SEAMS School Manila 2017: Topics on Elliptic Curves

July 30, 2017July 30, 2017 / Anton Hilado / Leave a comment

A few days ago, from July 17 to 25, I attended the SEAMS (Southeast Asian Mathematical Society) School held at the Institute of Mathematics, University of the Philippines Diliman, discussing topics on elliptic curves. The school was also partially supported by CIMPA (Centre International de Mathematiques Pures et Appliquees, or International Center for Pure and Applied Mathematics), and I believe also by the Roman Number Theory Association and the Number Theory Foundation. Here’s the official website for the event:

Southeast Asian Mathematical Society (SEAMS) School Manila 2017: Topics on Elliptic Curves

There were many participants from countries all over Southeast Asia, including Indonesia, Malaysia, Philippines, and Vietnam, as well as one participant from Austria and another from India. The lecturers came from Canada, France, Italy, and Philippines.

Jerome Dimabayao and Michel Waldschmidt started off the school, introducing the algebraic and analytic aspects of elliptic curves, respectively. We have tackled these subjects in this blog, in Elliptic Curves and The Moduli Space of Elliptic Curves, but the school discussed them in much more detail; for instance, we got a glimpse of how Karl Weierstrass might have come up with the function named after him, which relates the equation defining an elliptic curve to a lattice in the complex plane. This requires some complex analysis, which unfortunately we have not discussed that much in this blog yet.

Francesco Pappalardi then discussed some important theorems regarding rational points on elliptic curves, such as the Nagell-Lutz theorem and the famous Mordell-Weil theorem. Then, Julius Basilla discussed the counting of points of elliptic curves over finite fields, often making use of the Hasse-Weil inequality which we have discussed inThe Riemann Hypothesis for Curves over Finite Fields, and the applications of this theory to cryptography. Claude Levesque then introduced to us the fascinating theory of quadratic forms, which can be used to calculate the class number of a quadratic number field (see Algebraic Numbers), and the relation of this theory to elliptic curves.

Richell Celeste discussed the reduction of elliptic curves modulo primes, a subject which we have also discussed here in the post Reduction of Elliptic Curves Modulo Primes, and two famous problems related to elliptic curves, Fermat’s Last Theorem, which was solved by Andrew Wiles in 1995, and the still unsolved Birch and Swinnerton-Dyer conjecture regarding the rank of the group of rational points of elliptic curves. Fidel Nemenzo then discussed the classical problem of finding “congruent numbers“, rational numbers forming the sides of a right triangle whose area is given by an integer, and the rather surprising connection of this problem to elliptic curves.

On the last day of the school, Jerome Dimabayao discussed the fascinating connection between elliptic curves and Galois representations, which we have given a passing mention to at the end of the post Elliptic Curves. Finally, Jared Guissmo Asuncion gave a tutorial on the software PARI which we can use to make calculations related to elliptic curves.

Participants were also given the opportunity to present their research work or topics they were interested in. I gave a short presentation discussing certain aspects of algebraic geometry related to number theory, focusing on the spectrum of the integers, and a mention of related modern mathematical research, such as Arakelov theory, and the view of the integers as a curve (under the Zariski topology) and as a three-dimensional manifold (under the etale topology).

Aside from the lectures, we also had an excursion to the mountainous province of Rizal, which is a short distance away from Manila, but provides a nice getaway from the environment of the big city. We visited a couple of art museums (one of which was also a restaurant serving traditional Filipino cuisine), an underground cave system, and a waterfall. We used this time to relax and talk with each other, for instance about our cultures, and many other things. Of course we still talked about mathematics, and during this trip I learned about many interesting things from my fellow participants, such as the class field theory problem and the subject of real algebraic geometry .

I believe lecture notes will be put up on the school website at some point by some of the participants of the school. For now, some of the lecturers have put up useful references for their lectures.

SEAMS School Manila 2017 was actually the first summer school or conference of its kind that I attended in mathematics, and I enjoyed very much the time I spent there, not only in learning about elliptic curves but also making new friends among the mathematicians in attendance. At some point I also hope to make some posts on this blog regarding the interesting things I have learned at that school.

Some Useful Links: Quantum Gravity Seminar by John Baez

July 7, 2017July 7, 2017 / Anton Hilado / Leave a comment

I have not been able to make posts tackling physics in a while, since I have lately been focusing my efforts on some purely mathematical stuff which I’m trying very hard to understand. Hence my last few posts have been quite focused mostly on algebraic geometry and category theory. Such might perhaps be the trend in the coming days, although of course I still want to make more posts on physics at some point.

Of course, the “purely mathematical” stuff I’ve been posting about is still very much related to physics. For instance, in this post I’m going to link to a webpage collecting notes from seminars by mathematical physicist John Baez on the subject of quantum gravity – and much of it involves concepts from subjects like category theory and algebraic topology (for more on the basics of these subjects from this blog, see Category Theory, Homotopy Theory, and Homology and Cohomology).

Here’s the link:

Seminar by John Baez

As Baez himself says on the page, however, quantum gravity is not the only subject tackled on his seminars. Other subjects include topological quantum field theory, quantization, and gauge theory, among many others.

John Baez also has lots of other useful stuff on his website. One of the earliest mathematics and mathematical physics blogs on the internet is This Week’s Finds in Mathematical Physics, which apparently goes back all the way to 1995, and is one of the inspirations for this blog:

This Week’s Finds in Mathematical Physics by John Baez

Many of the posts on This Week’s Finds in Mathematical Physics show the countless fruitful, productive, and beautiful interactions between mathematics and physics. This is also one of the main goals of this blog – reflected even by the posts which have been focused on mostly “purely mathematical” stuff.

Monoidal Categories and Monoids

July 3, 2017July 3, 2017 / Anton Hilado / 1 Comment

A monoid is a concept in mathematics similar to that of a group (see Groups), except that every element need not have an inverse. Therefore, a monoid is a set, equipped with a law of composition which is associative, and an identity element. An example of a monoid is the natural numbers (including zero) with the law of composition given by addition.

In this post, we will introduce certain concepts in category theory (see Category Theory) that are abstractions of the classical idea of a monoid.

A monoidal category is given by a category $\mathbf{C}$ , a bifunctor $\Box: \mathbf{C}\times\mathbf{C}\rightarrow\mathbf{C}$ , an object $I$ of $\mathbf{C}$ , and three natural isomorphisms $\alpha$ (also known as the associator), $\lambda$ (also known as the left unitor), and $\rho$ (also known as the right unitor), with components

$\displaystyle \alpha_{A,B,C}:A\Box (B\Box C)\cong (A\Box B)\Box C$

$\displaystyle \lambda_{A}:I\Box A\cong A$

$\displaystyle \rho_{A}:A\Box I\cong A$

satisfying the conditions

$\displaystyle 1_{A}\Box\alpha_{A,B,C}\circ\alpha_{A,B\Box C,D}\circ\alpha_{A,B,C}\Box 1_{D}=\alpha_{A,B,C\Box D}\circ\alpha_{A\Box B,C,D}$

for any four objects $A$ , $B$ , $C$ , and $D$ of $\mathbf{C}$ , and

$\displaystyle \alpha_{A,I,B}\circ 1_{A}\Box \lambda_{B}=\rho_{A}\Box 1_{B}$

for any two objects $A$ and $B$ in $\mathbf{C}$ .

The following “commutative diagrams” courtesy of user IkamusumeFan of Wikipedia may help express these conditions better (the symbol $\otimes$ is used here instead of $\Box$ to denote the bifunctor; this is very common notation, but we use $\Box$ following the book Categories for the Working Mathematician by Saunders Mac Lane in order to differentiate it from the tensor product, which is just one specific example of the bifunctor in question; I hope this will not cause any confusion):

If the natural isomorphisms $\alpha$ , $\lambda$ , and $\rho$ are identities, then we have a strict monoidal category.

A monoid object, or monoid in a monoidal category $(\mathbf{C},\Box,I)$ is an object $M$ of $\mathbf{C}$ together with two morphisms $\mu:M\Box M\rightarrow M$ and $\eta:I\rightarrow M$ satisfying the conditions

$\displaystyle \mu\circ 1\Box\mu\circ\alpha=\mu\circ\mu\Box 1$

$\displaystyle \mu\circ \eta\Box 1=\lambda$

$\displaystyle \mu\circ 1\Box\eta=\rho$

Again we can use the following commutative diagrams made by User IkamusumeFan of Wikipedia to help express these conditions:

As examples of monoidal categories, we have the following:

$\displaystyle (\mathbf{Set},\times,1)$

$\displaystyle (\mathbf{Ab},\otimes,\mathbb{Z})$

$\displaystyle (K\mathbf{-Mod},\otimes_{K},K)$

$(\mathbf{Cat},\times,\mathbf{1})$

$(\mathbf{C}^{\mathbf{C}},\circ,\text{Id})$ ( $\mathbf{C}^{\mathbf{C}}$ denotes the category of functors from $\mathbf{C}$ to itself)

The monoids in these monoidal categories are given respectively by the following:

Ordinary monoids

Rings

$K$ -algebras

Strict monoidal categories

Monads (see Adjoint Functors and Monads)

Among the important kinds of monoidal categories with extra structure are braided monoidal categories and symmetric monoidal categories. A braided monoidal category $\mathbf{C}$ is a monoidal category equipped with a natural isomorphism $\gamma$ (also known as a commutativity constraint) with components $\gamma_{A,B}:A\Box B\cong B\Box A$ satisfying the following coherence conditions

$\displaystyle \alpha_{B,C,A}\circ\gamma_{A,B\Box C}\circ\alpha_{A,B,C}=1_{B}\Box\gamma_{A,C}\circ\alpha_{B,A,C}\circ \gamma_{A,B}\Box 1_{C}$

$\displaystyle \alpha_{C,A,B}^{-1}\circ\gamma_{A\Box B,C}\circ\alpha_{A,B,C}^{-1}=\gamma_{A,C}\Box 1_{B}\circ\alpha_{A,C,B}^{-1}\circ 1\Box\gamma_{A}\gamma_{A,B}$

which can be expressed in the following commutative diagrams (once again credit goes to User IkamusumeFan of Wikipedia):

The category $\mathbf{C}$ is a symmetric monoidal category if the isomorphisms $\gamma_{A,B}$ satisfy the condition $\gamma_{B,A}\circ\gamma_{A,B}=1_{A\Box B}$ . We have already encountered an example of this category in The Theory of Motives in the form of tensor categories, defined as a symmetric monoidal categories whose Hom-sets (the sets of morphisms from a fixed object $A$ to another object $B$ ) form a vector space (the term “tensor category” is sometimes used to refer to other concepts in mathematics though, including symmetric monoidal categories themselves).

Another important kind of monoidal category is a closed monoidal category. A closed monoidal category is a monoidal category where the functor $-\Box B$ has a right adjoint (see Adjoint Functors and Monads) also known as the “internal Hom functor”, which is like a Hom functor that takes values in the category itself instead of in sets, and is denoted by $(\ )^{B}$ . We have already seen an example of a closed monoidal category in Adjoint Functors and Monads, given by the category of $R$ -modules for a fixed commutative ring $R$ . There $A^{B}$ was given by $\text{Hom}(A,B)$ (this is the set of $R$ -linear transformations from $A$ to $B$ , which itself is an $R$ -module).

We see therefore that the concepts of monoidal categories and monoids can be found everywhere in mathematics. Studying these structures are not only interesting for their own sake, but can also help us find or construct other useful new concepts in mathematics.

References:

Monoidal Category on Wikipedia

Monoid on Wikipedia

Braided Monoidal Category on Wikipedia

Symmetric Monoidal Category on Wikipedia

Closed Monoidal Category on Wikipedia

Image by User IkamusumeFan of Wikipedia

Categories for the Working Mathematician by Saunders Mac Lane

The Yoneda Lemma

July 1, 2017July 3, 2017 / Anton Hilado / Leave a comment

Update: Some time after I published this post, I came across the following post on another blog that makes for a really nice intuitive introduction to the ideas expressed by the Yoneda lemma:

The Most Obvious Secret in Mathematics at Math3ma

I must admit that my own post might not offer much in the way of intuition (and really could have been written better), so I highly recommend reading the above link in conjunction with this one.

In Algebraic Spaces and Stacks we introduced the notion of a representable functor, and we made use of it to “transfer” the properties of schemes to functors and categories over some fixed category. In this short post we discuss an important related concept, one of the most important concepts in category theory, called the Yoneda lemma.

Let $\mathbf{C}$ be a category, and let $A$ be any object of $\mathbf{C}$ . The Yoneda lemma states that the set of natural transformations from the functor $\text{Hom}(-,A)$ to any contravariant functor $G$ from $\mathbf{C}$ to the category of sets is in bijection with the set $G(A)$ .

In the case that $G$ is the contravariant functor $\text{Hom}(-,B)$ , where $B$ is an element of $\mathbf{C}$ , the Yoneda lemma says that the set of natural transformations from $\text{Hom}(-,A)$ to $\text{Hom}(-,B)$ is in bijection with the set $\text{Hom}(A,B)$ .

We can treat the functor $\text{Hom}(-,-)$ as a covariant functor from the category $\mathbf{C}$ to the category $\textbf{Sets}^{\mathbf{C}^{\text{op}}}$ of contravariant functors from $\mathbf{C}$ to the category of sets, which sends an object $A$ of $\mathbf{C}$ to the contravariant functor $\text{Hom}(-,A)$ , and a morphism $f:A\rightarrow B$ of $\mathbf{C}$ to the natural transformation $\text{Hom}(-,f):\text{Hom}(-,A)\rightarrow\text{Hom}(-,B)$ . Then the Yoneda lemma, via the result given in the preceding paragraph, says that the functor $\text{Hom}(-,-): \mathbf{C}\rightarrow\mathbf{Sets}^{\mathbf{C}^{\text{op}}}$ is fully faithful. We also say that this functor is an embedding; in particular, it is called the Yoneda embedding. It embeds the category $\mathbf{C}$ into the category $\mathbf{Sets}^{\mathbf{C}^{\text{op}}}$ .

The Yoneda lemma is an important ingredient of the functor of points approach to the theory of schemes. Furthermore, the Yoneda lemma tells us that the category of schemes is embedded as a subcategory of the category of contravariant functors from the category of schemes to the category of sets, so we can also try looking at a bigger subcategory of the latter category, and see if we can come up with interesting objects to study – this actually leads us to the theory of algebraic spaces.

References:

Yoneda Lemma on Wikipedia

Yoneda Lemma on nLab

Representable Functors on Rigorous Trivialities

The Most Obvious Secret in Mathematics at Math3ma

Localization and Gromov-Witten Invariants by Kai Behrend

Categories for the Working Mathematician by Saunders Mac Lane

Theories and Theorems

Math and Physics for Everyone

Month: July 2017

SEAMS School Manila 2017: Topics on Elliptic Curves

Some Useful Links: Quantum Gravity Seminar by John Baez

Monoidal Categories and Monoids

The Yoneda Lemma