
Recent Posts
 Zeros of random polynomials
 Hopf fibration double covers circle bundle of sphere
 Euler’s formula e^ix = cos x + i sin x: a geometric approach
 An inequality for functions on the plane
 Weighted isoperimetric inequalities in warped product manifolds
 FaberKrahn inequality
 Why is a² + b² ≥ 2ab ?
 A remark on the divergence theorem
 The CauchySchwarz inequality and the Lagrange identity
 On the existence of a metric compatible with a given connection
Meta
Recent Comments
tong cheung yu on Toric perspective 1 Miodrag Mateljevic on Principle of subordination Lawrence G Mouille on Exponential maps of Lie g… tong cheung yu on On the existence of a metric c… Marco Barchiesi on Simple curves with a positive… tong cheung yu on Toric perspective 1 lamwk on Why is a² + b² ≥ 2ab ? Maurice OReilly on Spherical cosine law KKK on Sobolev and Isoperimetric… Anonymous on Sobolev and Isoperimetric… Categories
 Algebra
 Algebraic geometry
 Analysis
 Applied mathematics
 Calculus
 Combinatorics
 Complex analysis
 Differential equations
 Differential geometry
 Discrete Mathematics
 Dynamical system
 Fourier analysis
 Functional analysis
 General Relativity
 Geometry
 Group theory
 Inequalities
 Linear Algebra
 Miscellaneous
 Number Theory
 Operator Theory
 Optimization
 Potential theory
 Probability
 Set Theory
 Statistics
 Topology
 Uncategorized
Top Posts
 Complex analysis  Problem solving strategies.
 Martingale Theory III: Optional stopping theorem
 Martingale Theory II: Conditional expectation
 Sobolev and Isoperimetric Inequality
 Taylor expansion of metric
 Lie groups with biinvariant Riemannian metric
 Spherical cosine law
 AMGMHM Inequality: A Statistical Point of View
 Dual norm in R^n
 Sum of angle defects of polyhedrons
Archives
Category Archives: Algebra
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
Hopf fibration double covers circle bundle of sphere
Two days ago, I gave a seminar talk on Chern‘s proof of the generalized GaussBonnet theorem. Here I record the answer to a question asked by one of my colleague during the talk. Although not directly related to the proof … Continue reading
Posted in Algebra, Differential geometry, Group theory
Leave a comment
The CauchySchwarz inequality and the Lagrange identity
The classical Lagrange identity is the following: This can be proven by expanding and separating the terms into the crossterms part and the non crossterms part. The Lagrange identity implies the CauchySchwarz inequality in . And when , this can … Continue reading
Posted in Algebra, Group theory, Inequalities, Linear Algebra
Leave a comment
27 lines on a smooth cubic surface
Here describes two different proofs of a general smooth cubic surface containing exactly 27 lines. One approach uses blowups and the other one uses the Grassmannian. If I have time I will elaborate on the discussion.
Posted in Algebraic geometry
Leave a comment
Toric perspective 1
This series of posts will be about toric varieties. The author is not sure if there is a part 2, but he still calls this part 1. This post is about computing the dimensions and degrees of popular toric varieties. … Continue reading
Posted in Algebraic geometry, Combinatorics
2 Comments
A Fibonaccilike sequence
What’s the pattern of the following sequence? Can you guess the next term? The answer is
Posted in Algebra, Combinatorics
Leave a comment
High degree polynomial with few real roots
There are univariate polyomials that the number of nonreal roots is significantly larger than that of real roots. The simplest example is where is odd. It has one real root and nonreal roots. In this post we show an example … Continue reading
Posted in Algebra, Calculus, Miscellaneous
Leave a comment