Im A Topologist Now

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.
• Notes on the order of the group of homotopy spheres.
• Examples of handle attachments.
• Linear algebra of handle attachments.
• Defining higher K theory (Waldhausen construction).
• Summary of the key facts about Spectra.
• Definition of a spectral sequence.
• Examples of the Serre spectral sequence.
• 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).
• Basic facts about \(K(\mathbb{Z})\).
• 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.
• Computation of the first 17 - 19 homotopy groups of the Whitehead space, concordance groups and h-cobordism space.

Three reasons for \(\infty\)-categories

• Three Nerve functors
• The homotopy hypothesis
• Model categories

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

Notes from Seminar Our Group Ran on Atiyah-Singer Index Theorems

• 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.

Useful Tables and Lists

• Homotopy groups of the concordance space.
• Homotopy groups of the space of diffeomorphisms of the disc.

Misc Notes: Recent at Top

• Extension problems from exact sequences.
• My understanding of QFT.
• My understanding of electrodynamics.
• My understanding of quantum mechanics.
• Recovering a group from its local pieces .
• Homotopy theory of Schemes.
• Review of basic calculus.
• Orbit stabiliser theorem.
• Column space and full rank.
• Some basic examples of spectrum of operators.
• The Bruhat decomposition is just linear algebra.
• Basic eigenvalue notes.
• The equivalence of categories between vector spaces K/k with a Galois action.
• The equivalence of categories between real and analytic Lie groups
• The equivalence of categories between (representations) of reductive Lie groups and reductive LAG's.
• The equivalence of categories between Lie groups and Lie algebras.

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

\[\pi_i(X)\cong \pi_{i-1}(\Omega X)\cong \pi_{i+1}(BX)\]

NUMBER THEORY

Thesis:

• A paired back and revised version
• My initial one

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

High Quality and Pretty

• Geometric Groups Theory
• Functional Analysis
• Differential Geometry
• Measure Theory
• Complex Analysis and Riemann Surfaces notes and a summary sheet
• Representation Theory

High Quality and Ugly

• Groups, Categories and Homological Algebra

Garbage

• Vector Calculus
• Complex Analysis
• Probability for Inference
• Metric and Hilbert Spaces
• Algebraic Topology

RANDOM STUFF

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"

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

• The misrepresentation of JTB in intro philosophy courses
• The informality of model theory
• Reminders on how to write better

Links

• Brian Conrad: Has many notes from courses and for the express purpose of exposition
• Paul Garrett: Lots of notes.
• Nortonian mechanics