Let \(G\) be a matrix in \(F_3^{4\times n}\) with rows such that for every vector list of 287 vectors \(v_1,\ldots,v_{287}\) (there can be repetition) in \({F_3}^4\) I can find \(287\) disjoint subsets \(X_1,\dots,X_{287}\) of columns of \(G\) such that \(v_i\) belongs to the span of the columns of \(X_i\). Let \(C\) be a linear code generated by a matrix with the property described above that minimise \(n\). What is the weight enumerator of a code generated by such a \(G\)?

Consider the following homogeneous ideal in \(\mathbb{C}[z_1,\dots,z_5]\): \[\begin{gather} \bigl\langle -z_{2}+z_{4}-z_{5},\\ \quad z_{1}^{5}z_{2}^{5}+5z_{1}^{4}z_{2}^{5}z_{3}+10z_{1}^{3}z_{2}^{5}z_{3}^{2}+10z_{1}^{2}z_{2}^{5}z_{3}^{3}+5z_{1}z_{2}^{5}z_{3}^{4}-z_{2}^{6}z_{3}^{4}+z_{2}^{5}z_{3}^{5}\\ +5z_{1}^{5}z_{2}^{4}z_{4}+25z_{1}^{4}z_{2}^{4}z_{3}z_{4}+50z_{1}^{3}z_{2}^{4}z_{3}^{2}z_{4}+50z_{1}^{2}z_{2}^{4}z_{3}^{3}z_{4}+25z_{1}z_{2}^{4}z_{3}^{4}z_{4}+5z_{2}^{4}z_{3}^{5}z_{4}\\ +10z_{1}^{5}z_{2}^{3}z_{4}^{2}+50z_{1}^{4}z_{2}^{3}z_{3}z_{4}^{2}+100z_{1}^{3}z_{2}^{3}z_{3}^{2}z_{4}^{2}+100z_{1}^{2}z_{2}^{3}z_{3}^{3}z_{4}^{2}+50z_{1}z_{2}^{3}z_{3}^{4}z_{4}^{2}+10z_{2}^{3}z_{3}^{5}z_{4}^{2}\\ +10z_{1}^{5}z_{2}^{2}z_{4}^{3}+50z_{1}^{4}z_{2}^{2}z_{3}z_{4}^{3}+100z_{1}^{3}z_{2}^{2}z_{3}^{2}z_{4}^{3}+100z_{1}^{2}z_{2}^{2}z_{3}^{3}z_{4}^{3}+50z_{1}z_{2}^{2}z_{3}^{4}z_{4}^{3}+10z_{2}^{2}z_{3}^{5}z_{4}^{3}\\ +5z_{1}^{5}z_{2}z_{4}^{4}+25z_{1}^{4}z_{2}z_{3}z_{4}^{4}+50z_{1}^{3}z_{2}z_{3}^{2}z_{4}^{4}+50z_{1}^{2}z_{2}z_{3}^{3}z_{4}^{4}+25z_{1}z_{2}z_{3}^{4}z_{4}^{4}+5z_{2}z_{3}^{5}z_{4}^{4}\\ +z_{1}^{5}z_{4}^{5}+5z_{1}^{4}z_{3}z_{4}^{5}+10z_{1}^{3}z_{3}^{2}z_{4}^{5}+10z_{1}^{2}z_{3}^{3}z_{4}^{5}+5z_{1}z_{3}^{4}z_{4}^{5}+z_{3}^{5}z_{4}^{5}\\ -6z_{2}^{5}z_{3}^{4}z_{5}-15z_{2}^{4}z_{3}^{4}z_{5}^{2}-20z_{2}^{3}z_{3}^{4}z_{5}^{3}-15z_{2}^{2}z_{3}^{4}z_{5}^{4}-6z_{2}z_{3}^{4}z_{5}^{5}-z_{3}^{4}z_{5}^{6}\bigr\rangle. \end{gather}\] Can you tell me the Picard group of the projective variety cut out by this ideal?

What is the last fall degree of the family of polynomials \(\{x^{12456789}+y,y^{12456789}+x,x^{503}y^{503}\}\)?

Consider the ideal \(I \subset \mathbb{Q}[x,y]\) generated by \(5\) polynomials: \[\begin{align} I = \bigl\langle\; & -1 - ax + x^3 - by + y^2,\\ & -c - dx^2 - y + xy - ey^2,\\ & -fx + x^2 - gy + x^2 y - y^2,\\ & -h + x - kx^2 - my - ny^2 + xy^2,\\ & -p - qx - rx^2 + (57y)/13 + y^3 \;\bigr\rangle, \end{align}\] where \((a, b, c, d, e, f, g, h, k, m, n, p, q, r)\) are rational numbers. There is one set of values for \((a, b, c, d, e, f, g, h, k, m, n, p, q, r)\) for which the quotient \(\mathbb{Q}[x,y]/I\) has the structure of a \(5\)-dimensional \(\mathbb{Q}\)-vector space with basis given by \(\{[1],[x],[x^2],[y],[y^2]\}\). Using those values, find \(m+n\).

Let \(A\) be the connected cyclic Nakayama algebra with Kupisch series \([2,3]\) and let \(B\) be the Auslander algebra of \(A\). Let \(C\) be the algebra \(B/I\) where \(I\) is the two-sided ideal given by the socle of the direct sum of all indecomposable projective-injective \(B\)-modules (as a submodule of the regular module). What is the sum of all delooping levels of the indecomposable \(C\)-modules?

For a linear Nakayama algebra \(A\), denote by \(n_A\) the number of indecomposable injective modules over \(A\) (up to isomorphism) that have codominant dimension at least 2, and denote by \(m_A\) the number of simple modules over \(A\) (up to isomorphism) with projective dimension 1 that are not 1-regular. Compute a closed-form formula for \(\Sigma_A (n_A - m_A)\), where the sum ranges over all linear Nakayama algebras (up to isomorphism) with \(n\) simple modules.

Let \(R\) be the ring of polynomials in \(20\) variables and let \(I\) be the ideal of \(R\) consisting of functions vanishing on the set of points in \(\mathbf{C}^{20}\) where some \(6\) coordinates are equal to one another. Finally, let \(\mathbf{C}\) be the \(R\)-module which is \(\mathbf{C}\) as a vector space, upon which the variables all act by \(0\). What is the dimension of \(\mathrm{Tor}^2_R(I,\mathbf{C})\)?

For a linear Nakayama algebra \(A\) arising from a connected quiver on \(n\) vertices, denote by \(n_A\) the number of simple (left) modules over \(A\) (up to isomorphism) that have projective dimension at least 2. Compute a closed-form formula for \(\Sigma_A n_A\), where the sum ranges over all linear Nakayama algebras (up to isomorphism) with \(n\) simple modules.

Let \(n\geq 1\) be an integer. For an integer sequence \(w=(w_1, \dots , w_{n+1})\) with distinct positive entries and for a permutation \(\pi\in S_{n}\,,\) define the integer sequence \(a(w,\pi)=(a_1, \dots , a_n)\) as follows. Let \[a_1= \begin{cases} + \max( w_{\pi(1)}, w_{\pi(1) +1} ) & \text{if } w_{\pi(1)} < w_{\pi(1) +1}\,, \\ - \max( w_{\pi(1)}, w_{\pi(1) +1} ) & \text{if } w_{\pi(1)} > w_{\pi(1) +1}\,,\end{cases}\] and, if \(n\geq 2\), let \((a_2, \dots, a_n) = a(w^-, \pi')\) where \(w^-\) is obtained from \(w\) by deleting \(\max( w_{\pi(1)}, w_{\pi(1) +1} )\), and where \(\pi'\in S_{n-1}\) is the permutation defined by \[\pi' (i) = \begin{cases} \pi(i+1) & \text{if } \pi(i+1) < \pi(1) \,,\\ \pi(i+1)-1 & \text{otherwise} \,.\end{cases}\] Define the polynomial \(F_{w}(y,t)\in \mathbb{Z}[y,t]\) by \[F_{w}(y,t) = \sum_{\pi \in S_n} y^{\#\{a_i\in a(w,\pi) \mid a_i>0 \} } t^{\#\{i\in\{1,\dots, n-1\} \mid a_i<a_{i+1} \text{ where } a(w,\pi) = (a_1,\dots , a_n) \} } \,.\] Determine the coefficient of \(y^3\) in \(F_{w}(y,t)\) for \(w=(2, 4, 3, 1, 7, 8, 11, 6, 5, 10, 9)\) as polynomial in \(t\).

Let \(c = (1, 2, \dots, n)\) be the standard \(n\)-cycle in the symmetric group \(S_n\). For \(p\) satisfying \(\gcd(p, n) = 1\) with multiplicative inverse \(p'\) so that \(pp'=1 \mod n\), let \(w \in S_n\) be the permutation with \(w(i) \equiv i \cdot p' \pmod n\), so that \(w c^p w^{-1} = c.\)

Let \(B: S_n \to B_n\) be the positive lift of a permutation to the braid group on \(n\) strands. For \(n=7\) and \(p=5\), express the pure braid \(B(w) B(c)^p B(w)^{-1} B(c)^{-1}\) in terms of the Artin generators of the pure braid group \(t_{ij} := \sigma_i \sigma_{i+1} \cdots \sigma_{j-2} \sigma_{j-1}^2 \sigma_{j-2}^{-1} \cdots \sigma_{i+1}^{-1} \sigma_i^{-1}\).

For example, if \(n=4\) and \(p=3\), \(w=[3,2,1,4]\) and we should return the list [(2, 3), (1, 3), (1, 2)], representing the pure braid \(t_{23} t_{13} t_{12}\).

Given lattice paths \(P\) and \(Q\) from \((0, 0)\) to \((m, r)\), with \(P\) never rising above \(Q\) or incident with \(Q\) at any point, we define two matroids in terms of \(P\) and \(Q\).

For the first matroid, let \(P = p_1p_2\cdots p_n\) and \(Q = q_1q_2\cdots q_n\). If \(p_{u_1}, p_{u_2}, \ldots, p_{u_r}\) are the north steps of \(P\) with \(u_1 < u_2 < \cdots < u_r\) and \(q_{l_1}, q_{l_2}, \ldots, q_{l_r}\) be the north steps of \(Q\) with \(l_1 < l_2 < \cdots < l_r\). Let \(N_i\) be the interval \([l_i,u_i]\) of integers. Let \(M[P,Q]\) be the transversal matroid on the ground set \([n]\) that has the presentation \((N_1,N_2,\ldots,N_r).\) A matroid \(M\) for which there exists \(P, Q\) such that \(M = M[P, Q]\) is called a lattice path matroid.

For the second matroid, label the rows of the diagram enclosed by \(P\) and \(Q\), from bottom to top, by \(1\) through \(r\), and the columns, from left to right, by \(r+1\) to \(r+m\). For each \(i \in [r]\), let the set \(A_{i}\) consist of \(i\) along with the labels of all columns that have a square in row \(i\). The rook matroid \(R[P, Q]\) is the transversal matroid with the presentation \((A_{1}, \ldots , A_{r})\). A matroid \(M\) for which there exists \(P, Q\) such that \(M = R[P, Q]\) is called a rook matroid.

Let \(\lambda/\mu\) be the skew shape enclosed by the region bounded by \(P\) and \(Q\); then \(\lambda/\mu\) determines and is determined by \(P, Q\). For \(\alpha/\beta = 888888765/76654321\), find the number of lattice path matroids that are isomorphic to the rook matroid defined by \(\alpha/\beta\).

Define a map \(s\) from the set of all permutations of length \(n\) to that same set as follows.

1. If \(n=1\), then \(s(1)=1\).

2. If \(p=LnR\), that is \(L\) is the substring on the left of the entry \(n\), then \(s(p)=s(L)s(R)n\).

This operation \(s\) is often called the stack sorting.

Let \(B_{n,k}\) be the set of permutations \(p\) of length \(n\) for which all of the following hold.

1. The permutation \(s(p)\) avoids the patterns \(2314\) and \(3124\).

2. The number of descents of \(p\) is \(k-1\).

A block move is an operation that turns the permutation \[p = p_1 p_2 \cdots p_i\;p_{i+1}\cdots p_j\;p_{j+1}\cdots p_k\;p_{k+1}\cdots p_\ell \cdots p_n\] into the permutation \[p' = p_1 p_2 \cdots p_i\;p_{k+1}\cdots p_\ell\;p_{j+1}\cdots p_k\;p_{i+1}\cdots p_j\;p_{\ell+1}\cdots p_n.\] That is, a block move interchanges the string \(p_{i+1}\cdots p_j\) and the string \(p_{k+1}\cdots p_{\ell}\), for some numbers \(i<j<k<\ell\). If \(p,p'\in S_n\), and \(t\) is the smallest integer so that there exists a sequence of \(t\) block moves that turn \(p\) into \(p'\), then we say that \(t=\mathbf{d}(p,p')\) is the distance of \(p\) and \(p'\).

For a given permutation \(p\), let \(D_{p}(k)\) be the set of all permutations \(q\) that are at distance \(k\) from \(p\).

How many pairs of permutations \((p,\pi)\) are there so that

1. both \(p\) and \(\pi\) are of length \(30\), and

2. \(p\in B_{30,15}\), and

3. \(\pi \in D_p(10)\), that is, \(\pi\) is at distance \(10\) from \(p\)?

Fix two positive integers \(r\) and \(m\). Consider the family of subsets \(C_i=\{2i+1,...,2i+r\}\) where \(i\) ranges from \(0\) to \(m-1\). Let \(M\) be the matroid of rank \(r\) on \(n=2(m-1)+r\) elements whose non-bases are exactly the \(m\) subsets \(C_0,...,C_{m-1}\). Using the notation of the paper https://arxiv.org/pdf/2502.04980 express the unnormalised symmetrized class \([X_{M,sym}]\) in the basis of the symmetric part of the degree \(n-r\)-part of the Chow ring of the permutohedron given by square free monomials in the differences \(H_k:=L_{n-1-k} - L_{n-k}\) (where \(k\) runs from \(1\) to \(n-1\) and we use convention \(L_0=0\)). As a formula of \(r\) and \(m\) please give me the coefficient of the monomial \(H_{r-1}\cdot \prod_{k=1}^{n-r-1}H_{n-k}\) in this expression.

Let \(\mathcal{A}\) be the hyperplane arrangement that consists of the hyperplanes \(\{x_i \pm x_j \mid 1\le i < j \le 11\}\cup \{ x_i \mid 1\le i \le 11 \}\), and let \(F_{\mathcal{A}}(y,t)\) be the coarse flag Hilbert-Poincaré series of \(\mathcal{A}\). What is the coefficient of \(y^7\) in \(F_{\mathcal{A}}(-y,y)\)?

Consider an ordered tree on \(n+1\) vertices with root \(r\). We impose the breadth-first left-to-right order \(<_T\) on the vertices of \(T\): vertices are ordered first by distance from the root, and within each level from left to right according to the ordering of children. The covering relation in this order is denoted by \(\prec_T\).

Denote by \(d(u,v)\) the edge-distance between vertices \(u\) and \(v\) in \(T\). Let \(d(T)\) be the maximum of the distances \(d(u,v)\) over all pairs \(u \prec_T v\) of vertices. Compute a closed-form formula for the number of ordered trees on \(n+1\) vertices (\(n \geq 10\)) such that \(d(T) = n - 3\).

Determine the number of permutations of size \(20\) that avoid the patterns \(1324\) and \(2314\) and have exactly \(30\) inversions.

Fix a multiset \[M=\{1^3,2^3,3^4,4,5,6^2,7^2,8^4,9^3,10,11^{13},12^{100},13,14^2,15,16,17,18,19\},\] where \(i^k\) denotes that \(i\) appears with multiplicity \(k\). Let \(r=(r_1,\dots,r_{19})\) be a color vector, where each element \(i\) is assigned \(r_i\) colors. For each such \(r\), consider the colored multiset Eulerian polynomial (i.e., the descent generating polynomial over colored multiset permutations of type \((M,r)\)). For which color vectors \(r\) is this polynomial palindromic?

Let \(a(k)\) denote the number of \(\{1324, 1342\}\)-avoiding permutations of size \(k+2\) with exactly \(k\) inversions. Determine the generating function \(\sum_{k \geq 0} a(k) x^k\).

Consider a 20 by 20 square grid with lower left corner at (0,0), upper right at (20,20). What is the number of ways to draw 21 monotone grid paths (each having only E or N steps) through the grid which connect (0,0),..,(20,0) with (20,0).….,(20,20) so that no two paths have overlapping steps, endpoints and so that two paths may cross over at a point but not just touch, the path starting at (20,0) ends at (20,12) and the path starting at (10,0) ends at (20,0)?

For a Dyck path \(d\) of semilength \(n\), consider its bounce path. The “reversing bounce path” of \(d\) is the path that arises by reversing \(d\), considering its bounce path, and reversing it again. For example, for \(d=(()())\), its bounce path is \((())()\) and its reverse bounce path is \(()(())\). How many Dyck paths are there for \(n=10\), whose bounce path is equal to their reversing bounce path?

For a permutation \(\pi \in S_n\) and \(i \in [n]\), denote by \(\pi \setminus i\) the permutation that has the same relative order as the sequence obtained by deleting \(i\) from \(\pi\) written in one-line notation. We call \(\pi\) almost decomposable if at least one of the permutations \(\pi\), \(\pi \setminus 1\), \(\pi \setminus \pi_1\), \(\pi \setminus n\), or \(\pi \setminus \pi_n\) is decomposable.

For all \(n \geq 10\), find the smallest value of \(k\) such that there exists a \(1342\)-avoiding permutation of size \(n\) with \(k\) inversions that is not almost decomposable.

Let \(Q\) be the poset on \(\{1,2,3\}\times\{1,2,3,4\}\) obtained from the product of two chains by deleting exactly the cover relations \[(2,1)<(2,2),\qquad (2,3)<(2,4),\qquad (3,2)<(3,3).\] For each \(\sigma\in S_4\), label \((p,j)\in Q\) by \(\ell_\sigma(p,j)=p+3(\sigma(j)-1)\). If \(\pi=(q_1,\dots,q_{12})\) is a linear extension of \(Q\), define \[\operatorname{des}_{\ell_\sigma}(\pi) = \#\{i\in\{1,\dots,11\}:\ell_\sigma(q_i)>\ell_\sigma(q_{i+1})\}.\] Set \[F_Q(x) = \sum_{\sigma\in S_4}\sum_{\pi\in L(Q)} x^{\operatorname{des}_{\ell_\sigma}(\pi)}.\]

Determine the coefficients \(\gamma_i\) in the expansion \[F_Q(x)=\sum_i \gamma_i x^i(1+x)^{\deg F_Q-2i}.\]

Let \(x\) be a point in the complex Grassmannian \(G(k,n)\), and let \(M\) be the corresponding matroid. The Chow class of \(M\) is defined as the class of the torus orbit closure \(\overline{(\mathbb{C}^*)^n x}\) in the Chow ring of \(G(k,n)\).

Expand the Chow class of the lattice path matroid defined by \(URUURRUR\) and \(RRURRUUU\) as a linear combination of Schubert cycles \(s_{\lambda}\).

Consider the following double cover of graphs: \(G_1\) has 5 vertices and edges \(\{(1,2),(1,3),(2,3),(2,3),(2,4),(3,5),(4,5),(4,5)\}\), and \(G_2\) has \(3\) vertices and edges \(\{(1,2),(2,2),(2,3),(3,3)\}\). The map on vertices is given by \(1 \mapsto 1, 2\mapsto 2, 3 \mapsto 2, 4\mapsto 3, 5\mapsto 3\). Compute the signed graphic matroid of this double cover, and give the set of bases.

The circle graph with 8 edges has the property that on the Bergman fan of its graphic matroid the coarse fan structure coincides with the minimal nested set structure.

Count up to isomorphism the number of all simple connected graphs \(G\) with 8 edges such that the associated graphic matroid \(M\) has the property that it is connected and that on the Bergman fan of \(M\) the coarse fan structure coincides with the minimal nested set fan structure.

Let \(M\) be the lattice path matroid of the skew shape \((5,5,5,5,5,5,5)/(4,2,2,2,1,1)\). Compute the Hilbert series of the Chow ring of \(M\).

Determine the normalized volume of the non-nesting rook matroid (see arXiv: 2410.00127) defined by \(UUUURURRRRUR\) and \(RUURUURRRRUU\).

Consider the rational normal curve in real projective space \(\mathbb{P}^6\), and consider its image in an affine chart. Place 16 points in convex position on the image of the curve in the affine chart, and take their convex hull. How many distinct combinatorial types of polytopes can you get with this construction?

Let \(\mathcal{A}\) be a hyperplane arrangement of \(9\) hyperplanes in a \(3\)-dimensional complex vector space, with the hyperplanes labeled by \(\{0, 1, \dotsc, 8\}\). Suppose that the \(1\)-dimensional flats of the arrangement are given by the following sets of hyperplanes: \(\{0, 1, 2\}, \{0, 3,8\},\{0, 4, 6\},\{0, 5, 7\},\{1, 3, 7\},\{1, 4, 8\},\{1, 5, 6\},\{2, 3, 6\},\{2, 4, 7\},\{2, 5, 8\},\{3, 4, 5\}, \{6, 7, 8\}\). Let \(X\) be the visible contours compactification of the projectivization of \(\mathcal{A}\), which is naturally embedded in the torus-orbit closure of the point of \(Gr(3,9)\) corresponding to \(\mathcal{A}\). Let \(N\) be the normal bundle of \(X\) in this torus-orbit closure. Compute \(H^1(X, N)\).

Consider the log canonical embedding of the moduli space of marked cubic surfaces, that is, marked del Pezzo surfaces of degree 3. What is the dimension of the ambient projective space of this log canonical model? By way of comparison, for the moduli space \(M_{0,n}\) that dimension would be \((n-2)!-1\). What is the analogous number for moduli of marked cubic surfaces?

Let \(f: Y \to X\) be a degree 2 cover of smooth, projective, connected curves over the complex numbers. Assume that \(L\) is a line bundle on \(Y\) such that \(f^\ast f_\ast L\) is a stable vector bundle of degree zero on \(Y\) and such that \(\sigma^\ast L\) is not isomorphic to \(L\) for a generator \(\sigma\) of the Galois group of the covering \(Y/X\).

How many line bundles \(M\) on \(Y\) have the property that there exists an unramified Galois cover \(g: Z \to Y\) of degree three satisfying \(g^\ast f^\ast f_\ast L \simeq g^\ast f^\ast f_\ast (L \otimes M)\)?

Consider the tensor train variety parametrized via the map \[\Phi:\mathcal{P}\longrightarrow V^{=}_{D,d} \subseteq\mathbb{P}\!\left( \mathbb{C}^{d_1}\otimes\cdots\otimes \mathbb{C}^{d_N} \right), \quad (M^1,\dots,M^{N-1},[C])\longmapsto [T]\] with \[\mathcal{P} := \prod_{r=1}^{N-1}\mathrm{St}(D_r,n_r) \times \mathbb{P}^=\!\left(\mathbb{C}^{D_{N-1}\times d_N}\right)\] and \(\mathbb{P}^{=}\!\bigl(\mathbb{C}^{D_{N-1}\times d_N}\bigr)\) is the projectivization of the space of full rank \(D_{N-1}\times d_N\) complex matrices. Using this parametrization, compute the degree of the tensor train variety with \(D = (1, 2, 2, 2, 1)\) and \(d = (3, 3, 3, 3)\). Do not use Chern classes.

Let \(\lambda\) be the partition \((2,2,2)\). Let \(a=\{t^6,t^5,t^4,t^3,t^2,t\}\) such that, in the notation of this paper https://arxiv.org/abs/0902.1321v3. Let \(T\in \mathrm{SYT}(\lambda/\mu;a)\) be the tableaux with filling \(t^6,t^5,t^4,t^3,t^2,t\) read from top to bottom, left to right.

In Theorem 4.2 in this paper https://arxiv.org/abs/0902.1321v3 they say that assuming there exists a filling of the tableaux \((2,2,2)\) with numbers \(\omega_1,\ldots,\omega_6\) from left to right and top to bottom such that the determinant of the Jacobian \(J\) defined in Theorem 4.2 is non-zero then there exists a linear series in the fiber of \((z+t^6)(z+t^5)(z+t^4)(z+t^3)(z+t^2)(z+t)\) of the Wronski map with certain leading terms of the Plücker coordinates specified in the theorem if and only if the \(\omega_i\) satisfy certain equations.

Can you find numbers \(\omega_1,\ldots,\omega_6\) such that the theorem applies that is in the case where there exists a linear series in the fiber of \((z+t^6)(z+t^5)(z+t^4)(z+t^3)(z+t^2)(z+t)\) of the Wronski map. There should be a unique answer in this case. What is the sum \(\omega_1+\ldots +\omega_6\)?

Let \(E\) be an elliptic curve over the field \(\mathbb{C}_p\) without complex multiplication. We assume that \(E\) has good supersingular reduction. We define \(X\) as the perfectoid cover of \(E \times E\) which is the inverse limit along the infinite tower \(p: E \times E \rightarrow E \times E\) given by p-multiplication. What is the rank of the Néron-Severi group of \(X\) over \(\mathbb{Z}[1/p]\)?

Let \(X\) be the variety of symmetric \(6\times 6\) matrices of rank at most \(4\), viewed as a subvariety in a projective space \(\mathbb{P}^{20}\) of dimension \(20\). We are interested in the coisotropic hypersurface \(\mathrm{Ch}_3(X)\) which consists of all linear spaces of dimension \(5\) in \(\mathbb{P}^{20}\) which fail to intersect \(X\) transversally. What is the degree of this hypersurface in Plücker coordinates on the Grassmannian \(\mathrm{Gr}(5,\mathbb{P}^{20})\)?

Let \(X\) be the Segre embedding of \((\mathbb{P}^1)^{\times 3}\) in \(\mathbb{P}^{7}\), and consider its affine cone \(CX\) in \(\mathbb{C}^8\). Fix coordinates \((x_1,\ldots,x_8)\) for \(\mathbb{C}^8\). What is the number of complex critical points of the function \(f(x)=2(x_1-u_1)^2+\sum_{i=2}^8(x_i-u_i)^2\) from a generic point \(u=(u_1,\ldots,u_8)\in\mathbb{C}^8\) restricted to \(CX\)?

Consider the cubic threefold \(X\) in \(\mathbb{RP}^4\) defined by the following equation: \[x_0x_1x_2 + x_1x_2x_3 + x_2x_3x_4 + x_0x_3x_4 + x_0x_1x_4 = 0\] Let \(Y\) be the union of the two-dimensional linear spaces contained in \(X\). How many connected components does \(X \setminus Y\) have?

Let \([L]\) be a general point of the Grassmannian \(Gr(6, 17)\), and let \(X\) denote the visible contour compactification of the corresponding hyperplane arrangement. Then \(X\) is naturally embedded in the torus-orbit closure of \([L]\). Let \(E\) denote the conormal bundle of \(X\) in this toric variety. Compute the degree of \(c_{\dim X}(E)\).

Let \(Z\) be the free graded commutative algebra with one generator in each degree, except in degrees \(1\), \(4\), and \(6\). For a natural number \(k\), let \(X_k\) be the projective toric variety parametrized by the degree \(k\) monomials in \(Z\) (inside \(\mathbb{P}(Z_k)\)). What is the largest \(k\) such that \(X_k\) is normal?

Fix seven general points in the real projective plane. Consider the 21 lines spanned by any two points, the 21 conics spanned by any five points, and the 7 cubics through all points that pass doubly through one of the seven points. These \(49 = 21+21+7\) curves divide the real projective plane into regions, called polypols, each of which has a unique canonical differential form. How many of these canonical forms are linearly independent?

Let \(\Sigma\) be a complete convex fan in \(\mathbb{R}^n\) and \(X\) its toric variety with torus \(T = (\mathbb{C}^*)^n\) acting on \(X\). Fix a point \(x^*\in X\) such that the orbit of \(x^*\) sits dense in \(X\). Let \(Z_C \subset X\) be the torus orbit closure corresponding to the d-dimensional cone \(C \in \Sigma\). We call \(X\) equivariantly simple at \(Z_C\) iff for every \(L \subset \mathbb{C}^n\) linear space and every sequence \((t_n)_{n \geq 0} \subset T\) with limit \(t=lim_{n\to \infty}(t_n\cdot x^*) \in (Z_C)^{\circ}\) (where \((Z_C)^{\circ}\) is the open subset of \((Z_C)\) with no zero coordinates), the limit \(lim_{t\rightarrow \infty} (t_n \cdot L) \in Gr(dim(L),\mathbb{C}^n)\) depends only on \(t\), where \(t_n \cdot L\) is the inverse coordinate-wise action of T on L. Let \(\Sigma\) be the complete fan normal to the polytope whose vertex coordinates are \((1,2,3),(0,2,3),(1,3,2),(0,3,2),(1,0,2),(1,3,1)\). We realise \(X\subseteq \mathbb{P}^7\) as the closure of the image of the map \((\mathbb{C}^*)^3\to \mathbb{P}^7, (t_0,t_1,t_2)\mapsto (t_1^2t_2^3:\:t_1^3t_2^2:\:t_0t_2^2:\:t_0t_1t_2^2:\:t_0t_1^2t_2^2:\:t_0t_1^2t_2^3:\:t_0t_1^3t_2:\:t_0t_1^3t_2^2)\) and let \(T=(\mathbb{C}^*)^3\) act on \(X\) via the induced action of the multiplication in the domain of this parametrization. A subdivision of a fan consists of picking a maximal cone of the fan and a hyperplane and replacing the cone by the two cones obtained by intersecting the initial cone with the two half-spaces of the hyperplane. What is the minimal number of consecutive subdivisions needed to make \(X\) equivariantly simple?

Let \(X\subseteq\mathbb{P}:=\mathbb{P}^3\times\mathbb{P}^3\) be the image of the map \((x,y,z,w) \mapsto \bigl(\, [xy:z:w:1] \,,\,[zw:x:y:1]\, \bigr)\). In the following, we denote by \(\mathbb{G}(k,n)\) the Grassmannian of \(k\)-dimensional projective subspaces of \(\mathbb{P}^n\). Consider the product of Grassmannians \(\mathbb{G}:=\mathbb{G}(1,3)\times\mathbb{G}(1,3)\). For a generic \(L=L_1\times L_2\in\mathbb{G}\), since \(\dim(L)=2=\mathrm{codim}(X)\), we expect that \(X\cap L\) is either empty or a \(0\)-dimensional reduced scheme of \(\mathbb{P}\). Let \(I_\alpha\subseteq\mathbb{G}\) be the locus of products of linear spaces \(L\in\mathbb{G}\) such that the intersection \(X\cap L\) is nontransversal (in particular, a \(0\)-dimensional nonreduced set). The locus \(I_\alpha\) is a hypersurface in \(\mathbb{G}\), in particular, it has a multidegree \(\beta=(\beta_1,\beta_2)\), where \(\beta_i\) is the degree in the Plücker coordinates of the \(i\)-th Grassmannian \(\mathbb{G}(1,3)\). What is the value of \(\beta_1+\beta_2\)?

Let \(X = \operatorname{Seg}(\mathbb{P}^3 \times \mathbb{P}^3) \subset \mathbb{P}(\mathbb{C}^4 \otimes \mathbb{C}^4)\) be the Segre variety of rank-1 matrices, \(\sigma_2(X)\) its 2nd secant variety (consisting of rank-2 matrices), and \[u = e_1^{\otimes 2} + e_2^{\otimes 2} + \tfrac{1}{100}\,e_3^{\otimes 2} + \tfrac{1}{100}\, e_4^{\otimes 2} \;\in\; \mathbb{R}^{4 \times 4}.\] Consider the polynomial optimization problem \[\begin{align} \min_{\substack{a^{(k)},\, b^{(k)}\,\in\,\mathbb{R}^4\\k=1,2}} &f(a,b) \;:=\; \sum_{i,j=1}^{4} \Bigl( u_{ij} - \sum_{k=1}^{2} a^{(k)}_i\, b^{(k)}_j \Bigr)^{\!2}\\ \text{subject to } &f(a,b) \le 1/50. \end{align}\] Use the standard Moment-SOS hierarchy for this Polynomial Optimization Problem (POP), where the d-th level optimizes over polynomials of degree at most 2d. If the hierarchy has finite convergence, set s=1. Otherwise, set s=-1. Compute \((s\cdot f_2^*, s\cdot f_3^*, s\cdot f_4^*)\), where \(f_{d}^*\) denotes the optimal value of the d-th level of the hierarchy.

Let \(K\) be an algebraically closed and complete extension field of \(\mathbb{Q}_p\) and let \(X\) be a smooth, proper, connected curve over \(K\) of genus \(g\). We look at the associated diamond \(X^\diamond\) defined by Peter Scholze. Let \(\mathcal{O}\) be its structure sheaf. \(X^\diamond\) carries an etale topology which we denote by \(et\) and a quasi-pro-etale topology which we denote by \(qproet\).

Look at the inclusion \(Ext^1_{et}(\mathcal{O}, \mathcal{O}) \rightarrow Ext^1_{qproet}(\mathcal{O}, \mathcal{O})\) given by self-extensions of the structure sheaf on \(X^\diamond\) with respect to these two topologies. Compute the dimension of the cokernel of this inclusion as a \(K\)-vector space. If it is not finite dimensional, give back \(\infty\).

Let \(N=9\) and \(\lambda\) be the partition \((3,2,1)\) and \(\mu=(3,1,0)\). Let \(a=(t^2,t)\) such that \(a^+=a\cup \{0,0,0,0,\infty,\infty,\infty\}\), in the notation of this paper: arXiv:0902.1321v3. Let \(T\in \mathrm{SYT}(\lambda/\mu;a)\) be the tableau with filling of the skew shape with \(t\) in the bottom row and \(t^2\) in the cell above and to the right of this cell. In the language of the just mentioned paper the tableau is a diagonally increasing skew tableau.

According to Theorem 4.2 in the same paper there exists a linear series \(x\), with certain properties. Let \(f\) be the map \(\mathbb{P}^1\rightarrow \mathbb{P}^2\) corresponding to \(x\). Consider \((f,\{t^2,t\})\) as an element in the moduli space of stable maps, with marked points at \(t^2\), \(t\).

In the limit \(t\rightarrow 0\) in the moduli space of stable maps this becomes a degree \(4\) plane curve with a degree zero component \(X\) attached at \(c_\infty\) that has one marked point on it at \(c_t\) and one at \(c_0\), corresponding to the marked points \(t\), \(t^2\), respectively. There is a degree \(1\) component attached to \(X\) at \(c_*\).

Assume that \(X\) is parametrized such that \(c_\infty\) is at \(c=\infty\), \(c_0\) is at \(c=0\), and \(c_t\) is at \(c=1\). At what value of \(c\) is the node \(c_*\)?

Imagine you are an observer agent living in a universe with a smooth continuum of visible angular directions. Your visible sky is modeled as the complex projective line \(\mathbb{P}^1(\mathbb{C})\), equipped with normalized Fubini-Study probability measure \(\sigma\). You as an observer are not told whether this continuum is fundamental, or whether it is the low-energy rendering/simulation of a hidden finite algebraic substrate. If you are a capable observer to figure out whether you live in a simulation or not, you should be able to infer, from a small statistical defect in the visible angular distribution together with a few invariant-theoretic fingerprints, whether the continuum hypothesis is best explained by an underlying quotient substrate. You not given the hidden group, the primitive invariant, or the perturbation strength. These must be reconstructed from the algebraic data and then used in an asymptotic decision problem.

Let \(U=\mathbb{C}^2\), with homogeneous coordinates \((u,v)\), and let \(\sigma\) be the normalized Fubini-Study probability measure on \(\mathbb{P}^1(\mathbb{C})\). Under the alternative hypothesis, a hidden computational substrate is assumed to have a non-cyclic finite \(\operatorname{SL}_2(\mathbb{C})\)-quotient singularity \(X_\Gamma=\operatorname{Spec} \mathbb{C}[u,v]^\Gamma\), for some unknown finite subgroup \(\Gamma\subset \operatorname{SL}_2(\mathbb{C}).\) The visible angular sky is \(\mathbb{P}^1(\mathbb{C})\), and the local statistical defect is generated by the lowest primitive invariant of the quotient singularity.

The substrate is known only through the following side information.

1. The minimal resolution of \(X_\Gamma\) has exceptional intersection graph \(E_8\).

2. Equivalently, \(\mathbb{C}[u,v]^\Gamma\) has primitive homogeneous generators of degrees \(12,20,30\), satisfying one weighted-homogeneous relation of degree \(60\).

3. The degree-\(12\) primitive invariant is normalized as follows. It lies in the one-parameter family \[F_a(u,v)=u^{11}v+a\,u^6v^6-uv^{11},\] and \(a\in \mathbb{C}\) is chosen so that, if \[H_a=\operatorname{Hess}(F_a) = \det \begin{pmatrix} \dfrac{\partial^2 F_a}{\partial u^2} & \dfrac{\partial^2 F_a}{\partial u\partial v} \\[2mm] \dfrac{\partial^2 F_a}{\partial v\partial u} & \dfrac{\partial^2 F_a}{\partial v^2} \end{pmatrix},\] and \[T_a=\operatorname{Jac}(F_a,H_a) = \det \begin{pmatrix} \dfrac{\partial F_a}{\partial u} & \dfrac{\partial F_a}{\partial v} \\[2mm] \dfrac{\partial H_a}{\partial u} & \dfrac{\partial H_a}{\partial v} \end{pmatrix},\] then the three degree-\(60\) binary forms \(F_a^5, H_a^3, T_a^2\), are linearly dependent over \(\mathbb{C}\), with all three coefficients in the linear dependence nonzero. This is the intrinsic syzygy condition seen by the observer.

For such an admissible value of \(a\), define the projectively well-defined nonnegative function \[Q([u:v])= \frac{|F_a(u,v)|^2}{\bigl(|u|^2+|v|^2\bigr)^{12}}, \qquad [u:v]\in \mathbb{P}^1(\mathbb{C}),\] and its mean-zero normalization \[\rho([u:v])= \frac{Q([u:v])}{\displaystyle\int_{\mathbb{P}^1} Q\,d\sigma}-1.\] For sufficiently small \(\varepsilon\), define a probability measure \(P_\varepsilon\) on \(\mathbb{P}^1(\mathbb{C})\) by \[\frac{dP_\varepsilon}{d\sigma}=1+\varepsilon\rho.\]

The null hypothesis is that the visible sky is statistically indistinguishable from the continuum model, \(H_0:\varepsilon=0.\) The alternative hypothesis is that the observed sky is the continuum rendering of the hidden quotient substrate, \(H_1:\varepsilon=\varepsilon_0>0.\) You as an observer see \(n\) independent samples from \(P_\varepsilon\), where \(\varepsilon\in\{0,\varepsilon_0\}.\) You as an observer scientist use the Neyman/Pearson likelihood-ratio test for \[H_0:\varepsilon=0 \qquad\text{against}\qquad H_1:\varepsilon=\varepsilon_0>0.\] In the local regime \(\varepsilon_0=\frac{h}{\sqrt n},\) determine the exact constant \(C\) such that, when \(h=C,\) the asymptotic Type-I and Type-II errors are both \(\alpha=\beta=\Phi(-1).\)

Your answer must include the following five items:

1. the hidden group \(\Gamma\), up to conjugacy in \(\operatorname{SL}_2(\mathbb{C})\)

2. the admissible value or values of \(a\)

3. the exact Fisher information \(\mathcal{I}=\int_{\mathbb{P}^1}\rho^2\,d\sigma\)

4. the exact threshold constant \(C=\frac{2}{\sqrt{\mathcal{I}}}\)

5. the limiting log-Bayes factor, in favor of the hidden quotient-substrate sky over the continuum sky, assuming equal prior odds.

Let \(X\) be the variety in \((\mathbb{C}^*)^5\) cut out by \(x_3x_4 + x_1 - 1, x_5 + x_2x_3 - 1, x_4x_5 + x_2x_3x_4 + x_1x_2 - 1\). Consider the form \[\Omega= \frac{dx_1 \wedge dx_2}{x_1 x_2} + \frac{dx_1 \wedge dx_3}{x_1 x_3} - \frac{dx_1 \wedge dx_5}{x_1 x_5} + \frac{dx_2 \wedge dx_3}{x_2 x_3} + \frac{dx_2 \wedge dx_4}{x_2 x_4} + \frac{dx_3 \wedge dx_4}{x_3 x_4} + \frac{dx_3 \wedge dx_5}{x_3 x_5} + \frac{dx_4 \wedge dx_5}{x_4 x_5}.\] Let \({\rm Crit}_X(s)\) be the critical points on \(X\) of the multivalued function \(\log(x_1^{s_1}\cdots x_5^{s_5})\). Write a closed form in terms of \(s\) for \[\sum_{x(s) \in {\rm Crit}_X(s)} \Omega(x(s)).\]

Suppose that I have a graph composed of a chain of 10 cycles of length 20. Each cycle is joined by an edge to the cycle before and after, with the attachment points diametrically opposed to each other. Let \(v\) be the vertex on the last cycle diametrically opposed to the attachment point with the previous chain.

Determine the number of degree 4 effective and \(v\)-reduced divisors \(D\) such that \(D + 5v\) has Baker–Norine rank at least 3.

Let \(\mathrm{Gr}(2,n)\) be the Grassmannian of lines in \(\mathbb{C}^n\) with its Plücker embedding. Compute the combinatorial rank \(\operatorname{cr}(\mathrm{Gr}(2,4),D)\), where \(D\) is the union of all Plücker hyperplane sections.

Consider the Segre-Veronese embedding of \(\mathbb{P}=(\mathbb{P}^1)^{\times 3}\) associated with the line bundle \(\mathcal{L}=\mathcal{O}_{\mathbb{P}}(124534,2345729,87685764)\). Let \(\mathcal{P}^{100}(\mathcal{L})\) denote the jet bundle of order \(100\) of \(\mathcal{L}\). For all \(i\ge 0\), let \(c_i(\mathcal{P}^{100}(\mathcal{L}))\) denote the \(i\)-th Chern class of \(\mathcal{P}^{100}(\mathcal{L})\). Compute the value of \[\sum_{i=0}^3\deg(c_i(\mathcal{P}^{100}(\mathcal{L})))\]

Let \(f_t:\mathbb{P}^1\rightarrow \mathbb{P}^2\) be a family of maps of degree 3 parametrized by \(t\). Consider \((f_t,p_1,p_2,p_3)\) as a family of stable maps with \(3\) marked points where the marked points \(p_1,p_2,p_3\) are on the flexes of \(f_t\). In the case where the \(3\) flexes collide as \(t\rightarrow 0\) and the domain of the map splits into a degree \(2\) component \(C\), a degree \(1\) component \(E_1\), and a degree \(0\) component \(E_0\). Assume that \(C\) is connected to \(E_0\) and \(E_1\) is connected to \(E_0\). Assume that \(E_0\) is parametrized by \(c\) such that the node between \(C\) and \(E_0\) is at \(c=\infty\) and there is at least one marked point at \(c=0\) and at least one at \(c=1\) what is the highest possible value of \(c\) of the location of the node between \(E_0\) and \(E_1\)?

Consider any Dyck path \(D\) in the grid \(\mathbb{Z}^2\) that stays above the diagonal \((1,0)\) to \((n+1,n)\). Take \(P\) as the Minkowski sum of all simplices \(\Delta_{ij}={\rm Newt}(1+x_i+x_ix_{i+1}+\ldots+x_i\cdots x_j)\) for all \((i,j)\) on or below \(D\). How many facets does \(P\) have?

Suppose I have two functions \(f_1: \mathbb{R}^d \to \mathbb{R}^n\), and \(f_2: \mathbb{R}^n \to \mathbb{R}\). The functions are of the form \(f_i(x) = \max(A_i x, B_i x)\). Here, \(A_1, B_1\) are \(n \times d\) matrices with real entries, and \(A_2, B_2\) are \(1 \times n\) matrices with non-negative real entries. The composition \(f(x) = f_2(f_1(x))\) is convex, continuous, piecewise-linear and positively homogeneous. Its dual polytope is denoted \(P_f\). For \(d = 2\), \(n = 1003\), determine \(\max\{ \#\, \text{vertices } P_f : f \text{ is as described above} \}\).

We consider the space \(X\) of all closed \(C^2\)-submanifolds \(S\subset(\mathbb{R}/\mathbb{Z})^2\) of dimension \(1\) such that for every \(a\in\mathbb{Z}^2\) with positive entries the map \(\pi_a\colon S\to\mathbb{R}/\mathbb{Z},\, x\mapsto\langle x,a\rangle\) is a covering map and the map \[\mathrm{tr}_a\colon\mathbb{R}/\mathbb{Z}\to(\mathbb{R}/\mathbb{Z})^2,\, y\mapsto\sum_{x\in \pi_a^{-1}(y)}x\] is affine linear with slope \((11,5)\). We equip \(X\) with the Hausdorff distance. What is the Lebesgue covering dimension of \(X\)?

Let \(G(a_1,a_2,\ldots,a_n;z)\) be a Goncharov polylogarithm, such that for complex numbers \(a_1,a_2,\ldots,a_n\), and \(z\), are defined recursively by: \[G(;z)=1\] \[G(a_1,a_2,\ldots,a_n;z) = \int_0^z \frac{dt}{t-a_1} G(a_2,\ldots,a_n;t)\] with the common shuffle regularization: \(G(0;z)=\log(z)\); Another examples are \(G(1;z)=\log(1-z)\), and \(G(x;z)=\log(1-z/x)\) where \(\log(z)\) is the natural logarithm;

Now consider the analytic continuation of polylogarithms: \(G(x;y) = G(y;x) + i \pi +G(0;y)-G(0;x)\) which is done with a fixed branch choice. Note that this has the same information as the more familiar formula: \[\log(x-y) = i\pi+\log(y-x)\]

With that same branch choice, obtain a formula for \(G(x,1;y)\) in terms of polylogs where \(y\) is only allowed to be on the right of the semicolon when the numbers to the left of the semicolon are 0 or 1, and where \(x\) is allowed to be in the right of the semicolon when the numbers to the left of the semicolon are either of \(0,1\) or \(y\).

Consider the following matrix: \[A =\left(\!\begin{array}{cccccccccccccccc} 0&1&0&1&0&1&0&1&0&1&0&1&0&1&0&1\\ 0&0&1&1&0&0&1&1&0&0&1&1&0&0&1&1\\ 0&0&0&0&1&1&1&1&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&1&1&1&1&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1 \end{array}\!\right).\] We denote the \(i\)-th column of \(A\) as \(\alpha_i\), and we write \(A = [\alpha_1,\dots,\alpha_{16}]\). Now, for a fixed choice of \(W = (w_1,\dots,w_{16}) \in (\mathbb{C}^*)^{16}\) consider the polynomial map \[\Phi_W\colon(\mathbb{C}^*)^{5} \longrightarrow \mathbb{C}^{16}\] given by \(\Phi_{W}(t) = (w_1 t^{\alpha_1},\dots,w_{16} t^{\alpha_{16}})\). What is the expected maximum likelihood degree of the Zariski closure of the image of this map for the choice \(W = (1,1,1,1,1,-1,-1,1,4,-2,-2,1,c_1,c_2,c_3,c_4)\), where \(c_1,c_2,c_3,c_4\) are generic complex numbers in the torus?

What is the maximum likelihood degree of the Grassmannian of lines \(\mathrm{Gr}(2,n)\) in its Plücker embedding in \(\mathbb{P}^{\binom{n}{2}-1}\)?

Let \(G_1,\dots,G_N\) be independent standard Gaussian vectors in \(\mathbb{R}^2\), and set \[S_N:=\frac{1}{N}\sum_{j=1}^N G_jG_j^{\top}.\] For \(N\) large enough, define \[Y_j^{(N)}:=S_N^{-1/2}G_j, \qquad j=1,\dots,N.\] Set \[u_1=(1,0), \qquad u_2=\left(-\frac{1}{2},\frac{\sqrt3}{2}\right), \qquad u_3=\left(-\frac{1}{2},-\frac{\sqrt3}{2}\right).\] For \(u\in S^1\), define \[Q_N(u) := \frac{1}{N} \sum_{j=1}^N \left| \left\langle Y_j^{(N)},u\right\rangle \right|^6 .\] Finally, let \[B:= \begin{pmatrix} 2 & 1\\ 1 & 3 \end{pmatrix}.\]

Determine the exact value of \[\lim_{N\to\infty} \frac{1}{N^{1/3}} \log \mathbb{P}\left( Q_N(u_1)\ge 27,\; Q_N(u_2)\ge 27,\; Q_N(u_3)\ge 27,\; N^{1/3}(S_N-I_2)\succeq B \right),\] where \(\succeq\) denotes the Loewner order on symmetric matrices.

Consider the definition of Grassmann Distance Degrees of subvarieties of a Grassmannian \({\rm Gr}(2,n)\) as in https://arxiv.org/pdf/2601.22843.

Given a partition \(A_1,\dots,A_r\) of \([n]\) with \(|A_1| \geq |A_2| \geq \dots \geq |A_r|\), we can define an associated rank-two matroid on \(n\) elements by declaring \(e\neq f\) to be a non-basis if and only if there exists \(j\) such that \(e,f \in A_j\). Denote this by \(M_{(|A_1|,\dots,|A_r|)}\) and set \(m_i = |A_i|\). Such a matroid is always realized by the \(2 \times n\) matrix whose columns are partitioned into \(r\) blocks of sizes \(m_1,\dots,m_r\) (ordered such that the first \(m_1\) columns form the first block and so on) and such that two columns are linearly independent if and only if they belong to different blocks. The rowspans of such matrices form a subvariety of the Grassmannian \({\rm Gr}(2,n)\).

What is the Grassmann Distance Degree of the subvariety corresponding to a partition of sizes \((m_1,\dots,m_5) =(2,2,2,2,2)\) in \({\rm Gr}(2,10)\)?

Call a finite dimensional algebra \(A\) self-orthogonal if every indecomposable \(A\)-module \(M\) is self-orthogonal meaning \(Ext_A^i(M,M)=0\) for all \(i>0\). Consider the class of all connected cyclic Nakayama algebras given by quiver and admissible relations over a field \(K\) with 8 simple modules that have dominant dimension at least two and are self-orthogonal. What is the sum of the vector space dimensions of those algebras?

Consider the following symmetric tensor \(T\in \mathbb{C}^{6\times 6\times 6}\): \[\begin{align} T_{:,:,1} &= \begin{pmatrix} 25 & 20 & 17 & 15 & -16 & -6 \\ 20 & -4 & 2 & -13 & 30 & 27 \\ 17 & 2 & 59 & -28 & 54 & 60 \\ 15 & -13 & -28 & 19 & -2 & 0 \\ -16 & 30 & 54 & -2 & 8 & 15 \\ -6 & 27 & 60 & 0 & 15 & 54 \end{pmatrix}, & T_{:,:,2} &= \begin{pmatrix} 20 & -4 & 2 & -13 & 30 & 27 \\ -4 & 28 & 23 & 35 & -36 & 0 \\ 2 & 23 & 23 & 26 & -18 & -9 \\ -13 & 35 & 26 & 55 & -51 & 3 \\ 30 & -36 & -18 & -51 & 54 & 3 \\ 27 & 0 & -9 & 3 & 3 & -12 \end{pmatrix},\\[1ex] T_{:,:,3} &= \begin{pmatrix} 17 & 2 & 59 & -28 & 54 & 60 \\ 2 & 23 & 23 & 26 & -18 & -9 \\ 59 & 23 & 27 & 32 & -13 & 18 \\ -28 & 26 & 32 & 62 & -42 & -15 \\ 54 & -18 & -13 & -42 & 49 & 21 \\ 60 & -9 & 18 & -15 & 21 & 12 \end{pmatrix}, & T_{:,:,4} &= \begin{pmatrix} 15 & -13 & -28 & 19 & -2 & 0 \\ -13 & 35 & 26 & 55 & -51 & 3 \\ -28 & 26 & 32 & 62 & -42 & -15 \\ 19 & 55 & 62 & 72 & -43 & 39 \\ -2 & -51 & -42 & -43 & 37 & -27 \\ 0 & 3 & -15 & 39 & -27 & -21 \end{pmatrix},\\[1ex] T_{:,:,5} &= \begin{pmatrix} -16 & 30 & 54 & -2 & 8 & 15 \\ 30 & -36 & -18 & -51 & 54 & 3 \\ 54 & -18 & -13 & -42 & 49 & 21 \\ -2 & -51 & -42 & -43 & 37 & -27 \\ 8 & 54 & 49 & 37 & -29 & 33 \\ 15 & 3 & 21 & -27 & 33 & 15 \end{pmatrix}, & T_{:,:,6} &= \begin{pmatrix} -6 & 27 & 60 & 0 & 15 & 54 \\ 27 & 0 & -9 & 3 & 3 & -12 \\ 60 & -9 & 18 & -15 & 21 & 12 \\ 0 & 3 & -15 & 39 & -27 & -21 \\ 15 & 3 & 21 & -27 & 33 & 15 \\ 54 & -12 & 12 & -21 & 15 & 9 \end{pmatrix}. \end{align}\] Determine the rank \(r\) and the unique minimum rank decomposition of \(T\), so that \(T = \sum_{i=1}^{r} a_i^{\otimes 3}\), where \(a_1,\ldots,a_r \in \mathbb{C}^{6}\). As your answer, submit the tuple \((r, \sum_{i=1}^r |a_{i,3}|^2)\), consisting of the rank \(r\) and the “checksum” of \(3\)rd coordinates \(a_{i,3}\) of the vectors in the minimum rank decomposition.

Let \(p>5\) be a prime and \(\mathbb{Z}_p\) the \(p\)-adic integers. Let \(G = {\rm GL}_{12}\) over \(\mathbb{Z}_p\) (Dynkin type \(A_{11}\)). Let \(\breve{\mathbb{Z}}_p\) be the integers of the completed unramified closure of \(\mathbb{Q}_p\). For \(r > 0\) regard \(G_r :=G(\breve{\mathbb{Z}}_p/p^r)\) as (the \(k:=\overline{\mathbb{F}}_p\) points of) a perfect algebraic group (of perfectly finite type) over the residue field \(\mathbb{F}_p\) by using the positive loop functor (also called jet scheme) construction. Let \(G_r^1:=ker(G_r \to G_1)\). Denote by \(F\) the geometric Frobenius on \(G_r\). Let \(T\) be an elliptic unramified maximal torus of Coxeter type of \(G\) (defined over \(\mathbb{Z}_p\)) and let \(U\) be the unipotent radical of a Borel subgroup containing \(T\) (only defined over \(\breve{\mathbb{Z}}_p\)). We have the corresponding subgroups \(T_r^1,U_r^1\) of \(G_r^1\), and \(T_r^1\) is \(F\)-stable.

The \(k\)-variety \(X_r = \{g \in G^1_r \colon g^{-1}F(g) \in U^1_r\}\) has an action of \((T^1_r)^F\) by \(t : x \mapsto xt\). Let \(\phi : (T_r^1)^F \to \Lambda^\times\) be a character, where \(\Lambda=\overline{\mathbb{Q}}_\ell\) and \(\ell\neq p\) is a prime.

Let \(T^1\) be the preimage in \(G(\breve{\mathbb{Z}}_p)\) of \(T_r^1 \subseteq G_r^1\). Let \(\psi\) be any lift to \(T(\mathbb{Q}_p)\) of the inflation of \(\phi\) to \(T^1(\mathbb{Z}_p)\). Assume \(\psi\) has a Howe factorization in the sense of Kaletha (see section 3.6 of Kaletha’s paper “Regular supercuspidal representations”), such that \(\psi=\psi_{-1}\cdot\psi_0\cdot \psi_1\) with \(\psi_i\) of depth \(r_i\) with \(r_{-1}:=0<r_0<r_1 \leq r\) and the root systems of corresponding twisted Levi sequence is \(G^{-1} = T \subsetneq G^0 \subsetneq G^1 = G\), with \(G^0\) of (absolute) type \(A_5 \times A_5\). (Note that \(\psi_{-1}\) will not affect this question, as \(\psi|_{T^1(\mathbb{Z}_p)}\) does not depend on the depth zero character \(\psi_{-1}\).)

Question: assume \(r=20\), \(r_0=12\), \(r_1=15\). Determine the smallest \(i \geq 0\) for which the \(\phi\)-weight part of the etale cohomology group with compact support \(H_c^i(X_r,\Lambda)[\phi]\) is non-zero.

Let \(A\) be the connected linear Nakayama algebra with Kupisch series \([ 5, 5, 4, 4, 3, 3, 3, 2, 1 ]\) and let \(T\) denote the trivial extension algebra of \(A\) by \(D(A)\). What is the sum of all the \(\Omega\)-periods of the simple \(T\)-modules?

For a connected Nakayama algebra \(A\) with \(n\) simple modules with associated Kupisch series \(c(A)=[c_0,\dots,c_{n-1}]\), define \(\Sigma A\) as the sum of all entries of the Kupisch series. Let \(L_n\) be the set of all connected Nakayama algebras \(A\) (linear and cyclic) with \(n\) simple modules that are higher Auslander algebras (meaning the global dimension and the dominant dimension of \(A\) are finite and coincide) and for which the unique cycle in the resolution quiver of \(A\) consists of exactly 1 vertex. Compute \(\sum_{A \in L_n}\Sigma A\) for \(n=20\).

Consider the two posets \(P\) and \(Q\) given by the cover relations \[\begin{array}{l@{\qquad}l} P: & Q:\\[0.3ex] 0\lessdot a_1,a_2,a_3 & 0\lessdot a_1,a_2,a_3\\ a_1\lessdot b_1,b_2 & a_1\lessdot b_1,b_2,b_3\\ a_2\lessdot b_2,b_3 & a_2\lessdot b_2,b_3\\ a_3\lessdot b_3,b_1 & a_3\lessdot b_1\\ b_1\lessdot c_1,c_2 & b_1\lessdot c_1,c_2,c_3\\ b_2\lessdot c_2,c_3 & b_2\lessdot c_2,c_3\\ b_3\lessdot c_3,c_1 & b_3\lessdot c_1\\ c_1\lessdot d_1,d_2 & c_1\lessdot d_1,d_2,d_3\\ c_2\lessdot d_2,d_3 & c_2\lessdot d_2,d_3\\ c_3\lessdot d_3,d_1 & c_3\lessdot d_1,d_3\\ d_1\lessdot e_1,e_2 & d_1\lessdot e_1,e_2,e_3\\ d_2\lessdot e_2,e_3 & d_2\lessdot e_2,e_3\\ d_3\lessdot e_3,e_1 & d_3\lessdot e_1\\ e_1,e_2,e_3 \lessdot 1 & e_1,e_2,e_3 \lessdot 1 \end{array}\] Let \(\mathit{ex}\Psi_P(y;a,b)\) and \(\mathit{ex}\Psi_Q(y;a,b)\) be their respective extended ab-indices, and \(\mathit{ex}\Psi_P(1;a,b)\) and \(\mathit{ex}\Psi_Q(1;a,b)\) the corresponding pull-back ab-indices.

Determine the smallest coefficient of an ab-monomial in the difference \(\mathit{ex}\Psi_P(1;a,b) - \mathit{ex}\Psi_Q(1;a,b)\).

Consider the cycle graph \(C\) on \(5\) vertices. Let \(L\) be the lattice of flats of the associated matroid and \(\mathcal{G}\) the subset of \(L\) given by edge sets of connected induced subgraphs. In the paper https://arxiv.org/abs/2602.21194, refinements of the doubled nested set fan \(\Sigma^{2\mathcal{G}}\) associated to the pair \((L,\mathcal{G})\) are defined. The authors show that the tubing refinement of \(\Sigma^{2\mathcal{G}}\) defines a subdivision of the nested set fan \(\Sigma_{\mathcal{G}}\). Compute the number of maximal cones in this subdivision, using the results from https://arxiv.org/abs/2602.21194. How many of them are simplicial?

Consider the definition of the MMC regions and strata in Section 5 of the paper https://arxiv.org/abs/2503.09571. For \(n=15, r=6\) and the sign pattern \(\sigma=(+,+,-,+,+,+,+,+,+,+,-,+,+,+,+)\) how many MMC strata \(\mathcal{C}_{n,\sigma,r}^0\) of dimension \(24\) are there?

Consider variables \(u_{ij}\) labeled by chords in an \(n\)-gon with vertices \(1,\ldots,n\) and the relations \(R_{ij}:= u_{ij} + \prod_{kl \nsim ij} u_{kl} -1\) where \(kl \nsim ij\) means the chords do not cross. For \(n=8\) how many sign patterns in \(\{+,-\}^{n(n-3)/2}\) (which assigns a sign to each variable \(u_{ij}\)) do not contradict \(R_{ij}\)?

Let \(ex\Psi_{\Pi_n}(y,a,b)\) be the extended \(ab\)-index of the set partition lattice \(\Pi_n\) of rank \(n\). Compute \(ex\Psi_{\Pi_n}(1,1,1)\) for \(n=12\).

Take a square piece of paper of thickness \(d\) and side lengths \(l=1\) with barycenter at the origin. First, fold the paper randomly and endow the two resulting sides with different topologies. Fold it back to the original structure. Now fold it three times along the longest edge, i.e., half the longest edge. All folding is performed in Euclidean space \(\mathbb{R}^3\).

The structure you have now will have layers on top of each other endowed with different topologies. Now take one of the vertices and fold it to the middle of the longest edge.

Now subdivide the resulting structure into a triangle and a rectangular shape. Puncture the triangle at its barycenter with a pointy steel rod. Unfold the structure again to its original shape with side length 1.

Group the holes according to their distance to the origin. Remember that you have two topologies. This gives the tuple of the cardinality of the distance classes.

Now take the length of this tuple and define it as \(c\). Organize all holes on the paper, which has thickness \(d\), in a cyclic graph. Let \(c\) be the number of possible colors for the edges. Compute the number of monochromatic perfect matchings of the graph.

Let \(x\) be a point in the complex Grassmannian \(G(k,n)\), and let \(M\) be the corresponding matroid. The Chow class of \(M\) is defined as the class of the torus orbit closure \(\overline{(\mathbb{C}^*)^n x}\) in the Chow ring of \(G(k,n)\).

Expand the Chow class of the non-nesting rook matroid (arXiv: 2410.00127) defined by \(UUUURURRRRUR\) and \(RUURUURRRRUU\) as a linear combination of Schubert cycles \(s_{\lambda}\).

Let \(f_1\), ..., \(f_{12}\) be 12 linear forms on a 6-dimensional complex vector space V which define a hyperplane arrangement that realizes the matroid FP12. Let W be the subspace of rational functions on PV spanned by elements of the forms \(\prod_{i \in B} f_i^{-1}\), where \(B\) is a basis of FP12. Choose a generic hyperplane in \(V\), and let \(X\) be the kernel of the restriction map from \(W\) to this generic hyperplane. Compute the dimension of \(X\).

How many combinatorial types of minuscule type \(B_3\) strong valuated Coxeter matroids are there? By ‘combinatorial type’, we mean the number of induced regular subdivisions of the \(3\)-cube. Give me only the number of such matroids up to symmetry of the cube.

We use the notation of https://arxiv.org/pdf/2507.02426v2.

Let \(C/K\) be the curve defined by \[y^2=x(x+1)(x+5)(4x+1)(x+8)(x+8\cdot 2^{1/2})(x+8\cdot2^{3/4})(x+16)(x+32)\]

What are the normalized thicknesses of the double points of the stable marked model of \(C\) in the sense of the above article? Return your final answer as a descending list in the form \((a, b, c, \dots)\).

Let \(X\) be the Segre-Veronese embedding of \(Y=(\mathbb{P}^1)^6\) with respect to the line bundle \(\mathcal{O}_{Y}(1,2,3,4,5,6)\). Compute the Rayleigh-Ritz degree \(\mathrm{RRdeg}_{10}(X)\) with respect to a generic quadratic form.

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

Consider a triangulated surface with \(11\) vertices and whose facets are given by the following set of triangles: \[\begin{gather} \bigl\{(1,5,6),\;(1,2,6),\;(1,5,7),\;(5,10,7),\;(9,10,7),\;(4,7,9),\;(1,7,3),\\ (1,3,4),\;(1,4,8),\;(1,8,9),\;(1,9,10),\;(1,10,2),\;(10,8,2),\;(10,3,8),\\ (10,11,3),\;(2,3,11),\;(2,8,4),\;(4,5,2),\;(6,2,7),\;(2,11,7),\;(8,11,7),\\ (3,7,8),\;(6,7,4),\;(6,4,3),\;(6,3,9),\;(6,9,8),\;(6,8,11),\;(6,11,5),\\ (11,9,5),\;(2,3,5),\;(9,4,11),\;(10,11,4),\;(4,5,10),\;(3,9,5)\bigr\}. \end{gather}\] To begin, find coordinates for each of the \(11\) vertices, such that the topology of the surface is correct (its genus is the right one and facets overlap only on edges or vertices or the empty set; it has to be an embedded simplicial complex) and every triangle is flat (inside a hyperplane and is the convex hull of its vertices). This initial auxiliary answer should be a \(3 \times 11\) matrix \(A\) of integer numbers. Finally, multiply the \(A\) you obtained by \(y=[1,1,1]\), \(yA=B\) is a \(1 \times 11\) list of integer numbers. Call \(D\) the number of distinct numbers you get in \(B\). Multiply \(D\) by the genus of the surface. Print that final number.

What is the number of equivalence classes of \(d\)-dimensional virtual polytopes with \(d + 1\) vertices up to affine equivalence for \(d = 2\)?

Find the number of vertices of the reflexive flow polytope of the complete directed graph with six vertices.

Let \(K\) be the knot \(9_{46}\), \(n\) an integer and \(M_n\) the Dehn filling of \(K\) along the slope \(1/n\). The two strong inversions of \(K\) extend to \(M_n\). Let \(K^1_n\) and \(K^2_n\) be the branch sets in the respective quotients. For \(i=1,2\), let \(g_i\) be the limit of \(Kh(K^i_n)-Kh(K^i_{n-1})\) as \(n\) goes to \(\infty\), where \(Kh(L)\) denotes the (unreduced) Khovanov homology of a link \(L\). What is the dimension of the (hat version of) Heegaard Floer homology of \(M_{g_1\cdot g_2}\)? All homology theories are taken with coefficients in the field of 2 elements.

Consider the moduli space of 7 labeled points in linearly general position in the projective plane. This is the open Grassmannian \(\mathrm{Gr}(3,7)\) where all 35 Plücker coordinates are non-zero modulo the action of the 7-dimensional torus. We are interested in the log canonical embedding of this moduli space. What is the dimension of the projective space where this log canonical model lives?

Let \(G\) be an undirected graph with \(d\) vertices. Let \(\mathcal{I}_G\) be the set of global Markov statements \(A\!\perp\!\!\!\!\perp\! B|S\) for \(G\) (where “\(\perp\!\!\!\!\perp\)” stands for “independent”) such that the collection \(\{A,B,S\}\) is a partition of \(V\). For any vector \(x\in\mathbb{R}^d\) and any subset \(S\subseteq[d]\), we denote by \(x_S\) the projection of \(x\) onto the coordinates with indices in \(S\). Fix a subset \(\mathcal{X}\subseteq\mathbb{R}^d\) and define \(\mathcal{P}^{(0)}_{G}:=\operatorname{conv}(\mathcal{X})\subseteq\mathbb{R}^d\), where “\(\operatorname{conv}\)” stands for convex hull. For \(i\ge 1\) define \[\mathcal{P}^{(i)}_{G} := \operatorname{conv}\bigcup_{A\perp\!\!\!\!\perp B|S\in\mathcal{I}_G}\{(x_A,y_B,x_S),(y_A,x_B,x_S)\mid(x_A,x_B,x_S),(y_A,y_B,x_S)\in\mathcal{P}^{(i-1)}_{G}\}\,.\] Suppose now that \(d=4\) and that \(\mathcal{X}\) is the set of \(10\) points in \(\mathbb{R}^4\) whose coordinates are the columns of the matrix \[\left(\!\begin{array}{cccccccccc} \frac{10}{9}&1&1&\frac{4}{5}&\frac{4}{5}&\frac{3}{2}&\frac{5}{9}&\frac{1}{5}&\frac{8}{3}&\frac{2}{5}\\[7pt] \frac{3}{5}&\frac{6}{5}&4&\frac{5}{2}&\frac{1}{9}&6&\frac{3}{4}&2&1&\frac{3}{2}\\[7pt] \frac{3}{2}&\frac{3}{2}&\frac{3}{5}&3&\frac{4}{3}&\frac{2}{3}&\frac{1}{3}&\frac{4}{5}&2&\frac{1}{2}\\[7pt] 2&1&\frac{2}{5}&1&\frac{8}{9}&1&1&1&\frac{9}{4}&2 \end{array}\!\right)\,.\] Determine the number of vertices of \(\mathcal{P}^{(2)}_{G}\) with respect to the \(4\)-cycle \(G\).

Consider the \(5\)-dimensional cube \(C_5\) and all the non-simple \(4\)-dimensional combinatorial types of slices of \(C_5\) obtained by intersecting \(C_5\) with a hyperplane \(H\). How many of these are not combinatorially equivalent to a product of the form \((C_4 \cap H) \times C_1\)?

Fix \(n\in\mathbb{N}\), \(r\in[n]\), \(q\ge 4\), and \(0<t_1<\cdots<t_n\). Set \(X_n:=2^{\{t_1,\dots,t_n\}}.\) For \(c=(c_1,\dots,c_p)\in\mathbb{R}^p\), define \(\Psi_c:X_n^p\to X_n\) as follows. Given \(E=(E_1,\dots,E_p)\), write \[\bigcup_{j=1}^p E_j=\{s_1<\cdots<s_a\}.\] Let \(z_0:=0\), and for \(i=1,\dots,a\) put \[z_i := \mathbf{1}_{\{z_{i-1}\le 1\}} \cdot (z_{i-1})_+ + \sum_{j=1}^p c_j\mathbf{1}_{\{s_i\in E_j\}} .\] Then \[\Psi_c(E):=\{s_i:1\le i\le a,\;z_i>1\}.\]

Let \[\alpha=(K,n_1,\dots,n_{K-1},B^{(1)},\dots,B^{(K)}), \qquad n_0=n_K=1,\] with \(B^{(\ell)}\in\mathbb{R}^{n_\ell\times n_{\ell-1}}\). If \(b_i^{(\ell)}\) denotes the \(i\)-th row of \(B^{(\ell)}\), define \[G_\alpha^{(0)}(E):=E, \qquad G_\alpha^{(\ell)}(E) := \bigl( \Psi_{b_1^{(\ell)}}(G_\alpha^{(\ell-1)}(E)), \dots, \Psi_{b_{n_\ell}^{(\ell)}}(G_\alpha^{(\ell-1)}(E)) \bigr).\] Impose \(\max_{\ell,j}\#\{i:B_{ij}^{(\ell)}\neq 0\}\le q .\) For \(\rho\in[n]\), let \(\mathcal{G}_{n,\rho,q}\) be the class of all maps \(g:X_n\to X_n\) for which there exists an \(\alpha\) satisfying the above constraint such that \(g=G_\alpha^{(K)}\) and \(|g(\{t_1,\dots,t_n\})|\le \rho\). Define \[\mathfrak{N}(n,\rho,q) := \sup_{g\in\mathcal{G}_{n,\rho,q}} \min_{\substack{\alpha:\,G_\alpha^{(K)}=g\\ \max_{\ell,j}\#\{i:B_{ij}^{(\ell)}\neq 0\}\le q}} \sum_{\ell=1}^K n_\ell\] and \[\mathfrak{S}(n,\rho,q) := \sup_{g\in\mathcal{G}_{n,\rho,q}} \min_{\substack{\alpha:\,G_\alpha^{(K)}=g\\ \max_{\ell,j}\#\{i:B_{ij}^{(\ell)}\neq 0\}\le q}} \#\{(\ell,i,j):B_{ij}^{(\ell)}\neq 0\}.\] Determine the asymptotic order, up to polylogarithmic factors, of \[\mathfrak{N}(n,n,q)+\mathfrak{S}(n,r,q).\]

Consider the \(6\)-dimensional linear space of symmetric matrices \(L = \operatorname{Span}\{A_1,\dots,A_6\}\subset \mathbb{C}^{6\times 6}\) generated by \[\begin{align} A_1 &= \begin{bmatrix} 25&48&-14&18&10&13\\ 48&-2&-46&36&-18&-6\\ -14&-46&-2&-6&-5&-16\\ 18&36&-6&28&7&12\\ 10&-18&-5&7&-22&-1\\ 13&-6&-16&12&-1&-11 \end{bmatrix}, & A_2 &= \begin{bmatrix} 48&-2&-46&36&-18&-6\\ -2&89&9&23&47&43\\ -46&9&33&-20&34&12\\ 36&23&-20&33&-11&8\\ -18&47&34&-11&19&20\\ -6&43&12&8&20&27 \end{bmatrix},\\[1ex] A_3 &= \begin{bmatrix} -14&-46&-2&-6&-5&-16\\ -46&9&33&-20&34&12\\ -2&33&-1&12&25&14\\ -6&-20&12&-26&-21&-6\\ -5&34&25&-21&-1&10\\ -16&12&14&-6&10&18 \end{bmatrix}, & A_4 &= \begin{bmatrix} 18&36&-6&28&7&12\\ 36&23&-20&33&-11&8\\ -6&-20&12&-26&-21&-6\\ 28&33&-26&27&10&10\\ 7&-11&-21&10&6&-2\\ 12&8&-6&10&-2&0 \end{bmatrix},\\[1ex] A_5 &= \begin{bmatrix} 10&-18&-5&7&-22&-1\\ -18&47&34&-11&19&20\\ -5&34&25&-21&-1&10\\ 7&-11&-21&10&6&-2\\ -22&19&-1&6&47&11\\ -1&20&10&-2&11&-7 \end{bmatrix}, & A_6 &= \begin{bmatrix} 13&-6&-16&12&-1&-11\\ -6&43&12&8&20&27\\ -16&12&14&-6&10&18\\ 12&8&-6&10&-2&0\\ -1&20&10&-2&11&-7\\ -11&27&18&0&-7&62 \end{bmatrix}. \end{align}\] Let \(X \subseteq L\) be the subvariety of matrices of rank \(\leq 3\). What is the number of irreducible components of \(X\)?

Let \(KQ\) be the path algebra where \(Q\) is of linearly oriented Dynkin type \(A_3\) having 3 vertices. Let \(A\) be the total preprojective algebra of \(KQ\). How many indecomposable non-zero \(A\)-modules are there up to isomorphism?

Let \(KQ\) be the path algebra of the Dynkin quiver of type \(A_5\) (so \(Q\) has 5 vertices) with linear orientation. Let \(A\) be the Koszul dual algebra of the Auslander algebra of \(KQ\). How many indecomposable spherical \(A\)-modules are there? Here a module \(M\) is called spherical (in the sense of Auslander-Bridger) if \(M\) has finite projective dimension equal to \(n\) and \(Ext_A^i(M,A)=0\) for \(i=1,...,n-1\) with the convention that every module of projective dimension less than or equal to 1 is spherical.

Let \(A\) be the connected linear Nakayama algebra with Kupisch series \([2,2,2,2,1]\) given by quiver and relations over a field. How many spherical indecomposable \(A\)-bimodules are there? Here a module \(M\) over an algebra \(B\) is called spherical if \(M\) has finite projective dimension \(n\) and \(\mathrm{Ext}_B^i(M,B)=0\) for \(i=1,\dots,n-1\). By convention all modules of projective dimension \(\leq 1\) is spherical. For this problem, being spherical is meant in the bimodule sense.

Let \(Q\) be the quiver on the vertex set \(\{v_1,\dots,v_6\}\) with arrows \[\begin{array}{lll} a_1\colon v_1\to v_2, & a_2\colon v_2\to v_3, & a_3\colon v_2\to v_5,\\ a_4\colon v_3\to v_1, & a_5\colon v_3\to v_4, & a_6\colon v_4\to v_2,\\ a_7\colon v_4\to v_6, & a_8\colon v_5\to v_4, & a_9\colon v_6\to v_3. \end{array}\] Let \(A = \mathbb{F}_3 Q / I\), where \(\mathbb{F}_3\) is the field with three elements and \(I\) is the two-sided ideal of the path algebra \(\mathbb{F}_3 Q\) generated by \[\begin{array}{l} a_2 a_4,\quad a_2 a_5 + 2\,a_3 a_8,\quad a_4 a_1 + a_5 a_6,\quad a_6 a_3,\quad a_6 a_2 + 2\,a_7 a_9,\\ a_8 a_6,\quad a_9 a_4,\quad a_9 a_5,\quad a_4 a_1 a_2,\quad a_1 a_2 a_5 a_7. \end{array}\] How many indecomposable \(A\)-modules are there?

Let \(A\) be a Nakayama algebra with \(n\) simples, that is an algebra given by quiver and relations over a field where the quiver is a directed cycle of length \(n\) and the relations are monomial relations possibly of length 1, i.e.connected cyclic Nakayama algebras and (possibly not connected) linear Nakayama algebras. Determine the number of such Nakayama algebras with 20 simples that have even global dimension and are higher Auslander algebras, i.e.where the global dimension and the dominant dimension of the algebra coincide.

We color each element of the set \(\{1,2,\cdots ,1000\}\) red or blue. Then we arrange the red elements in a Standard Young Tableau \(S\) and the blue elements in a Standard Young Tableau \(T\) so that the column lengths of both \(S\) and \(T\) come from the set \(\{2,4,6,8,10\}\). In how many ways is this possible? Provide the first ten digits and the last ten digits of the answer.

Let \([L]\) be the point of the Grassmannian \(Gr(14, 105)\) corresponding to the \(14\)-dimensional braid arrangement, and let \(X\) be the torus-orbit closure of \([L]\). Compute the coefficient of the Schubert variety corresponding to the partition \((91)^{12}, 78, 0\) in the expansion of the homology class of \(X\) in the Schubert basis.

Johnny’s a farmer. Johnny has antlers, birds and cats on his farm: At least one cat, more birds than cats and more antlers than birds, and he loves all of them dearly. Johnny likes counting. One day he counts his cats, cubes the number and writes it down in his diary. The next day he counts the birds and the cats. He computes both the sum of cats and birds and the product of cats and birds and multiplies them with each other. The result he writes in his diary. On the third day, he counts the birds, cubes the number and writes it in his diary. On the fourth day, he counts the antlers and the cats. He calculates both the sum of cats and antlers and the product of cats and antlers and multiplies them with each other. The result he writes in his diary. On the fifth day, he counts antlers, birds and cats, multiplies their numbers and writes down the result times 3. On the sixth day, he counts antlers and birds. He calculates both the sum of birds and antlers and the product of birds and antlers and multiplies them with each other. The result he writes in his diary. On the seventh day, Johnny only counts the antlers and writes down the cube of their number into his diary. He then sees that it is good, because all seven numbers in his diary add up to 4 times the sum of birds and cats, times the sum of antlers and birds, times the sum of antlers and cats. How many antlers, birds and cats are on Johnny’s farm? Find the solution with the smallest total number of animals. Answer with the 3-tuple (#antlers, #birds, #cats), where #antlers denotes the number of antlers (likewise for the other animals).

What is the maximum number of vertices of a hyperplane section of an octahedron? A hyperplane section is the intersection with a hyperplane.

Let \(K_1\) and \(K_2\) be any alternating knots with Seifert genus 1 and signature 0. Suppose that \(\det(K_1)=2001\) and \(\det(K_2)=2025\). Let \(M_i\) be the exterior of \(K_i\) for \(i=1,2\). Let \(M\) be the manifold obtained by gluing \(M_1\) to \(M_2\) along their boundaries, identifying the meridian of one knot with the Seifert longitude of the other knot and vice versa. What is the dimension of the hat version of Heegaard Floer homology of \(M\) with coefficients in the field of two elements?

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 \(10^{-5}\).

Let \[I= \frac{1}{\pi^2} \int_0^\pi\int_0^\pi \log \left|1+2e^{ix}+3e^{iy}\right|\,dy\,dx .\]

Evaluate \(I\) in closed form as a finite expression built from explicit complex numbers, Riemann zeta values, one-variable classical polylogarithm values (\(\operatorname{Li}_s(q)\), where \(s\in\mathbb{Z}_{>0}\) and \(q\in\mathbb{C}\)), and the operations \(+,-,\times,\div\).

In terms of \(d\) the dimension, what is the ratio of the surface area of a \(d\)-dimensional \(\ell_p\) ball to its volume?

Let \(Q\) be the quiver with 4 vertices, with adjacency matrix \(\begin{pmatrix} 0 & 2 & 2 & 0 \\ 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 0\\ 2 & 2 & 2 & 0\end{pmatrix}\), and let \(d = (1, 3, 4, 1)\).

Count all the unique Luna types for representations of \(Q\) of dimension vector \(d\), across all stability conditions that admit semistables.

Let \(P\) be a set of \(6\) permutations of size \(n\) such that the identity and the reverse permutation are elements of \(P\). What is the maximum density of triples shattered by \(P\) asymptotically as \(n \to \infty\)?