Learning complex analysis with GPT o1

My journey of learning complex analysis only using the latest GPT available.

DALLE-3 Image of what is Complex Analysis

Why?

I like to learn areas of mathematics and use AI whenever I can, so I decided I wanted to use both and share my process. I want to test how good it currently stands to learn fields of mathematics from GPT o1. In this case, I have structured a big course in complex analysis with GPT o1, and it will be my lecturer and assign me problems as well as explaining to me the concepts.

Next Stories

In my next stories, I will be showcasing to you people my experience of note-taking, learning and problem-solving in this area of knowledge, this will be on the weekly basis. I will always give it a feedback grade and explore overall this notion of learning from AI.

Complex analysis study plan

Here I will detail the learning plan that I have agreed with GPT for the following weeks and that I will be publishing here as the course goes.

Weeks 1–2: Review of Fundamental Concepts

• Rigorous treatment of complex numbers as a field extension of the reals.
• Topology of the complex plane: open and closed sets, compactness, connectedness.
• Sequences and series of complex numbers; convergence tests.

Weeks 3–4: Holomorphic Functions

• Rigorous definition of holomorphic (analytic) functions.
• Differentiability in the complex sense vs. real differentiability.
• Complex differentiability implies analyticity.
• The role of the Cauchy-Riemann equations in different contexts.

Weeks 5–6: Complex Integration and Cauchy’s Theorems

• Path integrals in the complex plane.
• Detailed proofs of Cauchy’s integral theorem and Cauchy’s integral formula.
• Homotopy and simply connected domains.
• Winding numbers and their applications.

Weeks 7–8: Analytic Continuation and Monodromy

• Principles of analytic continuation.
• Monodromy theorem and its implications.
• Schwarz reflection principle.
• Applications to multivalued functions and branch cuts.

Weeks 9–10: Riemann Surfaces

• Introduction to Riemann surfaces.
• Holomorphic maps between Riemann surfaces.
• Classification of Riemann surfaces.
• Sheaf theory basics.

Weeks 11–12: Potential Theory and Harmonic Functions

• Harmonic functions and their relation to holomorphic functions.
• Dirichlet and Neumann problems.
• Green’s functions.
• Application of potential theory in complex analysis.

Weeks 13–14: Advanced Conformal Mapping

• Riemann Mapping Theorem: Proofs and implications.
• Uniformization theorem.
• Applications of conformal mapping in modern research.
• Teichmüller theory (introductory level).

Weeks 15–16: Elliptic Functions and Modular Forms

• Weierstrass ℘-function and Jacobi theta functions.
• Lattices in the complex plane.
• Modular forms and their properties.
• Connections to number theory and algebraic geometry.

Weeks 17–18: Entire and Meromorphic Functions

• Classification of entire functions (Liouville’s theorem, Picard’s theorems).
• Order and type of entire functions.
• Nevanlinna theory.
• Applications in differential equations.

Weeks 19–20: Several Complex Variables

• Extension to functions of several complex variables.
• Hartogs’ phenomenon.
• Domains of holomorphy.
• Coherence and Oka’s theorems.

Weeks 21–22: Current Research Topics

• Complex dynamics (iteration of holomorphic functions).
• Analytic number theory applications.
• Complex manifolds and Hodge theory.
• Complex algebraic geometry connections.

My Socials and Podcast

• Instagram: Tiago Veríssimo (@tiago_verissimo_krypton) • Instagram photos and videos
• Spotify: KryptonKast
• X: Tiago Verissimo (@KryptonPortugal) / X

See you next blog!

Newcastle Upon Tyne, England