# Learning complex analysis with GPT o1

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

# 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**