Recently, I strolled around an exciting fact about a difference of meaning of covariant and contravariant words in mathematics (category theory) and physics (tensor analysis). Well, that makes it harder for a mathematician and a physicist to talk about these two words without knowing in what sense.

In category theory, we can think of functors as the mapping of objects between categories. We can say that if a functor preserves the direction of morphism, then the functor is a covariant one. If it reverses the direction of the morphism, then it is a contravariant functor. John Baez has briefly mentioned them in his book "Gauge Fields, Knots and Gravity". An identity functor is a covariant functor, and so are tangent vectors. While cotangent vectors and 1-forms are contravariant. (1-form in this case is differential of a function, however, if a differential of a function is to be thought as a vector field then the vector fields are covariant.)

Suppose we have a map $\phi:M \rightarrow N$ from one manifold to another. On $N$, we have real valued functions defined from $\psi:N\rightarrow \mathbb{R}^n$. To get real valued functions on $M$ we have pullback $\psi$ from $N$ to $M$ by $\phi$. 

$$\phi * \psi = \psi \circ \phi$$

We see that real-valued functions on $M$ suffer a change of direction in their morphism. So they are contravariant.

In tensor analysis, one can say that $X_\mu$ is covariant and $X^\mu$ is contravariant. (It is important to dodge that $\partial_\mu$ is covariant while its component $v^\mu$ can be contravariant.)

I have been lately studying the random matrices and their application that widely defaults for the JT gravity. Though, random matrices need to be started with Wigner's idea of the random matrix in nuclear theory. Right now, random matrix theory can be considered an important subject, at least from my learning view. In the following (short), I present my rough ideas of random matrix theory, extracted from here.

Random matrix theory (RMT) is a classic example of statistical group theory in general physics. The most recent development for RMT is the equivalence of JT gravity with RMT, see here. From the correspondence of AdS/CFT, one learns that a bulk theory with gravity lives on the boundary of a quantum system. However, the equivalence of JT gravity is not given to a boundary theory (or a bulk theory). In fact, RMT shares the correspondence with JT gravity; hence JT gravity is dual to random matrix integral of Hamiltonian $ H $, where $ H $ is a random matrix.

(RMT is mainly concerned about groups' statistics, at least for us, whose applications are wide in physics, as indicated.)

Consider a matrix $ \sf M $, from linear algebra we know that $ \sf M$ holds eigenvalues $ {\sf {m}}_{ij} $ (for $ 2 \times 2 $ matrix). Suppose the elements of matrix $ \sf M $ are random variables, to which we say to obey some specific outlined properties. In that case, the study of those ($\mathsf{m}_{ij} $) eigenvalues are called the "random matrix theory'' problem. Now one can ask what the practical application of RMT is. Actually, there are many practical applications. Consider the well-studied example of the nucleus using these random matrices, which Wigner (and Dyson) developed. And the recent example of the success of RMT is JT gravity. I suggest the reader to read the most interesting book on this subject by  Madan Lal Mehta.

 People are often heard whispering "Tit for Tat", but reciprocally. Yet, what is the game theory behind Tit for Tat (TFT)? For that, what is TFT? 

TFT is the usual game for two/more individuals or two/more groups. It starts with a situation where one of the teams is given a chance to first act. This act can be of two types, either defection or cooperation. Then the second player acts according to the previous move. The most classic and well-known example is Chess. When white moves its piece, the black always makes a move according to the first. But this is not always right. In chess, if a move doesn't concern you much, you can follow your lead using your strategy without being in a situation where you have to act accordingly. 

However, there is one fascinating game theory called the Prisoner's Dilemma, which uses TFT (and its extensions) very much. It is intricate at first but self-realizing after you allow the logic to play itself. Merrill Flood and Melvin Dresher were the first ones to realise this game. Afterwards, Robert Axelrod influenced this game theory.

Prisoner's Dilemma is a situation concerning the two criminals who are arrested. Let's say A and B are criminals. Just for the sake of game theory, we give both the criminals a chance to get free from charges. Conditions are that A and B are given two choices. These choices are imagined on a ground where both are standing with each other. The options are to either defect/betray the opponent or remain silent. The catch is, they are not allowed to talk or get informed about one another. Rules are as follows,

• If A betrays B, and B betrays A, both will be charged with two years of the sentence.
• If A betrays B and B remains silent. Then A will get free, and B will be charged for three years.
• If B betrays A and A remains silent. Then B will get free, and A will be charged for three years.
• If neither A nor B defects each other, they get only one year of charge.