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. ...

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 $\hat{u}=u_{\hat{f}}$. ...

Let $v$ be a solution to the obstacle problem $$ \Delta v = h\chi_{\{v > 0\}}, \qquad v \geq 0 \qquad \hbox{in } B_1. $$ We assume $h \geq c_0 >0 $ is Lipschitz, and a Dirichlet data on $\partial B_1$ has been prescribed. ...

Consider the probability measure $\mathbb{P}_N^n(\lambda)=\frac{\dim E_\lambda}{N^n}$ on the set of Young diagrams with $n$ cells and at most $N$ rows, where $\dim E_\lambda$ is the dimension of the isotypic component corresponding to the Young diagram $\lambda$ in the tensor representation $(\mathbb{C}^N)^{\otimes n}$ of the group $GL(N,\mathbb{C})$. ...

If $E$ is a compact subset of ${\Bbb R}^d$, of Hausdorff dimension greater than $\frac{d+1}{2}$, then there exists a non-empty open interval $I$ such that, for any $t \in I$, there exist $x^1, x^2, \dots, x^{k+1} \in E$ such that $|x^{j+1}-x^j|=t$, $1 \leq j \leq k$. ...