Blog

Thoughts on mathematics, computational science,

and academic life

11 posts published

Filter by topic

Recent Posts

mathematics

mathematicscomplex analysispower series

The Hadamard Gap Theorem and Stein Manifolds

How do big gaps in power series relate to Stein manifolds? A look at the Hadamard Gap Theorem and why the unit disk is a Stein manifold.

June 5, 20251 min read

Diagram illustrating the complex analytic implicit function theorem for two variables

mathematics

mathematicscomplex analysisanalysis

The Complex Analytic Implicit Function Theorem: A Brief Revisit

A concise look at the complex analytic version of the Implicit Function Theorem (IFT), focusing on the case of two complex variables and its generalization.

May 20, 20251 min read

Illustration of a Stein manifold as a domain of holomorphy

mathematics

mathematicscomplex analysisStein manifolds

Stein Manifolds: Domains of Holomorphy in Complex Geometry

A brief introduction to Stein manifolds, their equivalent definitions, and the intuition behind domains of holomorphy in several complex variables.

April 20, 20251 min read

Illustration contrasting the unit ball and polydisc in complex dimensions

mathematics

mathematicscomplex analysis

Poincaré’s Theorem: The Ball vs. The Polydisc in Several Complex Variables

Why can’t we generalize conformal mappings to higher dimensions? Explore Poincaré’s theorem, which launched the study of several complex variables as a field of its own.

April 15, 20251 min read

A humorous illustration about confusing mathematical terminology

mathematics

mathematicshumorterminology

Mathematical Misnomers: A Light-hearted Reflection

A playful look at the inconsistent and misleading names in mathematics, and how terminology can confuse even the experts. Share your favorites in the comments!

April 10, 20251 min read

Diagram illustrating the trace operation and its dual/tensor characterization

mathematics

mathematicslinear algebra

The Trace of a Matrix: A Dual and Tensor Perspective

A fresh look at the trace of a square matrix: its linearity, role in the dual space of operators, unique tensor characterization, and its connection to inner products and the transpose via the dual.

April 5, 20251 min read

Portrait of Willian Fogg Osgood's theorem

mathematics

mathematicsdifferential equationsuniqueness theorems

Osgood’s Uniqueness Theorem: Beyond the Lipschitz Condition

How far can the classical Lipschitz condition for uniqueness in ODEs be relaxed? Discover Osgood’s theorem and its historical context in the theory of ordinary differential equations.

March 25, 20251 min read

Visualization of solution uniqueness in implicit differential equations

mathematics

mathematicsdifferential equationsanalysis

Cauchy-Lipschitz Theorem for Implicit ODEs: Existence and Uniqueness

Explore the Cauchy-Lipschitz (Picard-Lindelöf) theorem for implicit ordinary differential equations, which establishes rigorous conditions for existence and uniqueness of solutions in the general case.

March 20, 20251 min read

mathematics

mathematicspower seriesasymptotic analysis

Gevrey Classes of the Power Series

Explore Gevrey classes, a powerful framework for understanding the growth of power series coefficients, with insights from Edmond Maillet's theorem on formal solutions to analytical differential equations.

March 15, 20251 min read

mathematics

mathematicsdifferential geometry

The Submersion Theorem: Unveiling the Geometry of Manifold Mappings

Explore the elegant Submersion Theorem, a cornerstone of differential geometry that reveals the intricate relationship between smooth maps and manifold structures in mathematical analysis.

March 10, 20251 min read

Stay Updated

Subscribe to receive notifications about new blog posts and research updates

Subscribe via RSS•View all feeds