This is a classical problem in additive combinatorics that can be equivalently phrased as a problem in real analysis (this equivalence is due to Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa): what random variable $X$ with compact support on $\mathbb{R}$ has the property that the distribution of $X+X$ is as flat as possible? ...

Stanislaw Ulam, in a 1964 book, described what is now called the "Ulam sequence". It is defined by setting a_1 = 1, a_2 =2 and then picking a_n in a greedy manner as the smallest integer that can be uniquely written as the sum of two distinct earlier elements of the sequence. This results in the sequence

1,2,3,4,6,8,11, ... ...

The Hadamard maximal determinant problem is a fascinating combinatorial optimization problem with many concrete applications to signal processing, coding theory, experimental design theory, and cryptography . Let $M$ be an $n\times n$ matrix, all of whose entries are at most 1 in modulus. Note that $det(M)$ is equal to the volume of the $n$-dimensional parallelepiped spanned by the rows of $M$. By assumption, each row has Euclidean length at most $n^{1/2}$. Then $det(M) \leq n^{n/2}$, and equality holds if and only if every entry of $M$ is $\pm 1$ and the rows of $M$ are orthogonal. Hadamard established this upper bound which is only attainable if the order n equals $1, 2,$ or a multiple of $4.$ A matrix attaining the bound is called a Hadamard matrix. The classical and very old Hadamard conjecture asserts that a Hadamard matrix exists of every order divisible by $4.$ The smallest multiple of $4$ for which no such matrix is currently known is $668$, the value $428$ having been settled only in $2005$. Due to its extensive real life applications, it is still a very active line of research. ...

If $N = q^k n^2$ is an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$, then it is somewhat trivial to prove that $$I(q^k) < \frac{5}{4} < \sqrt{\frac{8}{5}} < I(n),$$ which led Dris to conjecture that $q^k < n$. Brown, Starni, and Dris has shown that $q^k < n$ follows from the truth of the Descartes-Frenicle-Sorli Conjecture that $k=1$. In fact, it is known (by work of Dris) that $$k = 1 \implies \sigma(q^k) < n,$$ where $\sigma(x)$ is the sum of divisors of the positive integer $x$. The problem is to prove $q^k < n$ unconditionally. (In other words, it would suffice to prove the implication $k > 1 \implies q^k < n$.) ...

Let $X$ be a matrix-valued Ito process $$dX_{t}=u_{t}dt+v_{t}dW_{t}$$ where $u$ and $v$ are $d\times d$-matrices, with $d\ge 2$, and $W$ is a real Brownian motion. Consider the exponential $$Q_{t}=\text{Exp}(X_{t}).$$ Under what conditions on $u,v$ is $Q$ a local martingale? ...

Let $A$ be a bounded linear operator on Hilbert space $\mathcal{H}$. For some fixed $x\in \mathcal{H}$, consider a system of vectors $$\left\{\frac{\|A^n x\|}{\|x\|} \right\}_{n=0,1,2,\dots}. $$ There are bounded (not-normal) operators for which a system of above form is an orthonormal basis (e.g. shift operator on $\mathcal{l}(\mathbb{N})$). ...

Let $$ \begin{cases} -\Delta u_f=f & \text{in }D,\\ u_f=0 &\text{on }\partial D. \end{cases} $$ It is well-known that the maximization of the functional $$ \Phi(f)=\int_D|\nabla u_f|^2dx, $$ over the set $$ \bar{\mathcal{R}}_\beta=\{f\,\,|\,\, 0\leq f\leq 1,\,\,\text{and}\,\, \int_Dfdx=\beta\} $$ has a solution $$ \hat{f}=\chi_{\{\hat{u}>\alpha\}}\in \mathcal{R}_\beta=\{\chi_E\,\,|\,\, |E|=\beta\}, $$ for some $\alpha>0$, and $\hat{u}=u_{\hat{f}}$. ...