9 problems

Let $d$ and $q$ be positive integers. For a continuous map $f:S^{d-1}\to\mathbb{R}^d$ and a point $p\notin f(S^{d-1})$, define the winding number $W(f,p)\in\mathbb{Z}$ by the homol…

Let $M_\varnothing\simeq\mathsf{Sp}(2)$ be the nodal manifold for class-$\textrm{CII}$ Hamiltonians with four occupied and four unoccupied bands. Its cell decomposition contains a…

In the chiral symmetry classes BDI, CII and AIII, consider a one-dimensional parameter space and the associated winding number, obtained by integrating the winding number density o…

Let $K_4$ be the complete graph on vertices $1,2,3,4$, let $f:K_4\to\mathbb R^2$ be an almost embedding, and let $\mathop{\mathrm{wu}}_f(123)$ and…

Realization conjecture. For any integers $n_1,n_2,n_3,n_4$ whose sum is odd, there exists an almost embedding $f:K_4\to\mathbb R^2$ such that

Winding-number conjecture. For any almost embedding $g:K_{3,3}-ab\to\mathbb{R}^2$,

Skopenkov–Kozlov's conjecture. For every integer $k$, there is an almost embedding $g:K_5-(4,5)\to\mathbb{R}^2$ such that

For two independent planar Brownian motions $X$ and $X'$, let $\theta_X(z)$ and $\theta_{X'}(z)$ denote their winding numbers around $z$, and define … Let $\ell$ be the intersectio…

Let $\mathcal{A}^{\eta}_4(\eta,1)$ be the four-arm event at mesh size $\eta$, let $\nu_4^{\eta}$ be the four-arm IIC measure, and let $\tilde{\theta}_{\eta}$ be the winding number…