Basics of Math and Physics

I’ve added another new page to the blog, called Basics of Math and Physics. As most of the posts lately have tackled subjects of increasing technical sophistication, I thought it would be a good idea to collect some of the more introductory posts in one page. It is not an exhaustive list, however, as this blog has not tackled all the basic subjects in math and physics; for example, there are no posts yet discussing real analysis or complex analysis. Perhaps if these subjects are tackled in future posts they will be added to that page too. It is also best supplemented with the references listed in Book List.

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Some Useful Links on the History of Algebraic Geometry

It’s been a while since I’ve posted on this blog, but there are some posts I’m currently working on about some subjects I’m also currently studying (that’s why it’s taking so long, as I’m trying to digest the ideas as much as I can before I can post about it). But anyway, for the moment, in this short post I’ll be putting up some links to articles on the history of algebraic geometry. Aside from telling an interesting story on its own, there is also much to be learned about a subject from studying its historical development.

We know that the origins of algebraic geometry can be traced back to Rene Descartes and Pierre de Fermat in the 17th century. This is the high school subject also known as “analytic geometry” (which, as we have mentioned in Basics of Algebraic Geometry, can be some rather confusing terminology, because in modern times the word “analytic” is usually used to refer to concepts in complex calculus).

The so-called “analytic geometry” seems to be a rather straightforward subject compared to modern-day algebraic geometry, which, as may be seen on many of the previous posts on this blog, is very abstract (but it is also this abstraction that gives it its power). How did this transformation come to be?

The mathematician Jean Dieudonne, while perhaps more known for his work in the branch of mathematics we call analysis (the more high-powered version of calculus), also served as adviser to Alexander Grothendieck, one of the most important names in the development of modern algebraic geometry. Together they wrote the influential work known as Elements de Geometrie Algebrique, often simply referred to as EGA. Dieudonne was also among the founding members of the “Bourbaki group”, a group of mathematicians who greatly influenced the development of modern mathematics. Himself a part of its development, Dieudonne wrote many works on the history of mathematics, among them the following article on the history of algebraic geometry which can be read for free on the website of the Mathematical Association of America:

The Historical Development of Algebraic Geometry by Jean Dieudonne

But before the sweeping developments instituted by Alexander Grothendieck, the modern revolution in algebraic geometry was first started by the mathematicians Oscar Zariski and Andre Weil (we discussed some of Weil’s work in The Riemann Hypothesis for Curves over Finite Fields). Zariski himself learned from the so-called “Italian school of algebraic geometry”, particularly the mathematicians Guido Castelnuovo, Federigo Enriques, and Francesco Severi.

At the International Congress of Mathematicians in 1950, both Zariski and Weil presented, separately, a survey of the developments in algebraic geometry at the time, explaining how the new “abstract algebraic geometry” was different from the old “classical algebraic geometry”, and the new advantages it presented. The proceedings of this conference are available for free online:

Proceedings of the 1950 International Congress of Mathematicians, Volume I

Proceedings of the 1950 International Congress of Mathematicians, Volume II

The articles by Weil and Zariski can be found in the second volume, but I included also the first volume for “completeness”.

All proceedings of the International Congress of Mathematicians, which is held every four years, are actually available for free online:

Proceedings of the International Congress of Mathematicians, 1983-2010

The proceedings of the 2014 International Congress of Mathematicians in Seoul, Korea, can be found here:

Proceedings of the 2014 International Congress of Mathematicians

Going back to algebraic geometry, a relatively easy to understand (for those with some basic mathematical background, anyway) summary of the work of Alexander Grothendieck’s work in algebraic geometry can be found in the following article by Colin McLarty, published in April 2016 issue of the Notices of the American Mathematical Society:

How Grothendieck Simplified Algebraic Geometry by Colin McLarty

Tangent Spaces in Algebraic Geometry

We have discussed the notion of a tangent space in Differentiable Manifolds Revisited in the context of differential geometry. In this post we take on the same topic, but this time in the context of algebraic geometry, where it is also known as the Zariski tangent space (when no confusion arises, however, it is often simply referred to as the tangent space).

This will present us with challenges, since the concept of the tangent space is perhaps best tackled using the methods of calculus, but in algebraic geometry, we want to have a notion of tangent spaces in cases where we would not usually think of calculus as being applicable, for instance in the case of varieties over finite fields. In other words, we want our treatment to be algebraic. Nevertheless, we will use the methods of calculus as an inspiration.

We don’t want to be too dependent on the parts of calculus that make use of properties of the real and complex numbers that will not carry over to the more general cases. Fortunately, if we are dealing with polynomials, we can just “borrow” the “power rule” of calculus, since that “rule” only makes use of algebraic procedures, and we need not make use of sequences, limits, and so on. Namely, if we have a polynomial given by

\displaystyle f=\sum_{j=1}^{n}ax^{j}

We set

\displaystyle \frac{\partial f}{\partial x}=\sum_{j=1}^{n}jax^{j-1}

We recall the rules for partial derivatives – in the case that we are differentiating over some variable x, we simply treat all the other variables as constants, and follow the usual rules of differential calculus. With these rules, we can now make the definition of the tangent space at the point P with coordinates (a_{1},a_{2},...,a_{n}) as the algebraic set which satisfies the equation

\displaystyle \sum_{j}\frac{\partial f}{\partial x_{j}}(P)(x_{j}-a_{j})=0

For example, consider the parabola given by the equation y-x^{2}=0. Let us take the tangent space at the point P with coordinates x=1, y=1. The procedure above gives us

\displaystyle \frac{\partial f}{\partial x}(P)(x-1)+\frac{\partial f}{\partial y}(P)(y-1)=0

Since

\displaystyle \frac{\partial f}{\partial x}=-2x

\displaystyle \frac{\partial f}{\partial y}=1

We then have

\displaystyle -2x|_{x=1,y=1}(x-1)+1|_{x=1,y=1}(y-1)=0

\displaystyle -2(1)(x-1)+1(y-1)=0

\displaystyle -2x+2+y-1=0

\displaystyle y-2x+1=0

The parabola is graphed (its real part, at least, using the Desmos graphing calculator) in the diagram below in red, with its tangent space, a line, in blue:

desmos-graph

In case the reader is not convinced by our “borrowing” of concepts from calculus and claiming that they are “algebraic” in the specific case we are dealing with, another way to look at things without making reference to calculus is the following procedure, which comes from basic high school-level “analytic geometry”. First we translate the coordinate system so that the origin is at the point P where we want to take the tangent space. Then we simply take the “linear part” of the polynomial equation, then translate again so that the origin is where it used to be originally. This gives the same results as the earlier procedure (the technical justification is given by the theory of Taylor series). More explicitly we have:

\displaystyle y-x^{2}=0

Translating the origin of coordinates to the point x=1, y=1, we have

\displaystyle (y+1)-(x+1)^{2}=0

\displaystyle y+1-(x^{2}+2x+1)=0

\displaystyle y+1-x^{2}-2x-1=0

\displaystyle y-x^{2}-2x=0

We take only the linear part, which is

\displaystyle y-2x=0

And then we translate the origin of coordinates back to the original one:

\displaystyle (y-1)-2(x-1)=0

\displaystyle y-1-2x+2=0

\displaystyle y-2x+1=0

which is the same result we had earlier.

But it may happen that the polynomial has no “linear part”. In this case the tangent space is the entirety of the ambient space. However, there is another related concept which may be useful in these cases, called the tangent cone. The tangent cone is the algebraic set which satisfies the equations we get by extracting the lowest degree part of the polynomial, which may or may not be the linear part. In the case that the lowest degree part is the linear part, the tangent space and the tangent cone coincide, and if this holds for all points of a variety, we say that the variety is nonsingular.

To give an explicit example, consider the curve y^{2}=x^{3}+x^{2}, as seen in the diagram below in red (its real part graphed once again using the Desmos graphing calculator):

desmos-graph (2)

The equation that defines this curve has no linear part. Therefore the tangent space at the origin consists of all x and y which satisfy the trivial equation 0=0; but then, all values of x and y satisfy this equation, and therefore the tangent space is the “affine plane” \mathbb{A}^{2}. However, the lowest order part is y^{2}=x^{2}, which is satisfied by all points which also satisfy either of the two equations y=x or y=-x. These points form the blue and orange diagonal lines in the diagram. Since the tangent space and the tangent cone do not agree, the curve is singular at the origin.

We can also define the tangent space in a more abstract manner, using the concepts we have discussed in Localization. Let \mathfrak{m} be the unique maximal ideal of the local ring O_{X,P}, and let \mathfrak{m}^{2} be the product ideal whose elements are the sums of products of elements of \mathfrak{m}. The quotient \mathfrak{m}/\mathfrak{m}^{2} is then a vector space over the residue field k. The tangent space of X at P is then defined as the dual of this vector space (the vector space of linear transformations from \mathfrak{m}/\mathfrak{m}^{2} to k). The vector space \mathfrak{m}/\mathfrak{m}^{2} itself is called the cotangent space of X at P. We can think of its elements as linear polynomial functions on the tangent space. There is an analogous abstract definition of the tangent cone, namely as the spectrum of the graded ring \oplus_{i\geq 0}\mathfrak{m}^{i}/\mathfrak{m}^{i+1}.

References:

Zariski Tangent Space on Wikipedia

Tangent Cone on Wikipedia

Desmos Graphing Calculator

Algebraic Geometry by J.S. Milne

Algebraic Geometry by Andreas Gathmann

Algebraic Geometry by Robin Hartshorne

Metric, Norm, and Inner Product

In Vector Spaces, Modules, and Linear Algebra, we defined vector spaces as sets closed under addition and scalar multiplication (in this case the scalars are the elements of a field; if they are elements of a ring which is not a field, we have not a vector space but a module). We have seen since then that the study of vector spaces, linear algebra, is very useful, interesting, and ubiquitous in mathematics.

In this post we discuss vector spaces with some more additional structure – which will give them a topology (Basics of Topology and Continuous Functions), giving rise to topological vector spaces. This also leads to the branch of mathematics called functional analysis, which has applications to subjects such as quantum mechanics, aside from being an interesting subject in itself. Two of the important objects of study in functional analysis that we will introduce by the end of this post are Banach spaces and Hilbert spaces.

I. Metric

We start with the concept of a metric. We have to get two things out of the way. First, this is not the same as the metric tensor in differential geometry, although it also gives us a notion of a “distance”. Second, the concept of metric is not limited to vector spaces only, unlike the other two concepts we will discuss in this post. It is actually something that we can put on a set to define a topology, called the metric topology.

As we discussed in Basics of Topology and Continuous Functions, we may think of a topology as an “arrangement”. The notion of “distance” provided by the metric gives us an intuitive such arrangement. We will make this concrete shortly, but first we give the technical definition of the metric. We quote from the book Topology by James R. Munkres:

A metric on a set X is a function

\displaystyle d: X\times X\rightarrow \mathbb{R}

having the following properties:

1) d(x, y)>0 for all x,y \in X; equality holds if and only if x=y.

2) d(x,y)=d(y,x) for all x,y \in X.

3) (Triangle inequality) d(x,y)+d(y,z)>d(x,z), for all x,y,z \in X.

We quote from the same book another important definition:

Given a metric d on X, the number d(x, y) is often called the distance between x and y in the metric d. Given \epsilon >0, consider the set

\displaystyle B_{d}(x,\epsilon)=\{y|d(x,y)<\epsilon\}

of all points у whose distance from x is less than \epsilon. It is called the \epsilon-ball centered at x. Sometimes we omit the metric d from the notation and write this ball simply as B(x,\epsilon) when no confusion will arise.

Finally, once more from the same book, we have the definition of the metric topology:

If d is a metric on the set X, then the collection of all \epsilon-balls B_{d}(x,\epsilon), for x\in X and \epsilon>0, is a basis for a topology on X, called the metric topology induced by d.

We recall that the basis of a topology is a collection of open sets such that every other open set can be described as a union of the elements of this collection. A set with a specific metric that makes it into a topological space with the metric topology is called a metric space.

An example of a metric on the set \mathbb{R}^{n} is given by the ordinary “distance formula”:

\displaystyle d(x,y)=\sqrt{\sum_{i=1}^{n}(x_{i}-y_{i})^{2}}

Note: We have followed the notation of the book of Munkres, which may be different from the usual notation. Here x and y are two different points on \mathbb{R}^{n}, and x_{i} and y_{i} are their respective coordinates.

The above metric is not the only one possible however. There are many others. For instance, we may simply put

\displaystyle d(x,y)=0 if \displaystyle x=y

\displaystyle d(x,y)=1 if \displaystyle x\neq y.

This is called the discrete metric, and one may check that it satisfies the definition of a metric. One may think of it as something that simply specifies the distance from a point to itself as “near”, and the distance to any other point that is not itself as “far”. There is also the taxicab metric, given by the following formula:

\displaystyle d(x,y)=\sum_{i=1}^{n}|x_{i}-y_{i}|

One way to think of the taxicab metric, which reflects the origins of the name, is that it is the “distance” important to taxi drivers (needed to calculate the fare) in a certain city with perpendicular roads. The ordinary distance formula is not very helpful since one needs to stay on the roads – therefore, for example, if one needs to go from point x to point y which are on opposite corners of a square, the distance traversed is not equal to the length of the diagonal, but is instead equal to the length of two sides. Again, one may check that the taxicab metric satisfies the definition of a metric.

II. Norm

Now we move on to vector spaces (we will consider in this post only vector spaces over the real or complex numbers), and some mathematical concepts that we can associate with them, as suggested in the beginning of this post. Being a set closed under addition and scalar multiplication is already a useful concept, as we have seen, but we can still add on some ideas that would make them even more interesting. The notion of metric that we have discussed earlier will show up repeatedly over this discussion.

We first discuss the notion of a norm, which gives us a notion of a “magnitude” of a vector. We quote from the book Introductory Functional Analysis with Applications by Erwin Kreyszig for the definition:

A norm on a (real or complex) vector space X is a real valued function on X whose value at an x\in X is denoted by

\displaystyle \|x\|    (read “norm of x“)

and which has the properties

(N1) \|x\|\geq 0

(N2) \|x\|=0\iff x=0

(N3) \|\alpha x\|=|\alpha|\|x\|

(N4) \|x+y\|\leq\|x\|+\|y\|    (triangle inequality)

here x and y are arbitrary vectors in X and \alpha is any scalar.

A vector space with a specified norm is called a normed space.

A norm automatically provides a vector space with a metric; in other words, a normed space is always a metric space. The metric is given in terms of the norm by the following equation:

\displaystyle d(x,y)=\|x-y\|

However, not all metrics come from a norm. An example is the discrete metric, which satisfies the properties of the metric but not the norm.

III. Inner Product

Next we discuss the inner product. The inner product gives us a notion of “orthogonality”, a concept which we already saw in action in Some Basics of Fourier Analysis. Intuitively, when two vectors are “orthogonal”, they are “perpendicular” in some sense. However, our geometric intuition may not be as useful when we are discussing, say, the infinite-dimensional vector space whose elements are functions. For this we need a more abstract notion of orthogonality, which is embodied by the inner product. Again, for the technical definition we quote from the book of Kreyszig:

With every pair of vectors x and y there is associated a scalar which is written

\displaystyle \langle x,y\rangle

and is called the inner product of x and y, such that for all vectors x, y, z and scalars \alpha we have

(IPl) \langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle

(IP2) \langle \alpha x,y\rangle=\alpha\langle x,y\rangle

(IP3) \langle x,y\rangle=\overline{\langle y,x\rangle}

(IP4) \langle x,x\rangle\geq 0,    \langle x,x\rangle=0 \iff x=0

A vector space with a specified inner product is called an inner product space.

One of the most basic examples, in the case of a finite-dimensional vector space, is given by the following procedure. Let x and y be elements (vectors) of some n-dimensional real vector space X, with respective components x_{1}, x_{2},...,x_{n} and y_{1},y_{2},...,y_{n} in some basis. Then we can set

\displaystyle \langle x,y\rangle=x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}

This is the familiar “dot product” taught in introductory university-level mathematics courses.

Let us now see how the inner product gives us a notion of “orthogonality”. To make things even easier to visualize, let us set n=2, so that we are dealing with vectors (which we can now think of as quantities with magnitude and direction) in the plane. A unit vector x pointing “east” has components x_{1}=1, x_{2}=0, while a unit vector y pointing “north” has components y_{1}=0, y_{2}=1. These two vectors are perpendicular, or orthogonal. Computing the inner product we discussed earlier, we have

\displaystyle \langle x,y\rangle=(1)(0)+(0)(1)=0.

We say, therefore, that two vectors are orthogonal when their inner product is zero. As we have mentioned earlier, we can extend this to cases where our geometric intuition may no longer be as useful to us. For example, consider the infinite dimensional vector space of (real-valued) functions which are “square integrable” over some interval (if we square them and integrate over this interval, we have a finite answer), say [0,1]. We set our inner product to be

\displaystyle \int_{0}^{1}f(x)g(x)dx.

As an example, let f(x)=\text{cos}(2\pi x) and g(x)=\text{sin}(2\pi x). We say that these functions are “orthogonal”, but it is hard to imagine in what way. But if we take the inner product, we will see that

\displaystyle \int_{0}^{1}\text{cos}(2\pi x)\text{sin}(2\pi x)dx=0.

Hence we see that \text{cos}(2\pi x) and \text{sin}(2\pi x) are orthogonal. Similarly, we have

\displaystyle \int_{0}^{1}\text{cos}(2\pi x)\text{cos}(4\pi x)dx=0

and \text{cos}(2\pi x) and \text{cos}(4\pi x) are also orthogonal. We have discussed this in more detail in Some Basics of Fourier Analysis. We have also seen in that post that orthogonality plays a big role in the subject of Fourier analysis.

Just as a norm always induces a metric, an inner product also induces a norm, and by extension also a metric. In other words, an inner product space is also a normed space, and also a metric space. The norm is given in terms of the inner product by the following expression:

\displaystyle \|x\|=\sqrt{\langle x,x\rangle}

Just as with the norm and the metric, although an inner product always induces a norm, not every norm is induced by an inner product.

IV. Banach Spaces and Hilbert Spaces

There is one more concept I want to discuss in this post. In Valuations and Completions, we discussed Cauchy sequences and completions. Those concepts still carry on here, because they are actually part of the study of metric spaces (in fact, the valuations discussed in that post actually serve as a metric on the fields that were discussed, showing how in number theory the concept of metric and metric spaces still make an appearance). If every Cauchy sequence in a metric space X converges to an element in X, then we say that X is a complete metric space.

Since normed spaces and inner product spaces are also metric spaces, the notion of a complete metric space still makes sense, and we have special names for them. A normed space which is also a complete metric space is called a Banach space, while an inner product space which is also a complete metric space is called a Hilbert space. Finite-dimensional vector spaces (over the real or complex numbers) are always complete, and therefore we only really need the distinction when we are dealing with infinite dimensional vector spaces.

Banach spaces and Hilbert spaces are important in quantum mechanics. We recall in Some Basics of Quantum Mechanics that the possible states of a system in quantum mechanics form a vector space. However, more is true – they actually form a Hilbert space, and the states that we can observe “classically” are orthogonal to each other. The Dirac “bra-ket” notation that we have discussed makes use of the inner product to express probabilities.

Meanwhile, Banach spaces often arise when studying operators, which correspond to observables such as position and momentum. Of course the states form Banach spaces too, since all Hilbert spaces are Banach spaces, but there is much motivation to study the Banach spaces formed by the operators as well instead of just that formed by the states. This is an important aspect of the more mathematically involved treatments of quantum mechanics.

References:

Topological Vector Space on Wikipedia

Functional Analysis on Wikipedia

Metric on Wikipedia

Norm on Wikipedia

Inner Product Space on Wikipedia

Complete Metric Space on Wikipedia

Banach Space on Wikipedia

Hilbert Space on Wikipedia

A Functional Analysis Primer on Bahcemizi Yetistermeliyiz

Topology by James R. Munkres

Introductory Functional Analysis with Applications by Erwin Kreyszig

Real Analysis by Halsey Royden