## The Lagrangian Method

What exactly is the Lagrangian method? It seems to be a popular method to solve Max/Min problems in Calculus. But generations of Calculus students may have found it troubling to understand why it works. We shall discuss this method today. This is a method of finding local maxima and minima. Clearly, derivatives don’t tell usContinue reading “The Lagrangian Method”

## Why manifolds

We know what complex manifolds are. They’re entities which “locally” look like . We also know about transition functions. However, today we’re going to ask an important question: a question that impedes all progress in modern math- **why** manifolds. We can sort of understand why manifolds have the condition that each point lies inside aContinue reading “Why manifolds”

## Notes on the Zero Forcing Algorithm

In this post I will try and understand the gist of the paper Zero Forcing Sets and the Minimum Rank of Graphs by Brualdi. Let be a field. The set of symmetric matrices of order containing entries from is called . The matrix corresponding to a graph is defined in the following way: if thereContinue reading “Notes on the Zero Forcing Algorithm”

## The Hurewicz Theorem

Here we talk about the Hurewicz theorem. Let be a path connected space with for . Then is determined up to homotopy by . What does “determined up to homotopy” mean? It means that all the spaces that satisfy the condition above are homotopic to each other. When two spaces are homotopic, what does thatContinue reading “The Hurewicz Theorem”

## A foray into Algebraic Combinatorics

I’m trying to understand this paper by Alexander Postnikov. This post is mainly a summary of some the concepts that I do not understand. Some examples. Grassmannian- A Grassmannian of a vector space is a space that parametrizes all the dimensional subspaces of . For instance, would be . Why do we need Grassmannians? BecauseContinue reading “A foray into Algebraic Combinatorics”

Today I’m going to be studying this paper by Theodor Christian Herwig to learn about -adic analysis. An absolute value on a field is a map which satisfies the usual absolute value conditions; namely and ; , and . An absolute value which is non-archimedian also satisfies the following additional property: . Why’re we doing allContinue reading “p-adic Analysis: A primer”

## Decomposition of Vector Spaces

Let be linear transformations on an -dimensional vector space such that and for . Then . How does this happen? Take the expression and multiply by any on both sides. We see that . Hence any vector can be expressed as a sum of elements in for . Why do we have a directContinue reading “Decomposition of Vector Spaces”

## Minimal Polynomials of Linear transformations

I’m prepared to embarrass myself by writing about something that should have been clear to me a long time ago. This is regarding something about the minimal polynomials of linear transformations that has always confused me. Let , where is an -dimensional vector space. Let us also assume that has as distinct eigenvectors, but theContinue reading “Minimal Polynomials of Linear transformations”

## Flat modules

This post is going to be about flat modules and flat families. A brief excursion into Commutative Algebra often brings up the following fact: a module over the ring is flat if for every inclusion of -modules the induced map is again an inclusion. Beyond this intuition often goes for a toss. Why the nameContinue reading “Flat modules”

## Open basis for Quasi-Projective Varieties

Today we’re going to generate a basis for a quasi-projective variety in the Zariski topology. These open sets will be affine charts (affine charts are open sets in the Zariski topology as the th affine chart, for instance, is the complement of the variety ). Hence, as every point lies in an affine chart, weContinue reading “Open basis for Quasi-Projective Varieties”