What does the future hold? Is there life beyond the stars? Will artificial intelligence take over the world? Is time travel possible? All of these questions and more are addressed every week in Futures, Nature’s science-fiction column.

The Internet is the decisive technology of the Information Age, and with the explosion of wireless communication in the early twenty-first century, we can say that humankind is now almost entirely connected, albeit with great levels of inequality in bandwidth, efficiency, and price.

Mirzakhani, who is Iranian, studies the geometry of moduli space, a complex geometric and algebraic entity that might be described as a universe in which every point is itself a universe. Mirzakhani described the number of ways a beam of light can travel a closed loop in a two-dimensional universe. To answer the question, it turns out, you cannot just stay in your “home” universe – you have to understand how to navigate the entire multiverse. Mirzakhani has shown mathematicians new ways to navigate these spaces.

I well remember a friend of mine in the math department of Sharif University of Technology back in 1990s who is now in MSU used to write his homework and research papers via LaTex but then I was more focused on humanities and didn’t need to learn this coding environment.

Given the fact that I am a first time user I was wondering that if someone could help me to locate the proper file for installation and also the basic help stuff for a beginner. Very soon, however, I learned that LaTeX is an open and free coding language. And that there is a jungle of downloadable files here and there!

As a first exercise today morning I coded via LaTeX a proof of Maschke’s Theorem which is an elegant result in modular representation of finite groups. Learning to do LaTeX was very much like my first time of riding a bicycle. It took a few hours of painful frustrations but the outcome is quite enjoyable and stylish.

\documentclass[12pt,a4paper]{article}

\usepackage{amsmath}

\usepackage{amsfonts}

\begin{document}

\title{Proof of Maschke Theorem}

\author {@VictorScenario First Exercise in \LaTeX{}}

\maketitle

Let G be a finite group, $F=\mathbb{R}$ or $F=\mathbb{C}$, and let V be an FG-module. If U is an FG-submodule of V, then there is an FG-submodule W of V such that

\[V= U \oplus W.\]

\textit{Proof:}

Choose any subspace $W_0$ of V such that $V=U \oplus W_0$.

Take a basis $(v_1, \dots , v_m)$ of U, extend it to a basis $(v_1, \dots, v_n)$ of V, and let \(W_0=sp(v_{m+1}, \dots ,v_n)\).

For $v \in V$, we have $v=u+w$ for unique vectors $u \in U$ and $w \in W_0$, we define $\phi : V \to V$ by setting $v\phi=u$.

Clearly $\phi$ is a projection of V with kernel $W_0$ and image U. Define $\vartheta:V \to V$ by

\[v\vartheta=\frac {1}{|G|} \sum_{g \in G} vg \phi g^{-1} \quad (v \in V). \]

Note that $\vartheta$ is an endomorphism of V and $Im \vartheta \subseteq U$. For $v \in V$ and $x \in G$,

\[(vx) \vartheta=\frac{1}{|G|}\sum_{g \in G}(vx)g \phi g^{-1}.\]

Suppose that $h=xg$. Thus

\[(vx) \vartheta=\frac {1}{|G|}\sum_{h \in G}vh \phi h^{-1}x

=\left(\frac {1}{|G|}\sum_{h \in G} vh \phi h^{-1}\right) x

=(v \vartheta)x.

\]

Hence $\vartheta$ is an FG-homomorphism. For $u \in U, g \in G \Rightarrow ug \in U \Rightarrow (ug) \phi = ug$. Therefore

\[u \vartheta=\frac {1}{|G|} \sum_{ g \in G} ug \phi g^{-1}=\frac {1}{|G|} \sum_{g \in G} (ug)g^{-1}=\frac {1}{|G|} \sum_{g \in G} u=u. \]

Let $v \in V$. Then $v \vartheta \in U \Rightarrow (v \vartheta) \vartheta = v \vartheta \Rightarrow \vartheta^{2}=\vartheta$. Hence $\vartheta$ is a projection.

Also $Im \vartheta = U$. Now if we let $W=Ker \vartheta$ $\Rightarrow$ W is an FG-submodule of V and $V=U \oplus W$. \

\textit{Source of the Proof: Gordon James, Martin Liebeck, (2001), Representations and Characters of Groups, Second Edition, Cambridge University Press.}

\end{document}

In science 2.0 my favorite is crowdsourcing astronomy. Imagine that you have 1 billion of people helping scientists to explore and make sense of the vastness of the cosmos.

This gives you a why and justification about should we encourage even more Earth population, that is approaching 10 billion and beyond it.

Clearly more population means that there will be more working brains to pave the way for the next breakthrough in science given the fact that there are billions of stars and galaxies over there to be dealt with.

Every person assigned to one star or one galaxy under the supervision of a great star scientist promises some unimagined revolutions.

December 2004

WASHINGTON DC—-“The tenuous peace inside Iraq that America had stitched together so laboriously came undone with the sudden re-igniting of the Sunni insurgency; the insurgents proclaimed themselves the true Caliphate and battled anew both Shia and the American garrisons.”

Source:

Pages 83-92, Mapping the Global Future, Report of the National Intelligence Council’s 2020 Project

June 2014

BAGHDAD—“The Sunni Islamist militant group whose three-week blitz through northern Iraq has nearly upended the country’s fragile unity announced itself as a new Islamist “caliphate” on Sunday, unilaterally declaring statehood and demanding allegiance from other Islamist groups.”

Source:

A huge expanse of water trapped in a layer of the Earth’s mantle could help explain the origin of our oceans

*status quo*. None of us can possibly question or cast doubt on the notion of change per se. Some people get the notion of the need to change but simply do not agree or accept its nature or associated process or are not committed to it.

But the lesson learned from quantum mechanics says that, on an ontological level, perhaps even if we are dealing with the same issue, say the future of women in the Middle East, it might be the case that in a particular mindset the change itself is not recognized at all let alone to be accepted or agreed and acted upon.

In quantum mechanics the evolution of a system is a tricky notion, it could mean that the particle remains at the same energy state indefinitely into the future, but mathematically it’s not constant and therefore changing, but in fact it is constant and not changing at all!

http://en.wikipedia.org/wiki/Stationary_state

Isn’t the history of our civilization obeying the same pattern of simultaneously changing and constant:

1914 - 2014: One hundred years time lapse http://vimeo.com/96057108