My name is Daniel Murfet, I am a mathematician in the School of Mathematics and Statistics at the University of Melbourne. My CV is here and you can contact me by email or on Discord. My papers are on the arXiv with the exception of my PhD thesis which you can find here. I post videos on a YouTube channel and hold seminars at metauni.

My research interests are in algebraic geometry, mathematical logic and deep learning. You can see some of the things I like to think about in my Spring 2013 article in Emissary on matrix factorisations and my survey of linear logic. I am part of the Melbourne Deep Learning Group. I have various code projects on GitHub including my joint project with Nils Carqueville on computing with matrix factorisations.

The title of this webpage refers to the following quote from Grothendieck, which I learned from Colin McLarty’s article “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.

News:

- Rohan Hitchcock’s masters thesis, October 2022, Differentiation, Division and the Bicategory of Landau-Ginzburg Models.
- Matthew Farrugia-Roberts’ masters thesis, October 2022, Structural Degeneracy in Neural Networks.
- Spencer Wong’s masters thesis, May 2022, From Analytic to Algebraic: The Algebraic Geometry of Two Layer Neural Networks.
- Elimination and cut-elimination in multiplicative linear logic, July 2022, preprint with Will Troiani.
- We taught an introductory algebraic geometry class at metauni (MAG) in June 2022.
- David Li’s summer scholar project, February 2022, AlphaFold.
- Jackson Godfrey’s masters thesis, October 2021, Superbicategories.
- Liam Carroll’s masters thesis, October 2021, Phase transitions in neural networks.
- Tom Waring’s masters thesis, October 2021, Geometric perspectives on program synthesis and semantics.
- The Rising Sea is now on metauni, a network of virtual spaces built on Roblox and Discord. Many of my talks this year will be hosted there.
- Deep Learning is Singular, and That’s Good, October 2020, preprint from MDLG on theory of deep learning.
- Geometry of Program Synthesis, October 2020, preprint with James Clift and James Wallbridge.
- Gentzen-Mints-Zucker duality, August 2020, preprint with William Troiani on sequent calculus, lambda calculus and the Curry-Howard correspondence (video).
- Isaac David Smith’s masters thesis, June 2020, Investigating Defect Topological Quantum Field Theories as Models for Quantum Error-Correction.
- Logic and the 2-Simplicial Transformer published at ICLR 2020 (code and videos | arXiv version).
- Will Troiani’s masters thesis, May 2019, Simplicial sets are algorithms.
- Constructing A-infinity categories of matrix factorisations preprint, March 2019 (my working notes are available).
- The notes and videos for a course on Metric and Hilbert spaces (and video HOWTO).
- Albert Zhang’s masters thesis, October 2018, Vertex algebras, Hopf algebras and lattices (and talk slides).
- Derivatives of Turing machines in Linear Logic preprint with James Clift, May 2018.
- Encodings of Turing machines in Linear Logic preprint with James Clift, May 2018.
- James Clift’s masters thesis, October 2017, Turing machines and differential linear logic.
- Patrick Elliott’s masters thesis, October 2017, A-infinity categories and matrix factorisations over hypersurface singularities.

Some talks:

- Matrix factorisations and Quantum Error Correcting Codes, July 2022 (notes).
- Introduction to singularity theory, March 2021 (notes)
- Theory of Deep Learning, January 2021, a general introduction to the topic in the Rising Sea metauni node (video).
- Introduction to matrix factorisations, April 2020 in University of Melbourne moduli spaces seminar (Part 1 notes | Part 2 notes)
- From critical points to extended TQFTs, April 2020 in Monash topology seminar (notes)
- A-infinity categories of matrix factorisations via A-infinity idempotents, January 2020 at KIAS workshop on Atiyah classes and related topics (notes)
- Mathematics of AlphaGo, November 2019 colloquium at Macquarie (notes | screencast)
- From critical points to A-infinity categories, October 2019 at Macquarie (notes)
- Monoidal bicategories of critical points, July 2019 (notes)
- Constructing A-infinity categories of matrix factorisations, July 2019 at RRAGE (slides)
- Proof synthesis and differential linear logic, June 2019 at CARMA (slides | screencast)
- Derivatives of Turing machines and inductive inference, November 2018 at Peking University (notes)
- The computational content of Landau-Ginzburg models, November 2018 at BICMR, Beijing (notes)
- Derivatives of Turing machines in linear logic, May 2018 in the Melbourne pure math seminar (notes)
- Algebra and Artificial Intelligence, May 2018 in the Melbourne logic seminar (slides | video)
- Bar versus Koszul, April 2018 in the Melbourne topology seminar (notes)
- Mini-course on A-infinity categories and matrix factorisations, September 2017 at the IBS in Korea (lecture 1, lecture 2, lecture 3).
- Turing machines and coalgebras, September 2017 at Neeman’s 60th conference at ANU (notes).
- Clifford algebras and 2D defect topological field theory, June 2017 at Tensor Categories and Field Theory (notes).
- A tour of well-generated triangulated categories, May 2017 at Neeman’s 60th conference (notes).
- Derivatives of proofs in linear logic (joint with James Clift), May 2017 in Melbourne (slides and transcript).
- The cobordism category, October 2016 in the TFT seminar (notes).
- The Curry-Howard principle, October 2016 in the CH seminar (notes).
- The category of simply-typed lambda terms, September 2016 in the CH seminar (lecture 1, lecture 2).
- Sheaves of A-infinity algebras from matrix factorisations, September 2016 at the IPMU (notes).
- Generalised orbifolding, September 2016 minicourse at the IPMU (lecture 1, lecture 2, lecture 3).
- Generalised orbifolding of simple singularities, August 2016 at Geometry at the ANU (notes).
- Two odd things about computation, October 2015 in Vienna and August 2016 in Melbourne (slides and transcript).
- Topological Quantum Field Theory in two dimensions, July 2016 in the TFT seminar (slides).
- Spectral sequences for vertex algebras, July 2016 in Melbourne (notes).
- Linear logic and deep learning (joint with Huiyi Hu), June 2016 at the AAL in Melbourne (slides and transcript).
- A-infinity algebras and matrix factorisations, June 2016 in Banff (notes and video).
- The Landau-Ginzburg/Conformal Field Theory correspondence, May 2016 in Melbourne (notes).
- The super-A-polynomial and knot differentials, May 2016 in Melbourne (notes).
- A-infinity algebras and minimal models [Part 1], May 2016 in Melbourne (notes and screencast).
- Reading group on proof-nets, April 2016 in the Melbourne logic seminar (handout 1, handout 2 and notes).
- Stratifications and complexity in linear logic, March 2016 in the Melbourne logic seminar (slides and screencast).
- An introduction to A-infinity algebras, November 2015 in Melbourne (notes).

I gave a mini-course at the IBS in Korea in January 2016 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 on GitHub.

- Lecture 1: 2D TFT with defects and matrix factorisations (notes and video).
- Lecture 2: The bicategory of Landau-Ginzburg models (notes and video).
- Lecture 3: The cut operation and computing fusions (notes and video).

These lectures were made with the excellent GoodNotes for the iPad Pro.

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 protect oneself from it (e.g. encryption) are themselves mathematical. Soon after the Snowden leaks my friend Nils Carqueville and I wrote an essay on this subject, which you might find interesting.

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”.

- Introduction to EGA I: The motivating ideas of modern algebraic geometry, presented beautifully by Grothendieck (translated with the help of Tamah Murfet, way back in 2003).
- Sheaves of Groups and Rings: (SGR) Sheaves of sets (incomplete), sheaves of abelian groups, stalks, sheaf Hom, tensor products, inverse and direct image, extension by zero.
- Modules over a Ringed Space: (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).
- Sheaves of Algebras: (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.
- Special Sheaves of Algebras: (SSA) Sheaves of tensor algebras, symmetric algebras, exterior algebras, polynomial algebras and ideal products. Complete study of these constructions as adjoints.
- Modules over a Scheme: (MOS) Ideals, special functors (extension by zero and coextension of scalars), locally free sheaves, sheaf Hom, extension of coherent sheaves.
- The Proj Construction: (TPC) Functorial properties, products, linear morphisms, projective morphisms, dimensions of some schemes, points of projective space.
- Modules over Projective Schemes: (MPS) Properties of the functor associating a graded module with a quasi-coherent sheaf on Proj.
- An Adjunction for Modules over Projective Schemes: (AAMPS) The adjoint for taking the associated sheaf of a graded module, the quasi-coherent case.
- An Equivalence for Modules over Projective Schemes: (AEMPS) The projective version of some important theorems in the affine case.
- Relative Affine Schemes: (RAS) Affine morphisms, the Spec construction, the sheaf associated to a sheaf of quasi-coherent modules over an algebra.
- The Segre Embedding: (SEM) Pullback of Proj schemes, properties of projective morphisms.
- Concentrated Schemes: (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.
- Section 2.6 - Divisors: (DIV) Weil divisors, divisors on curves, cartier divisors, invertible sheaves.
- Section 2.7 - Projective Morphisms: (PM) Morphisms to Pn, the duple embedding, ample invertible sheaves, linear systems.
- Section 2.7.1 - Blowing Up: (BU) Definition of the blow-up, blowing up of varieties.
- Section 2.8 - Differentials: (DIFF) Kahler differentials, sheaves of differentials, nonsingular varieties, rational maps, applications, some local algebra.
- Section 2.9 - Formal Schemes: (FS) Inverse limits, completion, adic rings (complete rings of fractions, local completion), affine formal schemes (this note is not yet complete).
- Section 3.2 - Cohomology of Sheaves: (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.
- Section 3.8 - Higher Direct Images of 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.
- Section 3.7 - Serre Duality: (SDT) Notes on Serre Duality and dualising sheaves as presented in Hartshorne.
- The Relative Proj Construction: (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.
- Ample Families: (AMF) Ample sheaves and ample families of sheaves on arbitrary schemes, as described in EGA and later SGA.
- Schemes via Noncommutative Localisation: (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.
- The Zariski site: (ZT) Definition of the Zariski site and the proof that schemes give sheaves on it. Probably directly from EGA, but I don’t recall.

My handwritten notes for Sections 1 and 2 of Hartshorne (there are some corrections):

- Section 1.1: Affine varieties
- Section 1.2: Projective varities
- Section 1.3: Morphisms
- Section 1.4: Rational maps
- Section 1.5: Nonsingular varieties
- Section 1.6: Nonsingular curves
- Section 1.7: Intersections in projective space
- Section 2.1: Sheaves
- Section 2.2: Schemes
- Section 2.3: First properties of schemes
- Section 2.4: Separated and proper morphisms
- Section 2.5: Sheaves of modules
- Section 2.5 proj: Sheaves of modules (Proj)

- Matsumura: (MAT) General rings, flatness, depth, Cohen-Macaulay rings, normal and regular rings, koszul complexes, unique factorisation.
- Matsumura Part II: (MAT2) Extension of a ring by a module, derivations and differentials, separability.
- Noether Normalisation: First introduction to various versions of Noether normalisation.
- More Noether Normalisation: A version of noether normalisation involving separability.
- Hensel’s Lemma: Hensel’s Lemma and a few small examples.
- Cohen’s Theorem.
- Graded Rings and Modules: (GRM) Definitions and basic properties, the category of graded modules, quasi-structures, grading tensor products.
- Tensor, Exterior and Symmetric algebras: (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.
- Automorphisms of Power Series Rings: (APSR) Constructing automorphisms of power series rings from an independent family of power series.
- Topological Rings: Topological groups and rings, fundamental systems of ideals and preparation for Gabriel topologies.

- Basic Set Theory: (BST) Ordinal numbers, transfinite induction, cardinal numbers, cardinal operations, regular cardinals.
- Foundations for Category Theory: (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.
- Abelian Categories: (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).
- Diagram Chasing in Abelian Categories: (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).
- Derived Functors: (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.
- Triangulated Categories Part I: (TRC) [Verdier quotients] Triangulated categories, triangulated functors, homotopy colimits, localising subcategories, right derived functors, left derived functors, portly considerations.
- Triangulated Categories Part II: (TRC2) [Thomason localisation] Finer localising subcategories, perfect classes, small objects, compact objects, portly considerations, morphisms in the quotient.
- Triangulated Categories Part III: (TRC3) [Brown representability] Representability theorems, compactly generated triangulated categories.
- Derived Categories Part I: (DTC) Homotopy categories, derived categories (extending functors, introduction to hearts, bounded derived categories), homotopy resolutions, homotopy direct limits, bousfield subcategories, existence of resolutions.
- Derived Categories Part II: (DTC2) Derived functors, derived Hom, derived Tensor, brown representability.
- Derived Categories Of Sheaves: (DCOS) Representing cohomology, derived direct image, derived sheaf Hom, derived Tensor, derived inverse image.
- Derived Categories Of Quasi-coherent Sheaves: (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) 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) Tor on the left and right and balancing the two, dimension shifting, Tor and colimits, Tor for commutative rings and bimodules, criteria for flatness.
- Dimensions: (DIM) Projective dimension, injective dimension, global dimension, flat dimension and change of rings.
- Spectral Sequences: (SS) Definition, basic convergence properties, the spectral sequence of a complex filtration, the two spectral sequences of a bicomplex, the Grothendieck spectral sequence.
- Algebra in a Category: (ALCAT) Groups, rings and modules in an arbitrary category. Sheaves of groups, rings and modules and graded versions.
- Linearised Categories: (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.
- Rings of Quotients: 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.
- Localisation of Ringoids: Additive Grothendieck topologies and localisation, the Gabriel-Popescu theorem.

These are the notes for the reading group on proof-nets in linear logic, spanning April 8 and 15 of the logic seminar at the University of Melbourne. The aim is to:

**April 8**(handout): 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**(handout): 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 previous talk without details.

The main references are:

- [G87] J.Y. Girard’s original paper “Linear logic“
- [G96] J.Y. Girard “Proof-nets: the parallel syntax for proof-theory“
- [J91] J. Davoren “A Lazy Logician’s Guide to Linear Logic“
- [BM09] P. Baillot and D. Mazza “Linear Logic by Levels and Bounded Time Complexity“
- [PTF09] M. Pagani and L. Tortora de Falco “Strong Normalization Property for Second Order Linear Logic“

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 sequent calculus proofs; 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 (slides and screencast).
- 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).

My interest in the mathematical theory of computation is primarily due to the work of the logician Jean-Yves Girard on linear logic. Below is a list of references for the parts of the theory of computation that I either know a little about, or am interested in.

General “big picture” references on computation and logic:

- The Nature of Computation, a textbook by Cristopher Moore and Stephan Mertens.
- Towards a geometry of interaction by Jean-Yves Girard.
- The Blind Spot also by Girard, quite eclectic and unpolished, but full of interesting ideas (especially the sections on Russell’s paradox and complexity).
- Constructive mathematics and computer programming by Martin-Lof.
- Two puzzles about computation by Samson Abramsky.

Some references on lambda-calculus, System F and proof theory:

- Proofs and types by Jean-Yves Girard.

Some references on computational complexity theory:

- Why philosophers should care about computational complexity by Scott Aaronson.
- Topological views on computational complexity by Michael Freedman.
- Light linear logic by Jean-Yves Girard.
- Light logic and polynomial time computation by Kazushige Terui.

Relations between computation and category theory:

- Lectures on the Curry-Howard isomorphism by Morten Heine Sorensen and Pawel Urzyczyn.
- Physics, Topology, Logic and Computation: A Rosetta Stone by John Baez and Mike Stay.
- Categorical semantics of linear logic by Paul-Andre Mellies.
- Notions of computation and monads by Eugenio Moggi.
- Homotopy type theory is a semantics of Martin-Lof type theory defined using homotopy theory.

Relations between computation and physics:

- The Feynman lectures on computation, 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).
- Complexity and phase transitions, Nature paper from 1999.
- There is a deep and interesting connection between Maxwell’s demon and topics in computation, for which see Szilard, Brillouin, Landauer and the papers of Bennett below.
- The thermodynamics of computation by Bennett 1982.
- Time-space tradeoffs for reversible computation by Bennett.

Some interesting videos:

- Lambda calculus, then and now by Dana Scott.
- Propositions as types by Philip Wadler.
- The future of programming by Bret Victor.