This page collects the course materials and student presentation files for the Texas State University Graduate Complex Analysis course in Spring 2026.
The main reference used in this course is the following:
https://arxiv.org/abs/2601.06868
Until the midterm exam, the course focuses on building concrete understanding of differential forms through explicit computations. In particular, we study the correspondence between gradient, curl, and divergence and differential forms, as well as concrete integrations of differential forms, integrations of the winding one-form, and, especially, computations of Gaussian curvature via first fundamental form on regular surfaces and the Euler characteristic on S2=CP1 and on the torus.
After the midterm, students prepare their own term presentations. During this stage, the course shifts toward applications of the Riemann–Roch theorem on compact Riemann surfaces, while also developing the details needed for final exam preparation.
To support this process, I provide the necessary notes and video materials in advance, so that class time can be used primarily for discussion and student presentations.
The below link is recorded lecture to this class.
1. Definitions of holomorphic functions, note
3. Consequences of Cauchy Integral formula, note
4. Identity theorem, Argument principle, Open mapping theorem, note
5. holomorphic map between compact Riemann surfaces is a branched covering, note
6. Residue Theorem, Riemann-Hurwitz Theorem, Principal Divisor and Canonical Divisor
7. Riemann-Roch Theorem and its elementary proof
8. Introduction to Sheaf, Cech Cohomology and Cohomological Interpretation of Riemann-Roch Theorem
9. Implication of Riemann-Roch Theorem to the genus 0 surface CP^1=S^2 : Rational Curves
10. Implication of Riemann-Roch Theorem to the genus 1 surface Torus : Elliptic Curves
Here are list of presentation from graduate students of TXST Math.
Frederick Abban - Fluid Dynamics via Complex Analysis and Riemann Surfaces
Michaela G Donnely - Field Extension corresponds to branched covering map on Riemann Surfaces
Eric Agyei - Delated Differential Equation and Riemann Surfaces
Matthew L McCutchen - Uniformization Theorem
Logan Greenland - TQFT, 2d Frobenius algebra and Cohomology of Torus
Cameron Poole - Riemann Mapping Theorem via Dirichlet Principle
Andrew Soule - Schwarz Lemma and complex hyperbolic geometry
Mason Mault - Minimal Surface via Weierstrass Representation
Landon Chambers - Riemann Hypothesis via Theta function
Jordan Anthony Guillory - Cohomological Approach of complex torus via Jacobian Variety