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Table 1: the Cayley–Dickson staircase from the reals to the sedenions, listing the property surrendered at each doubling of the dimension.

mathematics

mathematicsalgebraquaternions+2 more

The Cayley–Dickson Staircase: What Each Doubling Costs

The Cayley–Dickson staircase is usually presented as far as the octonions. One step further, at the sedenions of real dimension 16, alternativity, division and the multiplicative norm all fall at once.

November 16, 20251 min read

The de Bruijn–Springer theorem: the convex weights expressing the critical points in terms of the roots can be chosen with column sums equal to 1 - 1/n.

mathematics

mathematicspolynomialslinear algebra+1 more

From Gauss–Lucas to de Bruijn–Springer

Gauss–Lucas puts the roots of the derivative inside the convex hull of the roots. De Bruijn and Springer conjectured in 1948 that they are pulled inward on average — it took 55 years and two proofs.

November 2, 20252 min read

The Bochner–Martinelli formula: a universal (n, n-1) kernel reproducing f from its boundary values, with a volume term that vanishes precisely when f is holomorphic.

mathematics

mathematicscomplex analysisseveral complex variables+1 more

The Bochner–Martinelli Formula

Closing the cycle on the Cauchy integral formula: Bochner–Martinelli recovers a function from its boundary data, up to a term measuring the failure of holomorphicity, with one universal kernel.

October 25, 20252 min read

The Cauchy–Leray–Fantappiè theorem: a kernel of type (n, n-1) assembled from a defining function of the boundary, and the reproducing formula it yields on a bounded domain in C^n.

mathematics

mathematicscomplex analysisseveral complex variables

The Cauchy–Leray–Fantappiè Formula in Several Variables

In several complex variables there is no translation-invariant holomorphic kernel. The Cauchy–Leray–Fantappiè formula answers this by building its kernel from a defining function of the boundary.

October 18, 20251 min read

The classical Cauchy integral formula and its smooth counterpart, whose extra volume term measures the failure of holomorphicity.

mathematics

mathematicscomplex analysisintegral formulas

The Cauchy Integral Formula Beyond Analyticity

What survives of the Cauchy integral formula when the function is no longer analytical? A smooth version exists, and a single volume term measures exactly how far holomorphicity fails.

October 2, 20251 min read

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

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