Category Archives: Analysis

Real and complex analysis, functional analysis, PDE.

Noncommutative probability I: motivations and examples

In many areas of mathematics, one can usually assign a commutative algebraic structure to the “space” that one is studying. For example, instead of studying an (affine) algebraic variety, one can study algebraic functions on that variety, which gives rise … Continue reading

Posted in Functional analysis, Probability | Leave a comment

A simple proof of the Gauss-Bonnet theorem for geodesic ball

In this short note, we will give a simple proof of the Gauss-Bonnet theorem for a geodesic ball on a surface. The only prerequisite is the first variation formula and some knowledge of Jacobi field (second variation formula), in particular … Continue reading

Posted in Calculus, Differential equations, Differential geometry, Geometry | Leave a comment

Zeros of random polynomials

Given a polynomial , where the coefficients are random, what can we say about the distribution of the roots (on )? Of course, it would depend on what “random” means. Here, “random” means that the sequence is an i.i.d. sequence … Continue reading

Posted in Algebra, Complex analysis, Potential theory, Probability | Leave a comment

Euler’s formula e^ix = cos x + i sin x: a geometric approach

Today I mentioned the famous Euler’s formula briefly in my calculus class (when discussing hyperbolic functions, lecture notes here): where is a solution to (usually denoted by “”, but indeed there is no single-valued square root for complex numbers, or … Continue reading

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An inequality for functions on the plane

I accidentally came across a curious inequality for functions of two variables. I would like to know if this inequality is a special case of a more general result but I was unable to find a reference. It would also … Continue reading

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Weighted isoperimetric inequalities in warped product manifolds

1. Introduction The classical isoperimetric inequality on the plane states that for a simple closed curve on , we have , where is the length of the curve and is the area of the region enclosed by it. The equality … Continue reading

Posted in Analysis, Calculus, Differential equations, Functional analysis, Geometry, Inequalities | 2 Comments

Faber-Krahn inequality

I record a proof of the Faber-Krahn inequality here, mainly for my own benefit. Let be one of the standard space forms: the Euclidean space , the unit sphere , or the hyperbolic space . Suppose is a bounded domain … Continue reading

Posted in Analysis, Differential equations, Functional analysis, Geometry, Inequalities | Leave a comment