im a topologist now
basic topology notes
- proof of the les for a fibration
- defining stable homotopy groups
- the adjunction between the category of chains and graded abelian groups
- homology summary
- cohomology summary
- pontryagin construction
- constructions of vector bundles
- comparison of different notions of bundles and fibrations
- comparison of different notions of orientations
- becoming simply connected via van kampens.
- smooth structures on discs and spheres, their connections to the h-cobordism theorem and diffeomorphisms of discs.
- examples of handle attachments.
- linear algebra of handle attachments.
- summary of the key facts about spectra.
- definition of a spectral sequence.
- examples of the serre spectral sequence.
\(\infty\)-categories
- Motivation via three nerve functors
- Motivation via the homotopy hypothesis
- Motivation via model categories
notes on the literature
- notes on the order of the group of homotopy spheres.
- defining higher k theory (waldhausen construction).
- a sketch of the computation of the rational homotopy type of the space of diffeomorphisms of a disc.
- how the homotopy groups of the concordance space are related to psuedo-isotopy (cerfs theorem).
- some computations of the homotopy groups of the space of concordances.
- the geometry of the gromoll filtration, in terms of the existence of a fibration.
- notes on the parametrised h-cobordism theorem and the definition of \(\mathrm{Wh}^{\mathrm{Diff}}\).
- summary of wei wangs notes on the gromoll filtration.
numerics
- basic facts about \(K(\mathbb{Z})\).
- computation of the first 17 - 19 homotopy groups of the whitehead space, concordance groups and h-cobordism space.
- homotopy groups of the space of diffeomorphisms of the disc.
- some numerics from the work of burghelea and lashof.
notes from talks
- characteristic classes and the signature theorem
- reconstructing the signature theorem. sort of finishing of the previous talk.
- a 5 minute "lightning talk" given in the unimelb topology seminar: discussing km results on homotopy spheres
- talk in the index theory reading group, defining the topological index.
- define the analytic index (and discussing some basic k theory preliminaries).
- example computations for bundles over the point and circle.
misc notes
recent notes
- the equivalence of categories between lie groups and lie algebras.
- the equivalence of categories between (representations) of reductive lie groups and reductive lag's.
- the equivalence of categories between real and analytic lie groups
- the equivalence of categories between vector spaces k/k with a galois action.
- basic eigenvalue notes.
- the bruhat decomposition is just linear algebra.
- some basic examples of spectrum of operators.
- column space and full rank.
- orbit stabiliser theorem.
- review of basic calculus.
- homotopy theory of schemes.
- recovering a group from its local pieces .
- extension problems from exact sequences.
notes to write / in progress
- history of automorphic forms
- finish my intro to langlands work by tying together the two histories
- companion to langlands eisenstein series paper
- lie groups, lie algebras and algebraic groups: comparison and all the subgroups and classifications
- elliptic integrals and algebraic geometry
formulas i forget
\[\pi_i(X)\cong \pi_{i-1}(\Omega X)\cong \pi_{i+1}(BX)\]history, philosophy, science and coding
the history of mathematics
- (300 bc) euclid
- (250 ad) diophantus
- (1637 ad) descartes geometry
- (1684 ad) liebniz calculus
- (1748 ad) eulers intro to the infinite
- (1800 - 1850 ad) gauss, cauchy, louiville, abel and galois
- (1850 - 1900 ad) dedekind and riemann
- (1880 - 1920 ad) notes on grays "platos ghost"
physics
- my understanding of qft.
- my understanding of electrodynamics.
- my understanding of quantum mechanics.
- nortonian mechanics
code
- an analysis of the popularity of mathematical topics via the arxiv
- a terribly ugly way of plotting elliptic curves or varieties in one variable over finite fields
- you want a theorem, you got a theorem; raw zfc theorem generator
blog posts
older notes and research interests
number theory
thesis:
etymology:
- ramification in number theory and geometry
- a short note on the analytic origins of spec
- parabolics
- torus
- constant terms in the theory of automorphic forms
notes:
- some (bad) notes on the idea of smoothness
- adele definition
- adelic topology
- defining an automorphic form
- group scheme summary
- basic algebraic geometry
- very explicit computations of the constant term of an eisenstein series
- a sketch of the langlands conjectures
notes from talks
- definition and classification of reductive groups
- definition of automorphic forms; explanation of modular forms as a special case
- a talk on coherent sheaves in a student reading group
- etale site and etale cohomology
- slides from my talk describing my masters research
linear logic
my work:
- fixed point theorem for last
- russell's paradox and polynomials: a talk at the university of melbourne student seminar 2023
- russell's paradox and the y combinator
- sweedler semantics and the category of hilbert spaces
notes on the literature
- lawveres fixed point theorem
- goldblatts "categorical logic"
- ll, dill and stratified ll
- flow diagram of the proof of cut elimination referenced in the above
university notes: in rough order of quality
- geometric groups theory
- functional analysis
- differential geometry
- measure theory
- complex analysis and riemann surfaces notes and a summary sheet
- representation theory
- groups, categories and homological algebra
- vector calculus
- complex analysis
- probability for inference
- metric and hilbert spaces
- algebraic topology
i stole the background and crt effect from angs-corner
some real mathematicians have nice websites with helpful notes
- brian conrad: has many notes from courses and for the express purpose of exposition
- paul garrett: lots of notes.
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