How many mutually non-isomorphic extensions of degree 4 and Galois group of order at most $8$ does the the field $\mathbb{Q}_2$ of $2$-adic numbers admit?

The by codimension graded pieces of the Chow ring $A^\bullet (\mathcal{F})$ of the two-step flag variety $\mathcal{F}=\operatorname{Fl}(2,4;\mathbb{C}^6)$ are vector spaces generated by Schubert classes indexed by two subsets of $0,...,5$: The first of size $2$ and the second of size $4$ containing the former. Let $X$ be a variety of class $[1,5; 1,3,4,5]$ and $Y$ be of class $[3,5; 1,3,4,5]$. (For example, the former means lines $L\in X$ meet a $\mathbb{P}^1$, and there is no additional conditions on the projective $3$-planes containing $L$.) Suppose the intersection of $X$ and $Y$ is transversal. What is the class of the intersection $X\cap Y$ written in terms of the basis of $A^k(\mathcal{F})$ of correct codimension $k$?

Let $B$ be the unit ball in $\mathbb{R}^{65}$ with respect to the standard Euclidean norm. What is the smallest natural number $r$ such that there exist hermitian $r\times r$ matrices $A_0,\ldots,A_{65}$ with $B=\{p\in\mathbb{R}^{65}\mid A_0+p_1\cdot A_1+\cdots+p_{65}\cdot A_{65}\textrm{ is positive semidefinite}\}$?

Let $M = (E, \mathcal{B})$ be a matroid. Let $\mathbb{P}L_{M}$ be projectivization of the space of Lorentzian polynomials whose support is precisely equal to the collection $\mathcal{B}$. When $M = F_{7}$, the Fano matroid, find the dimension of $\mathbb{P}L_{M}$.

Let $Q = 1 \longrightarrow 2 \longrightarrow 3 \longleftarrow 4$. Consider $M = X_{[1,2]}^2 \oplus X_{[1,3]}^3 \oplus X_{[1,4]}^2 \oplus X_{[2]} \oplus X_{[2,3]}^2 \oplus X_{[2,4]}$. Calculate the generic Jordan form data of $M$.

In a card game each card has three attributes: color, number and shape. Each of these attributes has 4 possible variants. Color: red, blue, green, yellow. Number: 1, 2, 3, 4. Shape: circle, square, triangle, cross. All possible cards appear once in the deck. Define a "quartet" to be a set of four cards from the deck above such that for each of the three attributes, one of the following three possibilities holds for the four values: all four values are the same, all four values are different, or there are exactly two pairs of values. What is the minimum number of cards that must be randomly drawn from the deck so that one can guarantee that there is always at least one quartet among them?

Let $R=\mathbb{C}[x_1,\ldots,x_6,y_1,\ldots,y_6]$ with bigrading $\deg(x_i)=(1,0)$, $\deg(y_i)=(0,1).$ Let $I\subset R$ be the bihomogeneous ideal generated by the $2$-minors of the generic matrix \[\begin{pmatrix} x_1 & x_2 & \cdots & x_6 \\ y_1 & y_2 & \cdots & y_6\end{pmatrix}.\] What is the dimension of the $\mathbb{K}$-vector space $(I^2/I^3)_{(5,2)}$?

Let $B$ be the unit ball in $\mathbb{R}^{8193}$ with respect to the standard Euclidean norm. What is the smallest natural number $r$ such that there exist hermitian $r\times r$ matrices $A_0,\ldots,A_{8193}$ with $B=\{p\in\mathbb{R}^{1025}\mid A_0+p_1\cdot A_1+\cdots+p_{8193}\cdot A_{8193}\textrm{ is positive semidefinite}\}$?

Let $W$ be the Weyl algebra $\mathbb{C}\left<D,X \mid DX - XD = 1\right>$; this is the $\mathbb{C}$-algebra with two generators $D$ and $X$ and a single relation $DX - XD = 1$. Given an integer $n \ge 0$, we define an *$n$-monomial element* to be an element of $W$ that can be written as a product of $n$ generators from the set $\left\{D,X\right\}$, i.e., as $G_1G_2\cdots G_n$ where each $G_i \in \left\{D,X\right\}$. Note that some $n$-monomial elements can be written in several ways in such a form (for instance, $DXXD = XDDX$). Let $a_n$ denote the number of $n$-monomial elements. Find $a_{11} + a_{12}$.

What is the number of connected, simple graphs with exactly 1 cycle, 100 vertices and all vertex degrees at most 3? To clarify, the graphs are unlabelled and not necessarily planar.

Let $f_1,...,f_m$ be generic forms of degree 2 in n complex variables $x_1,...x_n$. Consider the *log-likelihood function* $\ell = \sum_{i=1}^m s_i \log(f_i)$. Here the $s_1,...,s_m$ are considered as complex variables too. This function is well-defined on the complement of the hypersurface arrangement defined by the $f_i$. The *likelihood correspondence* $L$ is the Zariski closure in $\mathbb{P}^{n-1}\times \mathbb{P}^{m-1}$ of the critical locus of the likelihood equations. Precisely \[ L = \overline{\left\lbrace (x, s)\in \mathbb{C}^{n}\times \mathbb{C}^{m} : \frac{\partial \ell}{\partial x_i}(x,s)=0, i=1,\dots,n, \prod_{i=1}^m f_i^{s_i(x)} \neq 0,\, F(x)\in X_{r} \right\rbrace}, \] where $X$ is the Zariski-closure of the image of $F\colon\mathbb{C}^n \rightarrow\mathbb{C}^{m}, x \mapsto (f_1(x),\dots, f_m(x))$, and $X_{r}$ is its set of nonsingular points. Determine the dimension of the likelihood correspondence for m=8 and n=5.

How many distinct combinatorial types of 3-dimensional polytopes can you obtain by intersecting the 4-dimensional cube with an affine hyperplane?

For $n$ a positive integer define $V(n)$ to be the integer obtained by using the base 10 digits of $n$ in base 11. I want to evaluate the series $\sum_{p \text{ prime}} \frac{1}{V(p)}$ to an accuracy of $10^{-5}$.

Among all possible Bruhat intervals in any Coxeter group, find an interval with the smallest number of elements whose Kazhdan-Lusztig polynomial does not equal $1$. How many cover relations does this interval have?

Let $L$ denote the log-canonical bundle on $\overline{M}_{0,20}$ over a field of characteristic $2$. Compute the dimension of $H^{17}(\overline{M}_{0,20}, L^{-1})$.