Classical to Quantum
504 views Aug 10, 2024
504 views • Aug 10, 2024
In this video Dr Kevin Limanta introduces the algebraic approach to circle integration over finite fields. This is largely his PhD thesis, written in 2022 under my supervision, and entitled "Super Catalan numbers and Fourier summation over Finite Fields, available at https://unsworks.unsw.edu.au/entities...In this video Kevin lays the algebraic groundwork for this novel approach in which the remarkable Super Catalan numbers are revealed to play an intimate role in harmonic analysis over finite fields, just as they do over the rational numbers. He also discusses the forms of the rotation groups in each of the blue, red and green geometries. This is involved in the Circular Integration functional which plays a central role in our theory. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... If you would like to support these new initiatives for mathematics education and research, you could also consider become a Patreon supporter. Thank you. Here are the Wild Egg Maths Playlists (some available only to Members!)
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Transcript
0:00
Hello everyone, well, it's a pleasure to have my former student Kevin Lamata join us today to tell us about, largely, his PhD thesis, which was completed in 2022 at UNSW
0:14
under my supervision. The title is "Super Catalan Numbers and F(q)-summations over Finite Fields." You can get it by going to the UNSW library. This is an important development,
0:28
I think, because, first of all, it explains these otherwise mysterious objects called Super Catalan Numbers that have intrigued combinatorialists for many decades,
0:40
but it also forms a bridge between the characteristic zero story—the familiar
0:45
unit circle integration we've been talking about—and a completely discrete version
0:51
over a finite field. It turns out that, in some sense, the same formulas, to a certain extent,
1:00
apply in all these situations. We'll see further down the road that, in fact, this algebraic
1:06
approach also allows us to create an integration theory over non-compact circles, for example,
1:14
relativistic circles either in the red or the green setting, where physicists usually encounter
1:21
divergence issues—things going to infinity. So, it's pretty interesting. In this initial talk,
1:29
Kevin will be setting the stage for this very interesting story. I do hope you enjoy his talk.
1:40
Hello everyone, I am Kevin Lamata, a former PhD student of Norman J. Wildberger. I will be
1:46
presenting a mini-lecture today, and maybe also in the next one or two, about algebraic integration
1:53
theory on circles. What I am presenting here is basically my PhD thesis, which I hope
2:00
will be a good way to introduce people to this beautiful theory of integration and, ultimately,
2:06
harmonic analysis presented in a more algebraic way, without reference to real numbers. In the
2:14
last video, Norman pretty much captured the motivation of this mini-lecture series, which
2:19
is to come up with a way of defining integration over circles that work in a more general field.
2:30
In this video, I will mainly talk about the setup before we delve deeper into the subject. So,
2:40
we start with any field F, although in our analysis we will exclude fields of characteristic
2:48
two and just work with either fields of characteristic p, where p is greater than two,
2:55
or fields of characteristic zero. Okay, so for now, let's just say that we have a field F and
3:01
consider the affine plane A, which consists of points of the form (x, y) where x and y
3:09
come from the field F. We consider three unit circles. The idea is to work with unit circles,
3:20
but we want to be more flexible and consider three at the same time instead of just one.
3:27
They are color-coded as blue, red, and green, and this is the very idea of chromogeometry,
3:35
which Norman introduced. The blue unit circle is defined to be the set of points (x,
3:44
y) in the plane where x² + y² = 1, and this 1 here is really just the multiplicative identity of the
3:54
field F. So this subscript F is sometimes used to emphasize that this is coming from
4:03
the field F. This is denoted by S₁ subscript B, and this is pretty much the unit circle that we
4:11
are most familiar with. We can think of this as the Euclidean unit circle. The red unit circle,
4:21
S₁ subscript R, is defined to be the collection of points (x, y) in A where x² - y² = 1,
4:29
so there is this subtle change of sign—instead of plus, we have a minus sign there. Also, the green
4:37
unit circle, S₁ subscript G, is defined to be the collection of points (x, y) where xy = 1. So,
4:45
the last two circles, the red and green circles, we can think of them as relativistic unit circles,
4:53
coming from relativistic geometry. In my PhD, I was mainly interested in the case where the field
5:04
F is finite of size pr, where p, which is the characteristic of the field, is greater than two.
5:18
Now, the essence is this integration functional. We want to define a linear functional I,
5:26
which mimics this integration functional, and we call this I. It takes polynomials or polynomial
5:36
numbers, say in alpha and beta, to the underlying field F and satisfies three conditions. So,
5:46
we impose three conditions on this functional I. The first one is just a mere normalization
5:52
condition, which states that the polynomial or polynomial number one, which is alpha to the 0,
5:58
beta to the 0, is mapped to one in the field. The second condition is called the locality condition,
6:08
and this is really saying that if my polynomial restricts to the zero function on the unit
6:15
circle (and here I put a subscript C to indicate the color—blue, red, or green),
6:36
then the integral of P ought to be zero. The third condition is called invariance,
6:44
specifically rotational invariance. Basically, if you have any polynomial P and any rotation
6:55
matrix X, then the new polynomial that you get by performing this action X on P,
7:05
the integral of that is the same as the integral of P. Of course, to make sense of this,
7:13
we have to define a few more things, such as what this action means and what this
7:24
restriction on the unit circle means. So, one is to carefully distinguish between polynomials and
7:34
polynomial functions. This is especially important if the field is finite. Another thing is to see
7:44
how the group action interacts, which is seen in the invariance condition. We can actually
7:54
look at what the rotation group looks like in geometry, in the blue, red, and green geometry.
8:06
Now, with this distinction between polynomials and polynomial functions, we recall again that we
8:15
have this field F, and we can associate with it an affine plane A. We can think of functions from the
8:27
plane A to the field F. This space of functions is denoted by F(A), and this is a vector space if you
8:39
think of the addition and scalar multiplication as being pointwise. If we agree that we can
8:47
multiply functions also pointwise, then this is really an algebra over F. Now, if I have,
8:59
say, a polynomial P₁, which is equal to alpha cubed minus 3 alpha beta squared plus ½ alpha
9:13
to the 4th beta, then the associated polynomial function is what you get if you replace alpha
9:25
with x and beta with y. Really, P₁(x, y) = x³ - 3xy² + ½ x⁴y. That's really just saying that
9:43
this is a function that takes (x, y) in the plane as input and spits out this element in the field,
9:53
x³ - 3xy² + ½ x⁴y. Whereas this polynomial P₁, we can think of it as a formal object. It is a linear
10:09
combination of the monomials alphak, betal, with coefficients coming from the field F. You can do
10:21
the same with, say, a second polynomial P₂: beta - 2 alpha squared + 3 alpha beta. The associated
10:29
polynomial function is P₂(x, y) = y - 2x² + 3xy. Now, this distinction is especially important
10:42
if you're dealing with fields that are finite. For example, if you have fields of size seven,
10:53
then, say, I have this polynomial P = alpha to the 7th beta squared minus alpha beta
10:59
squared. This is a non-zero polynomial because for some polynomial to be zero, all the coefficients
11:09
must be zero. However, if you look at what the associated polynomial function is, it's p(x,
11:24
y) = x7 y2 - xy², which we can write as x7 - x times y². But because we're in this field
11:36
with seven elements and this is due to Fermat, x7 - x is always zero for any x in the field.
11:47
Essentially, what we have here is a zero function because it takes any input (x,
11:56
y) and gives zero as the value. So, we start with a non-zero polynomial,
12:05
but the associated polynomial function is zero. This might not be very apparent if you are dealing
12:20
with fields of characteristic zero because you just replace alpha with x and beta with
12:29
y, and you don't actually need to distinguish carefully. However, with this setup, if you are
12:41
dealing with finite fields, we do have to be mindful of this distinction, so it pays to be
12:52
more formal and careful in our analysis. What we do here is we induce a map, okay? This conversion
13:05
from polynomial to polynomial functions is captured by this evaluation map, Epsilon.
13:15
Now, let's have a look at various group actions. We start with the group G or GL(2, F),
13:24
which is just the group of 2x2 invertible matrices with entries in the field F. What we have here is,
13:35
for example, if I have a matrix G that looks like [a, b; c, d], we can define
13:41
various actions. The first one is probably the most natural one. We can define a right action
13:49
of G on the plane A. If I have a point (x, y) here and I have a matrix G, then G acts on
13:59
A this way: (x, y) * G is just this new point (ax + cy, bx + dy). You can think of this as
14:10
really just multiplication. You can think of (x, y) as a row vector and G as a matrix. So, this is
14:20
what you get when you do that multiplication. Now, with this right action of G on the plane,
14:29
we can define a new action. This is now a left action of G on F(A), the function space, which
14:40
basically takes a function f in F(A) and gives you a new function g.f. This function g.f takes (x, y)
14:55
as input and spits out f((x, y) * G). So, the role of G here is that this G moves inside the bracket,
15:11
basically. What it does is, (x, y) is mapped to f(ax + cy, bx + dy). We can also define,
15:29
similarly, a left action of G on this polynomial space. Just as before, we have a polynomial or
15:38
poly P, then G.P is the new polynomial or poly. It basically replaces alpha with a alpha + c beta and
15:49
beta with b alpha + d beta. For example, if I have my matrix G to be [1, 1; 0, 1], and a polynomial
16:01
P that looks like alpha squared beta + 2 alpha beta squared, then G.P is this new polynomial,
16:11
which we get by replacing beta with alpha - beta, and we fix alpha. The reason for this is because,
16:21
well, the first column of G is [1, 0], so we don't change alpha, but we do change beta
16:29
to alpha - beta because of the second column being [1, -1]. So, this is really just alpha squared
16:37
times (alpha - beta) + 2 alpha times (alpha - beta) squared, and you get this new polynomial.
16:46
Now let's have a look if I consider, instead of P, Epsilon(P), which is the associated polynomial
16:53
function of P, we can check that this basically just simplifies to x² * (x - y) + 2x * (x - y)²,
17:08
and this can be checked to be, well, just the evaluation of this polynomial G.P. So, in general,
17:21
it works pretty much the same: if I have a polynomial P and a matrix G, then Epsilon(G.P), so
17:33
the evaluation of G.P, is really just the matrix G acting on the left on this function Epsilon(P).
17:49
Now, let's have a look at what the rotation groups look like in the blue, red, and green geometry.
17:57
This is pretty much explained extensively by Norman in his "Wild Linear Algebra" series,
18:06
and I encourage you to refer to that playlist for more detailed explanations. But in summary,
18:14
what we have here is this so-called dihedral algebra D over F. This D(F), which is really
18:26
just the algebra of 2x2 matrices over F. Now, as an algebra, especially as a vector space, it has
18:34
a basis, and we can choose one basis, a pleasant basis, to be consisting of these four matrices,
18:45
which we call 1, i, j, and k. So, 1 here (1 sub D) is really just the identity matrix [1,
18:56
1; 1, 1]. Okay, here I put 1 sub F just to emphasize that this 1 is coming from the field
19:04
F. Then there is this matrix i, which is really just [0, 1; -1, 0], matrix j, which is [0, 1;
19:12
1, 0], and matrix k [1, 0; -1, 1]. Now, together with this algebra, we can define a map called the
19:23
adjugate map. This adjugate map takes a matrix [a, b; c, d] and gives you a new matrix [d, -b; -c,
19:39
a]. So, we can check that if I have a matrix h in this dihedral algebra D, h * h star and h star * h
19:54
are equal, and really, it's just a multiple of the identity matrix, and the scalar in front
20:01
is really just the determinant of the matrix h. If you adjugate a matrix and adjugate it again,
20:12
you will end up with the same matrix you started with. If you multiply two matrices h₁ and h₂ and
20:22
then take the adjugate, it's just the same as doing the adjugates first and then multiplying
20:31
them, but doing so, you have to reverse the order. So, h₁ * h₂ star is h₂ star * h₁ star, and this is
20:42
true for all h₁ and h₂ in D. Together with this adjugate map, D is really a star algebra over F.
20:54
Now, there are three special subalgebras of interest here, and these are called the blue,
21:03
red, and green complex numbers, and you might guess that there is a connection with the blue,
21:12
red, and green unit circles that we defined at the very beginning, but we'll explain that
21:20
shortly. What you can see here is the blue complex number C sub B is really just the
21:29
span of 1 and i, so it's a two-dimensional subalgebra. In each case, C sub R and C sub G, the
21:47
matrix 1 is always present. What differentiates them is i, j, and k. So, R complex number C sub
21:57
R is the span of 1 and j, and green complex numbers is the span of 1 and k. Furthermore,
22:11
we can check that the matrices commute with each other in each subalgebra, and that makes C sub B,
22:19
C sub R, and C sub G commutative star subalgebras of D. Now, with this framework, you can see that
22:31
complex numbers now look like matrices. The matrices 1, i, j, and k are all matrices,
22:40
and this adjugate map really works as conjugation in the traditional sense. For example, if I have
22:48
a complex number z that looks like x * 1 + y * i, a typical complex number in C sub B, then z star,
22:59
which is x * 1 + y * i star, is really just x * 1 star + y * i star. It is not so hard to
23:09
check that 1 star is 1, but i star is -i. So, in the end, z star is just x * 1 - y * i. So,
23:21
this idea of conjugation is captured here at the level of matrices as adjugation. We can define
23:31
a symmetric bilinear form on this algebra as = ½ trace(h₁ star h₂). Again,
23:43
this defines a symmetric bilinear form, and we can also compute the associated quadratic form
23:54
with this bilinear form. If I have a matrix h and compute , I get what I call the "quadrance"
24:02
of h, which is the determinant of h. This is not very hard to check. With this framework, we can
24:13
define the unit circles from the perspective of matrices having quadrance 1. The blue unit
24:21
circle now can be defined as the collection of all blue complex numbers z such that the quadrance of
24:30
z equals 1. Similarly, for the red unit circle, defined as the collection of red complex numbers
24:39
z such that the quadrance of z equals 1, and the same for the green unit circle. What you can see
24:49
here, and this might not be easily seen if you look at the definition at the very beginning in
24:55
terms of points in the plane, but here you can perhaps appreciate the algebraic structure more
25:05
because, for example, this is actually a group, and this can actually be seen from the fact that
25:14
the quadrance is multiplicative because it's nothing but the determinant. So, for example,
25:21
if I have two matrices, or two elements in, say, blue complex numbers h₁ and h₂, since Q(h₁ * h₂),
25:34
the quadrance, is really just the determinant, we have the determinant of h₁ * h₂, which is
25:42
multiplicative. So, we have determinant of h₁ * determinant of h₂, which is just Q(h₁) * Q(h₂).
25:53
Now, how do we connect this with the unit circle that we see at the beginning?
25:58
Well, we can associate, for example, a blue complex number and a red complex
26:04
number x * 1 + yi and x * 1 + yj with just the point (x,
26:11
y) in the plane. The association for the green complex number is slightly different. Here,
26:20
we associate the complex number x * 1 + y * k not with (x, y) but with (x + y, x - y).
26:32
We can check that this association is actually a projection, and we do get the definition of
26:38
these unit circles from the beginning if you look at it from this point of view.
26:46
Now, the rotation group SO sub C (F) is really just the subgroup of GL(2) that preserves the
26:57
bilinear form that we defined previously. This is restricted to the subalgebra C of color
27:05
C with determinant 1, and this is really just the unit circle S₁. So, for example,
27:19
the blue rotation group SO sub B viewed this way is really just the set of matrices of the form [x,
27:28
y; -y, x]. So, this is really just x * 1 + yi, where the determinant of that is 1,
27:38
which is just x² + y² = 1. Now, this equation has a well-known rational parameterization. So,
27:54
think of what people usually call it—cosine, like trigonometric parameterization, like
28:03
cosine and sine—but now, because we want to avoid reference to real numbers and work with a more
28:11
general framework that works in any field without recourse to infinite processes, we want to avoid
28:20
that completely. We want to steer away from this mention of sine and cosine and purely work in the
28:31
field generally, without any mention of infinite processes, and this is actually possible. So,
28:43
say here, this equation x² + y² = 1 can be parameterized. Every point that satisfies
28:52
x² + y² = 1, except for one single point (-1, 0), can be written as (1 - t²)/(1 + t²) for the
29:03
x component and 2t/(1 + t²) for the y component, where t is in the field. This way, we can rewrite
29:15
this as just the collection of all matrices of the form [1/(1 + t²)] times [1 - t², 2t; -2t, 1 - t²],
29:28
where t is in the field, where 1 + t² is non-zero. This is important because, for example,
29:38
fields might be finite, and 1 + t² might actually be zero. Over rationals, 1 + t² is never zero,
29:48
so it is fine. So, 1 + t², we are not dividing by zero, but sometimes you do get situations
29:58
where 1 + t² equals zero if you're allowing yourself to work with more general fields. Because
30:09
the point (-1, 0) is not able to be written this way, we have to add this matrix that corresponds
30:19
to x = -1 and y = 0. So, this matrix is just [-1, 0; 0, -1], which is -1 times the identity matrix.
30:35
The same can be said about the red rotation group SO sub R. This is the set of all matrices of the
30:43
form [x, y; y, x] with determinant 1, which basically simplifies to x² - y² = 1. Similar
30:53
to the blue case, we can parameterize this. Every point that satisfies this equation x² - y² = 1,
31:05
except for one point (1, 0), can be written as x = (1 + t²)/(1 - t²) and y = 2t/(1 - t²). Therefore,
31:19
we can write SO sub R this way, where we have to add this special matrix [-1, 0; 0, -1] that
31:31
comes from this point (x, y) = (-1, 0). This time, the restriction is that 1 - t² is non-zero. So,
31:45
in any field, we get at least t = 1, and if your field is not of characteristic 2,
31:51
you also get t = -1. So, if we are working with fields of characteristic not equal to 2,
31:59
we get 1 and -1 to exclude. The easiest one is actually the green rotation group because this
32:10
is just the collection of matrices of the form [x, 0; 0, y], where xy = 1, and this defining equation
32:20
is very easy to tackle because if I let x = t, then y = 1/t. So, as long as your t is non-zero,
32:31
then I can write this as [t, 0; 0, 1/t], where t is coming from the field F, where t is non-zero.
32:42
I might also mention that, well, especially the size of these rotation groups and essentially the
32:51
size of the unit circle groups depends on the characteristic of the field,
32:59
especially if your field is finite. So, this set here is actually a finite set,
33:09
and what that means is I can find the cardinality of this set. If you are dealing with fields of
33:21
characteristic p, there is actually an interesting number-theoretic restriction because 1 + t² might
33:32
be zero for some finite field. In some finite fields, 1 + t² is never zero, so it behaves
33:40
pretty much the same as the rational numbers, but in some other fields, 1 + t² might be zero,
33:49
and that means we have to be more careful in the analysis. So, depending on the characteristic,
34:00
the size of these rotation groups and the size of the unit circle groups vary.
34:11
Now, just to conclude today's video, we will now more carefully define what our integral
34:19
is. We can define our integral, say, a linear functional I sub C. There is this
34:28
subscript C on the space of polynomials or Pols to F, and it is called a circular integral functional
34:38
with respect to S₁ sub C if this linear functional I sub C satisfies the three conditions we talked
34:49
about before—the normalization, locality, and invariance conditions. One of the main theorems
34:58
I proved in my PhD thesis is that there is one and only one circular integral functional I sub
35:06
C with respect to the unit circle S₁ with color C. So, we will have a look at three geometries,
35:18
and what this theorem is saying is that in each geometry, there is only one integral.
35:30
There are some natural questions we can ask. For example, this is a uniqueness theorem,
35:39
right? What does it look like? We can establish that it exists and is unique,
35:48
but what does it look like? Can we find an explicit formula that we can use?
35:58
If I have a polynomial P, for example, alpha² + beta² - 5 or whatever, can I compute the integral
36:12
easily? How do you prove its uniqueness? How can we be sure that there is no other integral that
36:25
satisfies these three conditions that we don't know of? Of course, with this field being finite
36:39
or of characteristic p, for example, greater than two or of characteristic zero, is there
36:46
some difference? What is different in that case? That is something that we can talk about in the
36:53
next video or two. Hopefully, you'll follow along as we talk about this beautiful theory
37:06
of integration in a more algebraic way. Thank you for watching, and I'll see you in the next video.
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