In the article [1], they study the KP-II equation: the function $\phi(t,x,r)=\partial_{r}^{2}\log(F)$ satisfies

$$\partial_{t}\phi+\frac{1}{2}\partial_{r}(\phi)^{2}+\frac{1}{12}\partial_{r}^{3}(\phi)+\frac{1}{4}\partial_{r}^{-1}\partial_{x}^{2}\phi=0$$

or in the Hirota-form ...

SLE$(\kappa)$ describes a curve growing in time describing the interface of many statistical models that fall within conformal field theory. The research problem here is to construct a family of loops using the Beltrami equation that might be related to the SLE-loop.

In the works [1] and [2] this was achieved for small enough $\kappa\leq \kappa_{0}$. ...

Let $a_1 \leq a_2 \leq \dots \leq a_n$ be a set of $n$ positive integers such that all $2^n$ subsets of the set are uniquely identified by the sum of their elements. An example of such a sequence are the powers of 2 (this is the uniqueness of the binary expansion in disguise). The question, due to Erdos in 1931 or 1932, is whether sets with such a property have to always have a large number. ...

Suppose we have a finite set of points $\mathcal{P} \subset [0,1]^d$ with the following nice equidistribution property: for any axis-parallel rectangle $R \subset [0,1]^d$ with one point anchored in the origin, the number of points from $\mathcal{P} $ that are contained in the rectangle $R$ is very nearly proportional to the volume of the rectangle ...

Create convexity in 3 (or 100?) steps only! Consider an integer $N$. Let us say that a compact subset $A$ of $\mathbb{R}^N$ is balanced if $x \in A$, $\lambda \in \mathbb{R}$, $|\lambda| \leq 1 \Rightarrow \lambda x \in A$. Let us denote by $\gamma_N$ the canonical Gaussian measure on $\mathbb{R}^N$. ...

Make 1000 USD from simple combinatorics. This is Research Problem 12.5.6 of the book Upper and Lower Bounds. You have to have a look at the book to get the connection between this problem and stochastic processes, but the problem can be understood without knowing anything about this book. ...

Is there a universal constant $c$ such that all translation and rotations of a cube $Q \subset \mathbb{R}^n$ with side-length $c$ always contains a point from the $\mathbb{Z}^n$? ...