondence, May 2016 in Melbourne ().
The super-A-polynomial and knot differentials, May 2016 in Melbourne ().
A-infinity algebras and minimal models [Part 1], May 2016 in Melbourne ( and ).
Reading group on proof-nets, April 2016 in the Melbourne logic seminar (,
Stratifications and complexity in linear logic, March 2016 in the Melbourne logic seminar ( and ).
An introduction to A-infinity algebras, November 2015 in Melbourne ().
on topological field theory with defects, specifically the fusion of defects in topological Landau-Ginzburg models. This included several computer demos, the source for which can be found .
Lecture 1: 2D TFT with defects and matrix factorisations (
Lecture 2: The bicategory of Landau-Ginzburg models ( and ).
Lecture 3: The cut operation and computing fusions ( and ).
These lectures were made with the excellent
for the iPad Pro.
Sat, Dec 26, 2015
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
from it (e.g. encryption) are themselves mathematical. Soon after the Snowden leaks my friend
and I wrote
on this subject, which you might find interesting.
Sat, Dec 26, 2015
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 cross-referenced 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.
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.
Weibel: Weibel&s &An Introduction to Homological Algebra&.
H & S: Hilton & Stammbach&s &A Course in Homological Algebra&.
A & M: Atiyah & Macdonald&s &Introduction to Commutative Algebra&.
Z & S: Zariski and Samuel&s books on commutative algebra.
EFT: My Elementary Field Theory notes, based on Z & S&s chapter on field theory.
Mitchell: B. Mitchell&s &Category Theory&.
H or Hartshorne: Hartshorne&s &Algebraic Geometry&.
Algebraic Geometry
: The motivating ideas of modern algebraic geometry, presented beautifully by Grothendieck (translated with the help of Tamah Murfet, way back in 2003).
: (SGR) Sheaves of sets (incomplete), sheaves of abelian groups, stalks, sheaf Hom, tensor products, inverse and direct image, extension by zero.
: (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, quasi-structures, modules over schemes, sheaves of algebras and sheaves of graded algebras (Quite rough in places, I&m in the process of typing written notes).
: (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.
: (SSA) Sheaves of tensor algebras, symmetric algebras, exterior algebras, polynomial algebras and ideal products. Complete study of these constructions as adjoints.
: (MOS) Ideals, special functors (extension by zero and coextension of scalars), locally free sheaves, sheaf Hom, extension of coherent sheaves.
: (TPC) Functorial properties, products, linear morphisms, projective morphisms, dimensions of some schemes, points of projective space.
: (MPS) Properties of the functor associating a graded module with a quasi-coherent sheaf on Proj.
: (AAMPS) The adjoint for taking the associated sheaf of a graded module, the quasi-coherent case.
: (AEMPS) The projective version of some important theorems in the affine case.
: (RAS) Affine morphisms, the Spec construction, the sheaf associated to a sheaf of quasi-coherent modules over an algebra.
: (SEM) Pullback of Proj schemes, properties of projective morphisms.
: (CON) Basic properties of quasi-compact and quasi-separated schemes and morphisms. In our notes a quasi-compact quasi-separated morphism (or scheme) is called concentrated.
: (DIV) Weil divisors, divisors on curves, cartier divisors, invertible sheaves.
: (PM) Morphisms to Pn, the duple embedding, ample invertible sheaves, linear systems.
: (BU) Definition of the blow-up, blowing up of varieties.
: (DIFF) Kahler differentials, sheaves of differentials, nonsingular varieties, rational maps, applications, some local algebra.
: (FS) Inverse limits, completion, adic rings (complete rings of fractions, local completion), affine formal schemes (this note is not yet complete).
: (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.
: (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 quasi-coherent sheaves, uniqueness of cohomology.
: (SDT) Notes on Serre Duality and dualising sheaves as presented in Hartshorne.
: (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 quasi-coherent sheaf, functorial properties, ideal sheaves and closed subschemes, the duple embedding, twisting with invertible sheaves.
: (AMF) Ample sheaves and ample families of sheaves on arbitrary schemes, as described in EGA and later SGA.
: (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.
: (ZT) Definition of the Zariski site and the proof that schemes give sheaves on it. Probably directly from EGA, but I don&t recall.
Commutative Algebra
: (MAT) General rings, flatness, depth, Cohen-Macaulay rings, normal and regular rings, koszul complexes, unique factorisation.
: (MAT2) Extension of a ring by a module, derivations and differentials, separability.
: First introduction to various versions of Noether normalisation.
: A version of noether normalisation involving separability.
: Hensel&s Lemma and a few small examples.
: (GRM) Definitions and basic properties, the category of graded modules, quasi-structures, grading tensor products.
: (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.
: (APSR) Constructing automorphisms of power series rings from an independent family of power series.
: Topological groups and rings, fundamental systems of ideals and preparation for Gabriel topologies.
Category Theory and Noncommutative Algebra
: (BST) Ordinal numbers, transfinite induction, cardinal numbers, cardinal operations, regular cardinals.
: (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.
: (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).
: (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 well-definedness issues for the connecting morphism even with the better embeddings).
: (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.
: (TRC) [Verdier quotients] Triangulated categories, triangulated functors, homotopy colimits, localising subcategories, right derived functors, left derived functors, portly considerations.
: (TRC2) [Thomason localisation] Finer localising subcategories, perfect classes, small objects, compact objects, portly considerations, morphisms in the quotient.
: (TRC3) [Brown representability] Representability theorems, compactly generated triangulated categories.
: (DTC) Homotopy categories, derived categories (extending functors, introduction to hearts, bounded derived categories), homotopy resolutions, homotopy direct limits, bousfield subcategories, existence of resolutions.
: (DTC2) Derived functors, derived Hom, derived Tensor, brown representability.
: (DCOS) Representing cohomology, derived direct image, derived sheaf Hom, derived Tensor, derived inverse image.
: (DCOQS) Derived direct image, Derived inverse image, Sheaves with quasi-coherent cohomology, local cohomology triangle, resolutions by Cech sheaves and the Cech triangles, Neeman&s unbounded Grothendieck duality, comparing the derived category of quasi-coherent sheaves with the derived category of sheaves with quasi-coherent cohomology, quasi-coherent hypercohomology, perfect complexes and compactness, projection formula and friends.
: (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.
: (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.
: (DIM) Projective dimension, injective dimension, global dimension, flat dimension and change of rings.
: (SS) Definition, basic convergence properties, the spectral sequence of a complex filtration, the two spectral sequences of a bicomplex, the Grothendieck spectral sequence.
: (ALCAT) Groups, rings and modules in an arbitrary category. Sheaves of groups, rings and modules and graded versions.
: (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.
: 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.
Mon, Oct 26, 2015
These are the notes for the reading group on proof-nets in linear logic, spanning April 8 and 15 of the
at the University of Melbourne. The aim is to:
April 8 (): understand the definition of proof-nets and their cut-elimination procedure, and see the statement of the two main theorems in the theory: the Sequentialisation Theorem (which identifies those proof-nets coming from sequent calculus proofs) and the Strong Normalisation Theorem.
April 15 (): work through details of the proof that in stratified linear logic, cut-elimination is achieved in polynomial time (Theorem 16 of Baillot and Mazza&s &Linear logic by levels&) as stated in my
without details.
The main references are:
[G87] J.Y. Girard&s original paper &&
[G96] J.Y. Girard &&
[J91] J. Davoren &&
[BM09] P. Baillot and D. Mazza &&
[PTF09] M. Pagani and L. Tortora de Falco &&
Here is a rough plan that makes sense to me, for the first seminar:
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.
Definition of proof-nets up to the definition of depth ([BM09] from p.8 to p.10).
Some examples of proof-nets (Church numerals from [G87] Section 5.3.2 p. 86 and binary integers from p.26 of [BM09]).
Definition of cut-elimination 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.
Sequentialisation of sequent calculus proofs to proof-nets ([BM09] p.11).
Examples of proof-nets that are not sequentialisable, and proof-nets that are the sequentialisation of multiple seq discussion of the advantages of proof-nets vs sequent calculus ([J91] p.140, p.156, p.157).
More complicated examples, with duplication of nested boxes.
Definition of switchings and statement of the Sequentialisation Theorem (very brief statement in [BM09] Proposition 2, details from [G96], examples from [J91] Section 6).
If there is any time remaining, some details of the proof of the Sequentialisation Theorem from [G96].
The canonical reference for proof-nets 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]).
A rough plan for the second seminar, which will be taken almost entirely from [BM09].
A brief recall of the relation between stratification and complexity from my earlier talk ( and ).
A brief recall of proof-nets and their cut-elimination steps from last time.
The definition of stratified proof-nets (mL3 in Baillot-Mazza) from Section 2.1 of [BM09].
Then Section 3 of [BM09] in its entirety, which has three parts (A) weak normalisation for cut-elimination in untyped stratified proof-nets (Proposition 13) (B) the characterisation of elementary time by stratification (Theorem 16) & the characterisation of polynomial time by stratification (Theorem 23).
Mon, Oct 26, 2015
My interest in the mathematical theory of computation is primarily due to the work of the brilliant logician
on . 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. The books can generally be downloaded in PDF form from , but do buy a copy as well.
General &big picture& references on computation and logic:
, a textbook by Cristopher Moore and Stephan Mertens.
by Jean-Yves Girard.
also by Girard, quite eclectic and unpolished, but full of interesting ideas (especially the sections on Russell&s paradox and complexity).
by Martin-Lof.
by Samson Abramsky.
Some references on lambda-calculus, System F and proof theory:
by Jean-Yves Girard.
Some references on computational complexity theory:
by Scott Aaronson.
by Michael Freedman (Fields medallist and now ).
by Jean-Yves Girard.
by Kazushige Terui.
Relations between computation and category theory:
by Morten Heine Sorensen and Pawel Urzyczyn.
by John Baez and Mike Stay.
by Paul-Andre Mellies.
by Eugenio Moggi.
is a semantics of Martin-Lof type theory defined using homotopy-theory.
Relations between computation and physics:
, 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).
, Nature paper from 1999.
There is a deep and interesting connection between Maxwell&s demon and topics in computation, for which see , ,
and the papers of Bennett below.
by Bennett 1982.
by Bennett.
Some interesting videos:
by Dana Scott.
by Philip Wadler.
by Bret Victor.
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