The Way Theorized

The Langlands program is vast and quite ambitious. Having started with Langlands in the 1960s with a letter to Weil in number fields, it [Langlands program] then moved to a similar setting in function fields, and it was eventually realized as a geometric correspondence by Beilinson-Drinfeld. Continuing this, we see the progress in mathematical physics with works, for instance, by Kapustin-Witten, Frenkel, and Gaiotto-Teshner. The proof of the categorical geometric Langlands conjecture by Gaitsgory et al., available here, is for the global, unramified case. Peter Scholze's work in the geometric Langlands program—which has been dubbed the Langlands Program 2.0—conjectures in the arithmetic Langlands in a form similar to the geometric Langlands conjecture. A talk by Scholze at the Bourbaki meeting is available here, and the write-up here. David Ben-Zvi gave a talk at the AMS Current Events Bulletin at the Joint Mathematical Meetings on 'What is the Geometric Langlands Correspondence about?' and this post contains philosophical elements from his talk and the survey article, which presents the program wonderfully.

The classification of objects (by isomorphism classes) is an important problem in mathematics. These objects can be representations of a quiver, subspaces $V \subseteq \mathbb{C}^n$ of dimension $k$, plane curves, and so on. We classify by studying the relevant moduli. Moduli have their earliest roots in Riemann's 1857 paper. Over time, it has influenced many areas of mathematics through the work of invariant theorists like Hilbert, Weil, Mumford (and his Geometric Invariant Theory), and, of course, Grothendieck, with his formalization of analytic moduli theory. The study of moduli has advanced significantly since Grothendieck's definition of representability and the introduction of (pre)stacks and descent theory.

This month's post is rather about an announcement of a note that I have written titled "Quiver Moduli, Motivic Hall Algebra, and Auslander-Reiten Triangles", which is available at this link. The note discusses quiver moduli and their passage to the motivic Hall algebra. This motivic Hall algebra already has data on extensions, which can provide a playground for AR theory in both abelian categories ($\text{Rep}_{k}Q$ or $\text{Mod-}kQ$) and the derived category/triangulated settings. But we use Happel's theorem to pass to another equivalent category of stable modules over the repetitive algebra of $kQ$, the path algebra of finite global dimension—for an exposition about Happel's theorem, you may read this talk.

ACT: Motive or Intention?

As I believe, one of the most significant developments in mathematics has been the study of dualities. There are many standard ones, like Hilbert's Nullstellensatz, Schemes, Gelfand duality, or Tannaka duality (on the dualities note, you may see Lecture 1 of Scholze's Gestalten notes). We will see the Tannaka (Grothendieck–Saavedra–Deligne–Milne) duality for a commutative Hopf algebra, however, in an incomplete way here. Other similar posts include theQuill20 and theQuill21. 

This post will be very specific (but basic in nature); for those who are not familiar with Quivers, please take a look here or here (or here, which is slightly advanced). Most (if not all!) of my knowledge about Auslander-Reiten theory of quivers comes from Prof. Amit Kuber.

The Auslander-Reiten (AR) quiver is a picture of a quiver with all of its indecomposable representations. For instance, in the $A_3$ quiver with linear orientation ($1 \to 2 \to 3$), we can draw the following AR quiver

AR graph for $A_3$

We are interested in finding an AR sequence in this quiver that is an exact sequence

In this post, we will quickly discuss the (basic) definition of triangulated categories and why there exist non-functorial cones in these categories. (The whole motivation, at least pointed in, to pass to some stable homotopy theory, like $\infty$-category, is built somewhat around this failure of triangulated categories.) We will discover more about these in the coming posts.

I was in a talk given on monodromy groups and the arithmetic (and thinness) property of them by Jitendra Bajpai. This post is a quick write-up on that problem.

I realize that there have been many posts in the Quill series about math. Many apologies for that.
Let $G$ be an abelian variety. We define the 1-shifted Cartier duality (following Ben-Zvi's notations) to be

Tannaka-Krein duality, in its classical form, says that given a compact group $G$, one can reconstruct the group $G$ from the category of the finite-dimensional complex representations $\Pi(G)$. And $\Pi(G)$ is a tensor category (thus having a monoidal structure). For a locally compact abelian group, it is a much easier case and from Pontryagin duality, one has that the dual group $\hat{G}$ is the space of 1-dimensional representations of G (namely characters), and the dual group determines the group $G$, please see this note from last year. However, for non-abelian cases, the category of representation $\Pi(G)$ contains all finite-dimensional $\mathbb{C}-$linear representations, not just 1-dimensional irreducible representations like abelian case. Actually, as I think, it is helpful to see that Pontryagin duality is an abelian prelude to Tannaka-Krein duality. The latter also generalized a lot of the framework later, which culminated in the Tannakian framework of Grothendieck, quantum groups by Drinfeld, Deligne's tensor categories, and so on.

But one can say that given
$$\Pi(G) \cong \Pi(H)$$
which is an equivalence of symmetric monoidal categories, $G \cong H$ will not always be true unless there are some conditions to observe. This was emphasized by introducing a faithful, exact, forgetful, $\mathbb{C}$-linear fiber functor which preserves the tensor products and forgets the group action $\omega$
$$\omega: \Pi(G) \rightarrow Vect_\mathbb{C}$$
which forgets the group action and remembers the underlying vector spaces. The group $G$ can now be associated with the automorphism of the fiber functor
$$G \cong \underline{Aut}^{\otimes}(\omega).$$
This essentially means that all the information about the group is encoded in $\Pi(G)$, and given this fiber functor, we can reconstruct our group. In more standard words, given any symmetrical monoidal category $\mathcal{C}$, if one has a fiber functor $\omega: \mathcal{C} \rightarrow Vect_\mathbb{C}$, then under the suitable conditions, there exists a compact group $G$ such that $\mathcal{C} \cong \Pi(G)$ and $G  \cong \underline{Aut}^{\otimes}(\omega)$.

The framework of Tannakian categories and fiber functor was introduced by Grothendieck-Deligne in the 1960s, while Tannaka originally used the ring of representative functions.

In this quick post, I wish to state a simple Lemma.

$\mathcal{D}$-modules are very important objects of algebraic analysis and algebraic geometry. We briefly talked about it in the last post. $\mathcal{D}$-modules are modules over the ring of differential operators. In this post, let us briefly revise the Weyl Algebra because it is very important to enter the world of $\mathcal{D}$-modules. Weyl Algebra can be constructed as a $\mathcal{D}$- module.

$\mathcal{D}$-modules are very important objects in algebraic geometry and mathematical physics (for example, in string theory). In a previous post, we saw that Hecke eigensheaves on a $Bun_G$ (a set of isomorphisms of vector bundles) are $\mathcal{D}$-modules. They are a quasi-coherent sheaf of the scheme of differential operators.

We take a finite field $\mathbb{F}_q$ and define a curve (smooth, projective) over it, $X$. Now $X$ can also be taken on a Riemann surface when doing geometric Langlands (that is, switching from the curves defined over the finite fields to curves defined over the Riemann surface). Anyway, we define 

With Vaibhav Kalvakota, I wrote some notes on string compactification starting from supergravity compactification and ending with whether there exists a de Sitter solution in string theory in non-classical cases. This review was meant to initiate discussions on the de Sitter case.

This will be a very quick act.
Grothendieck emphasized that properties of objects are better understood than the properties of morphism. But his relative perspective also lighted a switch of perspective. Vakil notes that it is a change of perspective while algebraic geometry is still intact. For the properties of a scheme, like quasi-compact, separated, affine, and so on, we can translate them to the properties of morphisms. For the affine case, we can use the Affine Communication Lemma, so let us take that example here.

A Quick ACT: We know that a ring morphism $R \to S$ induces a map in the opposite direction $Spec S \to Spec R$. Well, the scheme morphism $X \to Y$ induces a map of Z-valued points $X(Z) \to Y(Z)$. These Z-valued points of an affine scheme are helpful, for instance, the rational solutions to $x^2 + y^2 = 9$ are the $\mathbb{Q}$-valued points of scheme $Spec \mathbb{Z} [x,y]/(x^2+y^2=0)$ and similarly, the integral solutions are the $\mathbb{Z}$-valued points.

ACT A - What's in a Moduli?

ACT A - Why Calabi-Yau Manifolds?

For any map of rings $A \to B$, we want a map $Spec B \to Spec A$. There is also a geometric motivation that the map between two manifolds $\pi: M \to N$ has a pullback of a smooth function from $N$ to $M$ and for $\pi(x) = y$ if a function vanishes on $y$ then it will vanish on $x$ too.

With this, I restart my Quill notes. This report will be about two very different subjects, namely compactifications and the Langlands Program. And before you are excited or mistaken, I will mention them only independently.